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y(orungstheoretisc)n(he)d(En)n(t)n(wic)n(klung) 386 3167 y(k)n(on)n(v)n(ergiert,)i(und)k(v)n(or)d(allem,)i(wie)g (gro\031)e(die)i(P)n(oten)n(tialst)2272 3171 y(\177)2272 3167 y(ark)n(e)e(maximal)h(sein)h(darf,)g(damit)386 3267 y(diese)d(Eigensc)n(haften)g(v)n(orliegen.)486 3366 y(W)571 3370 y(\177)571 3366 y(ahrend)h(der)h(nac)n(h)f(Bro)n(wn)f(und)j(Ra)n (v)n(enhall)d(b)r(enann)n(te,)i(die)g(P)n(oten)n(tialst)2925 3370 y(\177)2925 3366 y(ark)n(e)d(bis)j(ma-)386 3466 y(ximal)36 b(erster)f(Ordn)n(ung)g(en)n(thaltende)i(Op)r(erator)e(ausf) 2194 3470 y(\177)2192 3466 y(uhrlic)n(h)g(mathematisc)n(h)h(analysiert) 386 3566 y(w)n(orden)23 b(ist,)i(ist)g(v)n(on)f(dem)h(v)n(on)f(Jansen)g (und)h(Hess)f(eingef)2240 3570 y(\177)2238 3566 y(uhrten)h(Op)r (erator,)e(der)h(zus)3137 3570 y(\177)3137 3566 y(atzlic)n(h)386 3665 y(den)d(quadratisc)n(hen)e(T)-7 b(erm)20 b(in)h(der)f(P)n(oten)n (tialst)1893 3669 y(\177)1893 3665 y(ark)n(e)e(en)n(th)2224 3669 y(\177)2224 3665 y(alt,)i(bisher)g(n)n(ur)g(seine)h(Besc)n(hr)3185 3669 y(\177)3185 3665 y(ankt-)386 3765 y(heit)30 b(nac)n(h)f(un)n(ten,) h(so)n(wie)f(seine)g(P)n(ositivit)1744 3769 y(\177)1744 3765 y(at)f(im)i(\014ktiv)n(en)g(F)-7 b(all)30 b(eines)f(einzigen,)g (masselosen)386 3865 y(T)-7 b(eilc)n(hens)27 b(gezeigt)f(w)n(orden.)g (Dab)r(ei)i(stimm)n(t)f(im)h(un)n(tersuc)n(h)n(ten)e(F)-7 b(all)27 b(des)g(reinen)g(Coulom)n(b-)386 3964 y(p)r(oten)n(tials)35 b(die)h(maximal)e(zul)1371 3968 y(\177)1371 3964 y(assige)g(P)n(oten)n (tialst)2010 3968 y(\177)2010 3964 y(ark)n(e)f(\(die)j(durc)n(h)f(die)g (Ladung)g(des)g(Zen-)386 4064 y(tralk)n(erns)25 b(festgelegt)i(ist\))g (fast)g(genau)f(mit)i(derjenigen)2196 4068 y(\177)2194 4064 y(ub)r(erein,)f(die)g(man)g(aus)f(einer)h(exak-)386 4163 y(ten)f(quan)n(tenmec)n(hanisc)n(hen)d(Rec)n(hn)n(ung)i(f)1745 4167 y(\177)1743 4163 y(ur)h(das)e(Nullw)n(erden)h(der)g (Grundzustandsenergie)386 4263 y(erh)502 4267 y(\177)502 4263 y(alt.)486 4363 y(In)g(der)f(v)n(orliegenden)g(Arb)r(eit)h(wird)g (ein)g(unit)1930 4367 y(\177)1930 4363 y(ares)f(T)-7 b(ransformationssc)n(hema)22 b(v)n(orgestellt,)386 4462 y(das)30 b(in)g(der)g(mathematisc)n(hen)g(F)-7 b(estk)1594 4466 y(\177)1594 4462 y(orp)r(erph)n(ysik)28 b(zur)i(Un)n(tersuc)n(h)n (ung)f(der)h(T)-7 b(eilc)n(hen)n(b)r(ew)n(e-)386 4562 y(gung)26 b(in)h(b)r(esc)n(hr)916 4566 y(\177)916 4562 y(ankten,)f(p)r(erio)r(disc)n(hen)g(P)n(oten)n(tialen)g(b)r(ereits)h (erfolgreic)n(h)e(angew)n(andt)g(w)n(or-)386 4662 y(den)k(ist.)h (Dieses)f(f)967 4666 y(\177)965 4662 y(uhrt)h(zu)f(einer)g(sehr)g (einfac)n(hen)g(Darstellung)f(der)h(transformierten)f(Op)r(e-)386 4761 y(ratoren)i(und)i(erm)990 4765 y(\177)990 4761 y(oglic)n(h)n(t)f (somit)g(eine)h(mathematisc)n(he)g(Analyse)f(auc)n(h)g(im)h(F)-7 b(all)32 b(mehrerer)386 4861 y(T)-7 b(eilc)n(hen.)486 4960 y(Zun)629 4964 y(\177)629 4960 y(ac)n(hst)27 b(wird)h(im)g(Ein)n (teilc)n(henfall)g(die)h(unit)1980 4964 y(\177)1980 4960 y(are)2130 4943 y(\177)2119 4960 y(Aquiv)-5 b(alenz)29 b(des)f(neuen)g(T)-7 b(ransforma-)386 5060 y(tionssc)n(hemas)32 b(zu)h(der)f(v)n(on)g(Douglas)g(und)i(Kroll)e(eingef)2240 5064 y(\177)2238 5060 y(uhrten)h(T)-7 b(ransformation)31 b(gezeigt.)386 5160 y(So)r(dann)38 b(wird)g(die)g(F)-7 b(orm)n(b)r(esc)n(hr)1459 5164 y(\177)1459 5160 y(anktheit)36 b(und)j(P)n(ositivit)2281 5164 y(\177)2281 5160 y(at)e(des)h (Jansen-Hess-Op)r(erators)386 5259 y(f)413 5263 y(\177)411 5259 y(ur)21 b(masseb)r(ehaftete)h(T)-7 b(eilc)n(hen)21 b(un)n(tersuc)n(h)n(t)g(und)h(sein)f(w)n(esen)n(tlic)n(hes)f(Sp)r (ektrum)i(und)g(Punkt-)386 5359 y(sp)r(ektrum)34 b(lok)-5 b(alisiert.)33 b(Betre\013s)h(der)g(Kon)n(v)n(ergenz)e(der)h(st)2302 5363 y(\177)2302 5359 y(orungstheoretisc)n(hen)f(En)n(t)n(wic)n(k-)386 5459 y(lung)h(im)h(hier)f(un)n(tersuc)n(h)n(ten)g(F)-7 b(all)33 b(des)h(reinen)f(Coulom)n(bfeldes)g(k)-5 b(ann)33 b(n)n(ur)g(die)h(Dominanz)p eop %%Page: 2 8 2 7 bop 386 259 a FC(2)386 459 y FG(des)24 b(An)n(teils)g(erster)g 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b(En)n(t)n(wic)n(klung)e(k)n(on)n(v)n(ergiert.)486 1056 y(Die)20 b(gleic)n(hen)f(Un)n(tersuc)n(h)n(ungsmetho)r(den)f(w)n (erden)h(so)r(dann)h(auf)f(den)h(Zw)n(eiteilc)n(henfall)f(an-)386 1156 y(gew)n(andt,)27 b(w)n(ob)r(ei)f(zum)i(ersten)f(Mal)g(auc)n(h)g (die)g(Zw)n(eiteilc)n(hen)n(w)n(ec)n(hselwirkung)e(bis)i(inklusiv)n(e) 386 1256 y(zw)n(eiter)k(Ordn)n(ung)g(in)h(der)g(Kopplungsk)n(onstan)n (ten)d(b)r(er)2210 1260 y(\177)2208 1256 y(uc)n(ksic)n(h)n(tigt)i (wird.)h(Auc)n(h)g(hier)f(l)3259 1260 y(\177)3259 1256 y(a\031t)386 1355 y(sic)n(h)25 b(F)-7 b(orm)n(b)r(esc)n(hr)969 1359 y(\177)969 1355 y(anktheit,)25 b(Dominanz)h(der)g(linearen)2169 1359 y(\177)2167 1355 y(ub)r(er)g(die)g(quadratisc)n(hen)e(P)n(oten)n (tial-)386 1455 y(terme,)g(so)n(wie)g(P)n(ositivit)1165 1459 y(\177)1165 1455 y(at)f(zeigen.)h(Die)h(im)g(Rahmen)g(der)f(hier)g (angew)n(andten)f(Absc)n(h)3144 1459 y(\177)3144 1455 y(atzun-)386 1555 y(gen)i(sic)n(h)g(ergeb)r(ende)g(kritisc)n(he)g(P)n (oten)n(tialst)1812 1559 y(\177)1812 1555 y(ark)n(e)e(f)2016 1559 y(\177)2014 1555 y(ur)j(die)f(G)2317 1559 y(\177)2315 1555 y(ultigk)n(eit)h(dieser)f(Eigensc)n(haften)386 1654 y(liegt)i(leic)n(h)n(t)h(un)n(terhalb)f(derjenigen)g(des)h(Ein)n(teilc) n(henfalls.)486 1754 y(F)542 1758 y(\177)540 1754 y(ur)j(das)h(Zw)n (eiteilc)n(hensp)r(ektrum)f(k)-5 b(ann)32 b(gezeigt)g(w)n(erden,)f (da\031)g(das)h(w)n(esen)n(tlic)n(he)f(Sp)r(ek-)386 1853 y(trum)e(dasjenige)f(zw)n(eier)g(freier)h(T)-7 b(eilc)n(hen)28 b(umfa\031t)h(und)h(sic)n(h)e(nic)n(h)n(t)2579 1857 y(\177)2579 1853 y(andert,)h(w)n(enn)g(man)g(die)386 1953 y(Zw)n(eiteilc)n(hen)n(w) n(ec)n(hselwirkung)j(zw)n(eiter)i(Ordn)n(ung)f(w)n(egl)2233 1957 y(\177)2233 1953 y(a\031t.)g(Au\031erdem)h(existieren)g(k)n(eine) 386 2053 y(Eigen)n(w)n(erte,)26 b(w)n(enn)h(die)h(T)-7 b(eilc)n(henmasse)26 b(zu)i(n)n(ull)g(gesetzt)f(wird.)486 2152 y(Absc)n(hlie\031end)c(wird)h(ein)g(kurzer)e(Ausblic)n(k)i(auf)g (den)g(transformierten)e(Op)r(erator)g(zw)n(eiter)386 2252 y(Ordn)n(ung)32 b(im)h(allgemeinen)f(F)-7 b(all)33 b(v)n(on)f FD(N)42 b FG(T)-7 b(eilc)n(hen)33 b(gegeb)r(en.)f(Dieser)h (Op)r(erator)e(k)-5 b(ann)33 b(ex-)386 2352 y(plizit)d(angegeb)r(en)e (und)i(seine)f(P)n(ositivit)1685 2356 y(\177)1685 2352 y(at)g(b)r(estimm)n(t)h(w)n(erden.)f(Die)h(sic)n(h)f(dab)r(ei)g(ergeb)r (ende)386 2451 y(kritisc)n(he)j(P)n(oten)n(tialst)1119 2455 y(\177)1119 2451 y(ark)n(e)d(f)1329 2455 y(\177)1327 2451 y(ur)k(ein)g(neutrales)e(A)n(tom)i(ist)f(deutlic)n(h)h(niedriger)f (als)g(im)g(Ein-)386 2551 y(teilc)n(henfall.)j(W)-7 b(enngleic)n(h)34 b(die)h(hier)g(v)n(erw)n(endeten)e(Metho)r(den)j(k)n(eine)e(optimale)g (Sc)n(hrank)n(e)386 2651 y(f)413 2655 y(\177)411 2651 y(ur)29 b(die)g(zul)760 2655 y(\177)760 2651 y(assige)f(P)n(oten)n (tialst)1393 2655 y(\177)1393 2651 y(ark)n(e)e(liefern,)j(ist)g(dies)g (do)r(c)n(h)g(ein)g(Indiz)h(daf)2796 2655 y(\177)2794 2651 y(ur,)e(da\031)h(man)g(die)386 2750 y(Zw)n(eiteilc)n(henp)r(oten)n (tiale)g(h)1269 2754 y(\177)1269 2750 y(oherer)g(Ordn)n(ung)g(nic)n(h)n (t,)i(wie)f(bislang)2548 2754 y(\177)2545 2750 y(ublic)n(h,)h(v)n (ernac)n(hl)3122 2754 y(\177)3122 2750 y(assigen)386 2850 y(sollte.)p eop %%Page: 3 9 3 8 bop 3342 259 a FC(3)386 459 y FE(Abstract)486 676 y FG(By)24 b(means)h(of)f(a)h(unitary)f(transformation)g(sc)n(heme)g(b) r(orro)n(w)n(ed)f(from)h(the)i(study)f(of)f(quan-)386 776 y(tum)k(lattice)g(systems,)f(the)h(Dirac)f(op)r(erator)f(of)h(a)h (one-electron)e(ion)h(is)g(transformed)g(in)n(to)g(a)386 876 y(pseudo-relativistic)h(op)r(erator)f(whic)n(h)i(easily)f(allo)n (ws)g(for)g(the)i(elimination)f(of)g(the)h(p)r(ositron)386 975 y(degrees)d(of)h(freedom.)39 b(This)29 b(op)r(erator)d(is)j(blo)r (c)n(k-diagonal)d(with)j(resp)r(ect)f(to)g(the)h(pro)5 b(jection)386 1075 y(on)n(to)23 b(the)i(p)r(ositiv)n(e)f(\(resp)r (ectiv)n(e)f(negativ)n(e\))h(sp)r(ectral)g(subspace)f(of)h(the)h(free)f (Dirac)f(op)r(erator,)386 1175 y(to)34 b(a)h(\014xed)f(order)g(in)h 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(pseudo-relativistic)f(op)r(erators)f(are)h(relativ)n(ely)g(b)r(ounded) 386 1972 y(with)34 b(resp)r(ect)g(to)g(the)g(kinetic)g(energy)f(op)r (erator)f(for)i(sub)r(critical)f(p)r(oten)n(tial)h(strength.)56 b(In)386 2071 y(the)34 b(case)f(of)g(a)g(mo)r(di\014ed)h(Coulom)n(b)f (p)r(oten)n(tial,)i FD(V)52 b FG(=)33 b FA(\000)p FD(\015)5 b(x)2335 2041 y Fz(\000)p FC(1+)p FB(\017)2502 2071 y FD(;)80 b FG(0)33 b FD(<)g(\017)g FA(\034)g FG(1)p FD(;)66 b FG(one)33 b(can)386 2171 y(ev)n(en)e(pro)n(v)n(e)g(sub)r(ordinacy)f (of)i(the)h(higher-order)c(p)r(oten)n(tial)j(terms)g(and)f(th)n(us)h (con)n(v)n(ergence)386 2270 y(of)27 b(the)h(p)r(erturbation)g(series.) 486 2370 y(The)33 b(in)n(v)n(estigations)g(of)g(Ev)-5 b(ans,)35 b(P)n(erry)d(and)i(Sieden)n(top)f(and)h(of)g(Balinsky)f(and)h (Ev)-5 b(ans,)386 2470 y(concerning)17 b(the)i(single-particle)e(Bro)n (wn-Ra)n(v)n(enhall)e(op)r(erator)i(are)g(extended)i(to)f(the)h (Jansen-)386 2569 y(Hess)25 b(op)r(erator.)34 b(It)25 b(is)g(sho)n(wn)f(that)h(its)g(essen)n(tial)f(sp)r(ectrum)h(is)g(giv)n (en)f(b)n(y)g([)p FD(m;)14 b FA(1)p FG(\))25 b(for)g(p)r(oten-)386 2669 y(tial)i(strengths)f FD(\015)i(<)22 b FG(1)p FD(:)p FG(006)p FD(;)49 b FG(that)27 b(the)g(singular)f(con)n(tin)n(uous)g(sp) r(ectrum)h(is)g(empt)n(y)g(and)g(that)386 2769 y(for)g FD(\015)h(<)22 b FG(0)p FD(:)p FG(29)p FD(;)27 b FG(there)g(are)g(no)g (em)n(b)r(edded)h(eigen)n(v)-5 b(alues)27 b(in)g([)p FD(m;)14 b FA(1)p FG(\);)51 b(also,)27 b(that)h(for)f(massless)386 2868 y(particles,)c(the)f(sp)r(ectrum)h(is)f(absolutely)g(con)n(tin)n (uous.)34 b(Whereas)22 b(p)r(ositivit)n(y)g(of)g(the)h(massless)386 2968 y(Jansen-Hess)29 b(op)r(erator)f(w)n(as)h(pro)n(v)n(ed)g(b)n(y)h (Brummelh)n(uis,)g(Sieden)n(top)g(and)g(Sto)r(c)n(kmey)n(er)f(for)386 3067 y FD(\015)i FA(\024)26 b FG(1)p FD(:)p FG(006)p FD(;)i FG(w)n(e)h(w)n(ere)g(in)g(the)h(massiv)n(e)f(case)g(only)g(able) g(to)g(sho)n(w)g(p)r(ositivit)n(y)h(for)f FD(\015)i FA(\024)26 b FG(0)p FD(:)p FG(83)p FD(:)386 3167 y FG(F)-7 b(or)23 b(the)i(t)n(w)n(o-electron)d(ion)i(and)g FD(N)9 b FG(-electron)23 b(atom,)h(p)r(ositivit)n(y)g(is)g(established)g(for)g FD(\015)j FA(\024)c FG(0)p FD(:)p FG(82)386 3267 y(and)g FD(\015)28 b FA(\024)22 b FG(0)p FD(:)p FG(44)p FD(;)h FG(resp)r(ectiv)n(ely)-7 b(.)34 b(The)24 b(large)e(reduction)h(of)g (the)h(critical)f(p)r(oten)n(tial)g(strength)g(for)386 3366 y(the)k FD(N)9 b FG(-electron)26 b(atom)h(is)g(attributed)g(to)g (the)h(t)n(w)n(o-particle)d(second-order)g(p)r(oten)n(tial)i(terms)386 3466 y(in)j(the)g(pseudo-relativistic)f(op)r(erator.)41 b(Although)30 b(our)f(b)r(ounds)h(on)g FD(\015)k FG(are)29 b(not)h(sharp,)f(this)386 3566 y(is)e(a)h(c)n(hallenge)e(to)i(the)f (quan)n(tum)h(c)n(hemists)f(who)h(are)e(usually)h(neglecting)g(these)h (terms.)486 3665 y(Apart)19 b(from)h(p)r(ositivit)n(y)-7 b(,)22 b(also)d(the)h(relativ)n(e)f(b)r(oundedness)h(of)h(the)f (second-order)e(p)r(oten)n(tial)386 3765 y(terms)36 b(with)g(resp)r (ect)g(to)f(the)i(\014rst-order)c(p)r(oten)n(tial)j(terms)g(is)g(in)n (v)n(estigated)e(for)i(the)g(t)n(w)n(o-)386 3865 y(particle)31 b(pseudo-relativistic)f(op)r(erator,)i(as)f(w)n(ell)g(as)h(its)g(sp)r (ectrum.)49 b(It)32 b(is)g(found)g(that)g(the)386 3964 y(free-particle)26 b(p)r(ositiv)n(e)g(sp)r(ectrum)i([2)p FD(m;)14 b FA(1)p FG(\))27 b(is)g(a)f(subset)h(of)g(the)h(essen)n(tial) e(sp)r(ectrum)h(of)g(the)386 4064 y(full)i(t)n(w)n(o-particle)d (pseudo-relativistic)h(op)r(erator)f(\(for)i FD(\015)g(<)c FG(0)p FD(:)p FG(89\))p FD(;)51 b FG(and)28 b(that)g(the)h(essen)n (tial)386 4163 y(sp)r(ectrum)23 b(do)r(es)g(not)g(c)n(hange)f(when)h (the)h(t)n(w)n(o-particle)d(second-order)g(in)n(teraction)h(terms)h (are)386 4263 y(dropp)r(ed)31 b(\(for)f FD(\015)j(<)28 b FG(0)p FD(:)p FG(65\))p FD(:)58 b FG(Again,)32 b(eigen)n(v)-5 b(alues)29 b(are)h(absen)n(t)g(in)h(the)h(massless)d(case.)46 b(This)386 4363 y(prop)r(ert)n(y)26 b(holds)i(for)f FD(\015)g(<)c FG(0)p FD(:)p FG(98)p FD(:)p eop %%Page: 4 10 4 9 bop 386 259 a FC(4)386 459 y FE(In)m(tro)s(duction)486 676 y FG(With)27 b(the)f(adv)n(en)n(t)g(of)g(relativistic)f(quan)n(tum) h(mec)n(hanics)g(\(Dirac)g(1928\),)e(the)j(analysis)e(of)386 776 y(Dirac)k(op)r(erators)e(has)i(pla)n(y)n(ed)g(an)g(imp)r(ortan)n(t) g(role)g(in)g(mathematical)g(ph)n(ysics.)42 b(Relativit)n(y)386 876 y(co)n(v)n(ers)36 b(t)n(w)n(o)i(asp)r(ects,)j(particles)c(mo)n (ving)h(at)g(high)g(v)n(elo)r(cit)n(y)g(\(close)g(to)g(the)h(v)n(elo)r (cit)n(y)e FD(c)i FG(of)386 975 y(ligh)n(t\))24 b(or)g(b)r(eing)g(exp)r (osed)g(to)g(v)n(ery)f(strong)h(p)r(oten)n(tials.)35 b(F)-7 b(or)24 b(a)g(single)g(particle)f(with)i(mass)f FD(m)386 1075 y FG(and)j(spin)731 1042 y FC(1)p 731 1056 34 4 v 731 1104 a(2)801 1075 y FG(in)g(an)g(electric)f(p)r(oten)n(tial) h FD(V)46 b FG(the)27 b(Dirac)f(op)r(erator)f(reads)h(\(in)h (relativistic)f(units)386 1175 y Fx(~)d FG(=)f FD(c)h FG(=)g(1\))1115 1331 y FD(H)52 b FG(=)46 b FD(D)1416 1343 y FC(0)1485 1331 y FG(+)32 b FD(V)125 b FG(=)46 b FA(\000)p FD(i)14 b Fw(\013)2094 1275 y FD(@)p 2069 1312 100 4 v 2069 1388 a(@)5 b Fv(x)2220 1331 y FG(+)41 b FD(\014)t(m)32 b FG(+)g FD(V)386 1490 y FG(where)c FD(D)696 1502 y FC(0)762 1490 y FG(is)g(the)h(Dirac)f(op)r(erator)f(of) i(a)f(free)h(particle)f(\(i.e.)40 b(its)29 b(kinetic)g(energy)e(op)r (erator\),)386 1590 y(and)g FD(\014)t(;)61 b(\013)735 1602 y FB(i)762 1590 y FD(;)37 b(i)23 b FG(=)g(1)p FD(;)14 b FG(2)p FD(;)g FG(3)49 b(are)26 b(the)i(Dirac)f(matrices)g(in)h Fx(C)2201 1560 y FC(4)p FB(;)p FC(4)2348 1590 y FG(\(Rose)f(1961\))1110 1788 y FD(\013)1163 1800 y FB(i)1237 1788 y FG(=)1348 1671 y Fu(\022)1467 1737 y FG(0)100 b FD(\033)1656 1749 y FB(i)1451 1837 y FD(\033)1498 1849 y FB(i)1625 1837 y FG(0)1725 1671 y Fu(\023)1800 1788 y FD(;)180 b(\014)51 b FG(=)2211 1671 y Fu(\022)2314 1737 y FD(I)156 b FG(0)2315 1837 y(0)83 b FA(\000)p FD(I)2589 1671 y Fu(\023)751 2058 y FD(\033)798 2070 y FC(1)882 2058 y FG(=)992 1941 y Fu(\022)1095 2007 y FG(0)f(1)1095 2107 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FG(-form)h(b)r(oundedness)g(of)g FD(V)42 b FG(leads)22 b(to)h(a)g(restriction)f(of)h(the)g(p)r(oten)n (tial)g(strength)386 4483 y FD(\015)5 b(:)486 4583 y FG(The)23 b(sp)r(ectrum)h(of)f(an)h(op)r(erator)d FD(H)31 b FG(giv)n(es)22 b(information)h(ab)r(out)g(the)h(energy)f(v)-5 b(alues)23 b(whic)n(h)386 4683 y(the)28 b(particle)f(can)g(ha)n(v)n(e)g (when)g(it)h(mo)n(v)n(es)f(in)g(the)h(p)r(oten)n(tial)g FD(V)19 b FG(.)37 b(It)28 b(is)f(the)h(set)807 4831 y FD(\033)s FG(\()p FD(H)7 b FG(\))24 b(:=)46 b FA(f)p FD(\025)23 b FA(2)g Fx(R)29 b FG(:)46 b FD(H)39 b FA(\000)32 b FD(\025)14 b FG(id)79 b(is)27 b(not)h(b)r(oundedly)g(in)n(v)n (ertible)o FA(g)p FD(;)386 4979 y FG(i.e.,)35 b FD(\033)s FG(\()p FD(H)7 b FG(\))35 b(is)e(the)h(complemen)n(t)g(of)f(the)h (subset)g FD(\025)f FA(2)h Fx(R)39 b FG(for)33 b(whic)n(h)h FD(H)29 b FA(\000)22 b FD(\025)14 b FG(id)34 b(is)g(bijectiv)n(e)386 5079 y(with)c(a)g(b)r(ounded)g(in)n(v)n(erse)f(\(id)h(is)g(the)h(iden)n (tit)n(y)f(op)r(erator\).)42 b(F)-7 b(or)29 b(a)h(free)g(particle)f(\() p FD(V)46 b FG(=)27 b(0\))p FD(;)386 5179 y FG(the)h(sp)r(ectrum)g(of)f (the)h(Dirac)f(op)r(erator)f(consists)h(of)g(t)n(w)n(o)g(half-lines,)51 b(\()p FA(\0001)p FD(;)14 b(m)p FG(])41 b FA([)h FG([)p FD(m;)14 b FA(1)p FG(\))p 569 5625 914 4 v 569 5616 V 569 5633 V 1523 5625 665 4 v 2221 5625 914 4 v 2221 5616 V 2221 5633 V 1407 5789 a FA(\000)p FD(m)618 b(m)1498 5623 y Fs(r)681 b(r)p 1854 5623 4 50 v 1831 5789 a FG(0)p eop %%Page: 5 11 5 10 bop 3342 259 a FC(5)386 459 y FG(separated)25 b(b)n(y)h(a)g(gap)g (of)h(2)p FD(m)49 b FG(\(in)27 b(our)e(units,)j FD(m)e FG(is)g(the)h(rest)f(energy)g(of)g(the)h(particle;)f(states)386 558 y(ab)r(o)n(v)n(e)k FD(m)i FG(are)e(electronic)h(states,)h(while)g (states)f(b)r(elo)n(w)g FA(\000)p FD(m)h FG(are)e(allo)r(cated)h(to)g (p)r(ositrons\).)386 658 y(F)-7 b(rom)20 b(this)h(it)g(follo)n(ws)e (that)i(the)g(Dirac)f(op)r(erator)e(is)j(un)n(b)r(ounded)g(b)r(oth)g (from)f(ab)r(o)n(v)n(e)f(and)h(from)386 758 y(b)r(elo)n(w.)36 b(When)28 b(the)g(p)r(oten)n(tial)f(is)g(switc)n(hed)h(on,)f(b)r(ound)g (eigenstates)g(app)r(ear)f(with)i(energies)386 857 y FD(E)447 869 y FB(i)505 857 y FG(lying)i(in)g(the)h(gap.)43 b(The)30 b(eigen)n(v)-5 b(alue)29 b(equation)h FD(H)7 b(')2198 869 y FB(i)2267 857 y FG(=)26 b FD(E)2419 869 y FB(i)2447 857 y FD(')2501 869 y FB(i)2587 857 y FG(is)k(exactly)f (solv)-5 b(able)29 b(for)386 957 y(the)k(p)r(oin)n(t-n)n(ucleus)g (Coulom)n(b)f(p)r(oten)n(tial,)j(with)e(its)h(lo)n(w)n(est)e (\(ground-state\))g(energy)g(giv)n(en)386 1056 y(b)n(y)27 b(\(Darwin)h(1928,)e(see)h(also)f(Rose)h(1961\))1485 1265 y FD(E)1546 1277 y FC(0)1630 1265 y FG(=)46 b FD(m)1837 1183 y Fu(p)p 1920 1183 319 4 v 82 x FG(1)18 b FA(\000)g FD(Z)2126 1241 y FC(2)2162 1265 y FD(e)2201 1241 y FC(4)2252 1265 y FD(;)386 1456 y FG(yielding)30 b FD(E)761 1468 y FC(0)827 1456 y FA(\025)e FG(0)i(for)g FD(\015)j FG(=)28 b FD(Z)6 b(e)1393 1426 y FC(2)1471 1456 y FA(\024)28 b FG(1)p FD(:)59 b FG(This)30 b(exact)h(reference)e(v)-5 b(alue)31 b(can)f(b)r(e)h(used)g(to)f(test)386 1555 y(our)23 b(p)r(erturbativ)n(e)h(approac)n(h)f(giv)n(en)g(b)r(elo)n(w.)35 b(Bey)n(ond)24 b(that)g(our)g(metho)r(d)h(can)f(b)r(e)h(applied)f(to) 386 1655 y(Coulom)n(b-t)n(yp)r(e)d(p)r(oten)n(tials)h(where)f(no)h (exact)f(solutions)h(exist)f(\(suc)n(h)h(as)g(sums)g(of)g(Coulom)n(bic) 386 1755 y(and)g(short-range)e(p)r(oten)n(tials)j(o)r(ccurring)e(in)i (the)g(single-particle)e(mo)r(dels)h(for)g(m)n(ulti-electron)386 1854 y(ions\).)486 1954 y(In)38 b(electron)f(sp)r(ectroscop)n(y)f(the)j (energies)d(in)n(v)n(olv)n(ed)h(are)g(usually)h(m)n(uc)n(h)g(smaller)f (than)386 2053 y(the)i(gap)f(suc)n(h)h(that)g(the)g(creation)e(of)i (electron-p)r(ositron)e(pairs,)k(describ)r(ed)d(in)h(terms)g(of)386 2153 y(excitation)28 b(of)g(a)g(state)h(with)g(energy)e(smaller)g(than) i FA(\000)p FD(m)f FG(to)g(a)g(state)g(lying)h(ab)r(o)n(v)n(e)e FD(m)h FG(\(or)g(to)386 2253 y(an)f(uno)r(ccupied)h(b)r(ound)g(state)f (with)i(p)r(ositiv)n(e)e(energy\),)g(is)g(negligible.)486 2352 y(As)32 b(long)f(as)h(only)f(electrons)g(\(but)i(not)f(p)r (ositrons\))g(are)f(considered,)h(it)h(is)f(of)g(disadv)-5 b(an-)386 2452 y(tage)24 b(to)h(describ)r(e)g(them)h(in)f(terms)g(of)g (an)f(op)r(erator)g(whic)n(h)h(is)g(un)n(b)r(ounded)g(from)g(b)r(elo)n (w.)35 b(F)-7 b(or)386 2552 y(the)28 b(determination)g(of)g(the)h (energy)d(eigenstates)i(the)g(common)g(simple)g(v)-5 b(ariational)26 b(princi-)386 2651 y(ples,)h(based)h(on)f(minimising)h (the)g(energy)e(exp)r(ectation)h(v)-5 b(alue,)28 b(cannot)f(b)r(e)h (used)g(an)n(y)f(more.)386 2751 y(Instead,)32 b(so-called)e(minimax)h (pro)r(cedures)g(are)f(required)g(\(Dolb)r(eault,)j(Esteban)e(and)g(S)n (\023)-39 b(er)n(\023)g(e)386 2850 y(2000)26 b(and)h(references)g (therein\),)g(or)g(di\013eren)n(t)h(metho)r(ds)g(ha)n(v)n(e)e(to)i(b)r (e)g(found.)486 2950 y(Historically)-7 b(,)31 b(sev)n(eral)f(to)r(ols)h (w)n(ere)g(emplo)n(y)n(ed)g(to)g(get)h(rid)f(of)h(the)g(negativ)n (e-energy)d(con-)386 3050 y(tin)n(uum)f(and)e(to)h(deriv)n(e)g(from)f (the)i(Dirac)e(op)r(erator)f(a)i(pseudo-relativistic)f(op)r(erator)f (whic)n(h)386 3149 y(is)f(b)r(ounded)h(from)f(b)r(elo)n(w.)36 b(P)n(auli,)24 b(based)g(on)g(his)g(t)n(w)n(o-comp)r(onen)n(t)f (pre-Dirac)g(theory)h(whic)n(h)386 3249 y(incorp)r(orates)f (relativistic)i(e\013ects)g(\(P)n(auli)g(1927\),)f(in)n(tro)r(duced)h (a)f(systematic)h(pro)r(cedure)f(for)386 3349 y(the)h(elimination)f(of) g(the)h(t)n(w)n(o)e(small)h(\(p)r(ositron-lik)n(e\))f(comp)r(onen)n(ts) h(of)g(the)h(4-spinor)d(ob)r(eying)386 3448 y(the)30 b(Dirac)f(eigen)n(v)-5 b(alue)28 b(equation)h(\(see,)h(e.g.,)f(P)n (auli)g(1958\).)40 b(The)30 b(resulting)f(equation)f(for)h(a)386 3548 y(function)34 b(comp)r(osed)f(of)h(the)g(remaining)e(t)n(w)n(o)h (\(large,)h(i.e.,)h(electron-lik)n(e\))e(comp)r(onen)n(ts)g(of)386 3647 y FD(')h FG(is)f(of)g(\(nonrelativistic\))g(Sc)n(hr)1436 3651 y(\177)1436 3647 y(odinger)f(t)n(yp)r(e.)54 b(Ho)n(w)n(ev)n(er,)33 b(the)h(op)r(erator)e(de\014ned)h(b)n(y)g(this)386 3747 y(Sc)n(hr)546 3751 y(\177)546 3747 y(odinger-t)n(yp)r(e)c(equation)h (has)h(some)f(serious)f(dra)n(wbac)n(ks,)h(e.g.,)h(the)h(op)r(erators)d (in)i(cer-)386 3847 y(tain)21 b(orders)f(of)h FD(Z)6 b(e)984 3817 y FC(2)1042 3847 y FG(are)20 b(no)h(longer)f(symmetric.)34 b(The)22 b(idea)e(of)i(reducing)e(the)i(Dirac)f(op)r(erator)386 3946 y(b)n(y)i(means)g(of)h(elimination)f(and)h(substitution)g(metho)r (ds)g(to)f(a)g(semib)r(ounded)h(op)r(erator)e(\(act-)386 4046 y(ing)32 b(on)g(2-spinors\))f(with)h(the)h(same)f(ground-state)e (prop)r(erties)i(w)n(as)f(pursued)h(further,)h(see)386 4146 y(for)27 b(example)g(Duran)n(t)h(and)f(Malrieu)g(\(1987\),)f(DES)i (\(2000\).)486 4245 y(Another)42 b(approac)n(h)f(consisted)i(in)g (minimising)h(the)f(square)f(of)h(the)g(Dirac)g(op)r(erator)386 4345 y(\(Ba)n(ylis)27 b(and)g(P)n(eel)g(1983\),)f(a)h(metho)r(d)h(whic) n(h)g(w)n(orks)e(w)n(ell)h(if)h(the)g(p)r(oten)n(tial)g(is)f(nonp)r (ositiv)n(e.)486 4444 y(Concerning)33 b(the)i(free)g(\()p FD(V)54 b FG(=)34 b(0\))h(Dirac)f(op)r(erator,)h(a)f(ma)5 b(jor)34 b(step)h(forw)n(ard)e(w)n(as)g(tak)n(en)386 4544 y(b)n(y)27 b(F)-7 b(oldy)26 b(and)h(W)-7 b(outh)n(uysen)27 b(\(1950\).)35 b(They)27 b(decoupled)g(the)g(p)r(ositiv)n(e)g(and)f (negativ)n(e)g(sp)r(ec-)386 4644 y(tral)38 b(subspaces)g(b)n(y)h(means) g(of)g(a)f(certain)g(unitary)h(transformation)e(and)i(so)f(obtained)h (a)386 4743 y(semib)r(ounded,)30 b(2)19 b FA(\002)g FG(2)28 b(matrix-v)-5 b(alued)28 b(op)r(erator)f(in)j(the)f(electronic)f (subspace.)41 b(Also)28 b(other)386 4843 y(p)r(eople)g(got)f(in)n(v)n (olv)n(ed)f(in)i(this)g(pro)5 b(ject.)37 b(De)29 b(V)-7 b(ries)27 b(\(1970\))f(giv)n(es)h(an)h(o)n(v)n(erview)d(o)n(v)n(er)h (related)386 4943 y(transformations)g(and)h(the)h(prop)r(erties)f(of)g (the)h(transformed)f(op)r(erators.)486 5042 y(In)h(the)g(presence)g(of) g(an)f(external)h(\014eld)g(the)g(F)-7 b(oldy-W)g(outh)n(uysen)28 b(transformation)e(leads)386 5142 y(no)31 b(longer)f(to)h(a)g(complete) g(decoupling)g(of)h(the)f(t)n(w)n(o)g(sp)r(ectral)g(subspaces.)47 b(So)31 b(Bro)n(wn)f(and)386 5241 y(Ra)n(v)n(enhall)23 b(\(1951\))f(suggested)h(to)h(pro)5 b(ject)23 b(the)i(Dirac)e(op)r (erator)f(on)n(to)h(the)i(p)r(ositiv)n(e)e(sp)r(ectral)386 5341 y(subspace)k(of)g(the)h(free)g(Dirac)f(op)r(erator)f FD(D)1759 5353 y FC(0)1823 5341 y FG(b)n(y)i(means)f(of)1439 5577 y(\003)1497 5589 y FC(+)1597 5577 y FG(=)1718 5521 y(1)p 1718 5558 42 4 v 1718 5634 a(2)1806 5460 y Fu(\022)1868 5577 y FG(1)k(+)2071 5521 y FD(D)2140 5533 y FC(0)p 2048 5558 153 4 v 2048 5634 a FA(j)p FD(D)2140 5646 y FC(0)2177 5634 y FA(j)2224 5460 y Fu(\023)2299 5577 y FD(:)p eop %%Page: 6 12 6 11 bop 386 259 a FC(6)386 459 y FG(They)27 b(obtained)h(the)g(op)r (erator)1400 643 y FD(H)1476 609 y FC(\(1\))1588 643 y FG(:=)46 b(\003)1780 655 y FC(+)1858 643 y FG(\()p FD(D)1959 655 y FC(0)2028 643 y FG(+)32 b FD(V)19 b FG(\))24 b(\003)2306 655 y FC(+)386 825 y FG(whic)n(h)29 b(is)g(b)r(ounded)g (from)g(b)r(elo)n(w)f(for)h(not)f(to)r(o)h(strong)f(p)r(oten)n(tials.) 40 b(F)-7 b(ormally)g(,)29 b(an)g(op)r(erator)386 924 y(semib)r(ounded)24 b(for)g(all)g(p)r(oten)n(tials)g(\(whic)n(h)h (admit)f(a)g(sp)r(ectral)g(gap\))f(w)n(ould)h(arise)f(from)h(using)386 1024 y(instead)34 b(of)g(\003)838 1036 y FC(+)927 1024 y FG(an)g(exact)f(pro)5 b(jector)33 b FD(P)1692 1036 y FC(+)1782 1024 y FG(whic)n(h)h(pro)5 b(jects)33 b(on)n(to)g(the)i(ab) r(o)n(v)n(e-gap)c(sp)r(ectral)386 1124 y(subspace)38 b(of)g(the)h(Dirac)f(op)r(erator)f(with)i(p)r(oten)n(tial,)i FD(D)2241 1136 y FC(0)2304 1124 y FG(+)25 b FD(V)5 b(:)80 b FG(Since)38 b(this)h(pro)5 b(jector)37 b(is)386 1223 y(unkno)n(wn)32 b(in)g(the)h(case)e(of)i(a)e(general)g(p)r(oten)n (tial,)j(Douglas)d(and)h(Kroll)f(\(1974\))g(in)n(tro)r(duced)386 1323 y(a)d(p)r(erturbativ)n(e)f(approac)n(h)g(whic)n(h)h(is)g(based)g (on)g(the)g(ideas)g(of)g(F)-7 b(oldy)28 b(and)h(W)-7 b(outh)n(uysen.)38 b(It)386 1423 y(aims)28 b(at)f(decoupling)h(the)g (sp)r(ectral)g(subspaces)f(of)h FD(D)2098 1435 y FC(0)2163 1423 y FG(up)g(to)g(a)g(giv)n(en)f(\(arbitrarily)f(c)n(hosen\))386 1522 y(order)g FD(n)i FG(in)g(the)g(p)r(oten)n(tial)g(strength)f FD(\015)5 b(:)51 b FG(This)28 b(is)g(ac)n(hiev)n(ed)e(b)n(y)i(means)f (of)h(a)f(series)g(of)h FD(n)18 b FG(+)g(1)386 1622 y(consecutiv)n(e)23 b(unitary)h(transformations)e(of)i(the)h(Dirac)e(op)r(erator,)h (starting)f(with)i(the)f(F)-7 b(oldy-)386 1722 y(W)g(outh)n(uysen)19 b(one.)34 b(In)19 b(eac)n(h)f(successiv)n(e)g(step,)j(the)e (transformation)e(op)r(erator)h(is)h(determined)386 1821 y(b)n(y)28 b(the)h(requiremen)n(t)e(that)i(the)f(terms)g(of)g(corresp)r (ondingly)f(lo)n(w)n(est)g(order)g(in)h FD(\015)33 b FG(whic)n(h)28 b(still)386 1921 y(couple)f(the)h(electron-p)r(ositron)e (subspaces)h(are)f(eliminated.)486 2020 y(In)f(the)h(presen)n(t)f(w)n (ork)f(it)h(is)h(sho)n(wn)e(that)i(the)g(Douglas-Kroll)c (transformation)i(sc)n(heme)h(is)386 2120 y(a)i(sp)r(ecial)h(case)e(of) i(a)f(m)n(uc)n(h)h(more)f(general)f(transformation)g(sc)n(heme)h(deriv) n(ed)g(from)g(p)r(ertur-)386 2220 y(bation)k(theory)g(\(Morse)f(and)i (F)-7 b(esh)n(bac)n(h)30 b(1953,)h(p.1018\))f(and)h(dev)n(elop)r(ed)g (for)g(the)h(study)g(of)386 2319 y(quan)n(tum)27 b(lattice)h(systems.) 486 2419 y(The)f(basic)g(idea)g(is)h(nicely)f(displa)n(y)n(ed)f(in)i (the)g(w)n(ork)e(of)i(Datta,)f(F)-7 b(ern\023)-42 b(andez)27 b(and)g(F)-7 b(r)3160 2423 y(\177)3160 2419 y(ohlic)n(h)386 2520 y(\(1999\).)33 b(Assume)19 b(a)g(self-adjoin)n(t)g(op)r(erator)f (of)h(the)h(form)f FD(H)29 b FG(=)23 b FD(H)2444 2532 y FC(0)2483 2520 y FG(+)r FD(\015)2610 2499 y FG(~)2598 2520 y FD(V)61 b FG(suc)n(h)19 b(that)g(the)h(sp)r(ec-)386 2619 y(trum)j(of)g FD(H)748 2631 y FC(0)808 2619 y FG(has)f(b)r(ounded) h(subsets)g(separated)e(b)n(y)i(sp)r(ectral)f(gaps)g(and)g(let)h FD(P)2864 2631 y FB(i)2892 2619 y FD(;)60 b(i)23 b FG(=)f(1)p FD(;)14 b(:::;)g(N)386 2719 y FG(b)r(e)33 b(a)g(partition)f(of)h(unit)n (y)-7 b(,)34 b(where)f FD(P)1568 2731 y FB(i)1629 2719 y FG(are)e(sp)r(ectral)i(pro)5 b(jections)31 b(on)n(to)h(disjoin)n(t)h (subsets)g(of)386 2819 y FD(\033)s FG(\()p FD(H)537 2831 y FC(0)575 2819 y FG(\))p FD(:)59 b FG(By)30 b(means)g(of)h(a)f(series) f(of)i FD(n)f FG(unitary)g(transformations,)g FD(H)37 b FG(is)31 b(transformed)e(in)n(to)386 2918 y(an)d(op)r(erator)e(whic)n (h)i(is)g(blo)r(c)n(k-diagonal)d(up)k(to)e(order)g FD(n)h FG(in)g FD(\015)31 b FG(with)26 b(resp)r(ect)g(to)g(the)g(sp)r(ectral) 386 3018 y(pro)5 b(jections)31 b FD(P)871 3030 y FB(i)899 3018 y FD(:)65 b FG(In)33 b(the)g(case)f(of)g(a)h(lattice)g(where)f (one)g(deals)g(with)i(b)r(ounded)f(op)r(erators,)386 3119 y(it)i(can)e(b)r(e)i(sho)n(wn)f(that)g(for)g FD(H)1402 3131 y FC(0)1439 3119 y FG(-b)r(ounded)g(p)r(oten)n(tials)g FD(\015)2263 3098 y FG(~)2250 3119 y FD(V)53 b FG(the)35 b(p)r(erturbation)e(series)h(in)g FD(\015)386 3218 y FG(con)n(v)n(erges.)486 3318 y(In)27 b(the)g(lattice)g(case)f(of)g(a)h (discrete)f(sp)r(ectrum,)h(the)g(transformation)f(op)r(erators)e(can)j (eas-)386 3418 y(ily)f(b)r(e)h(computed.)36 b(Since)27 b(only)e(\(discrete\))i(sums)f(are)f(in)n(v)n(olv)n(ed,)g(the)i (de\014ning)f(equation)f(for)386 3517 y(the)31 b(transformation)e(op)r (erator)f(is)j(algebraic)d(with)j(an)f(explicit)h(solution)f(\(see)h (e.g.)45 b(Datta,)386 3617 y(F)-7 b(ern\023)-42 b(andez)31 b(and)h(F)-7 b(r)1034 3621 y(\177)1034 3617 y(ohlic)n(h)31 b(1999\).)47 b(If,)34 b(on)d(the)h(other)f(hand,)i(the)f(sp)r(ectrum)g (is)g(con)n(tin)n(uous)386 3717 y(as)i(for)g(the)h(Dirac)f(op)r (erator,)h(the)g(corresp)r(onding)e(sums)i(will)g(b)r(e)g(singular,)g (and)g(a)f(di\013er-)386 3816 y(en)n(t)k(metho)r(d)g(for)f(solving)g (the)h(de\014ning)g(op)r(erator)e(equation)i(has)f(to)h(b)r(e)g(used.) 67 b(Sob)r(olev)386 3916 y(\(2003,2004\),)23 b(in)j(his)f(study)h(of)g (p)r(erio)r(dic)f(Sc)n(hr)1873 3920 y(\177)1873 3916 y(odinger)f(op)r(erators)f(on)j(a)f(lattice,)h(in)n(tro)r(duced)386 4015 y(pseudo)r(di\013eren)n(tial)j(op)r(erator)f(tec)n(hniques)h(whic) n(h)h(are)f(readily)f(applicable)h(to)h(the)g(in)n(tegral)386 4115 y(op)r(erators)25 b(o)r(ccurring)h(for)h(Coulom)n(b-t)n(yp)r(e)g (p)r(oten)n(tials.)37 b(Changing)26 b(from)h(co)r(ordinate)g(space)386 4215 y(to)g(momen)n(tum)h(space)f(where)g FD(D)1454 4227 y FC(0)1519 4215 y FG(is)g(diagonal,)1471 4396 y FD(D)1540 4408 y FC(0)1577 4396 y FG(\()p Fv(p)p FG(\))46 b(=)g Fw(\013)14 b Fv(p)42 b FG(+)f FD(\014)18 b(m;)386 4578 y FG(one)34 b(can)h(represen)n(t)f(the)h(p)r(oten)n(tials)f FD(V)89 b FG(\(as)35 b(w)n(ell)g(as)f(the)h(transformation)e(op)r (erators\))h(in)386 4678 y(terms)27 b(of)h(pseudo)r(di\013eren)n(tial)f (op)r(erators,)f(de\014ned)i(b)n(y)f(their)g(sym)n(b)r(ols)g FD(v)s FG(,)h(i.e.,)1051 4897 y(\()p FD(V)33 b(')p FG(\)\()p Fv(x)p FG(\))25 b(:=)1642 4841 y(1)p 1532 4878 261 4 v 1532 4956 a(\(2)p FD(\031)s FG(\))1688 4932 y FC(3)p FB(=)p FC(2)1817 4784 y Fu(Z)1863 4973 y Fy(R)1910 4956 y Ft(3)1955 4897 y FD(e)1994 4863 y FB(i)p Fr(p)n(x)2124 4897 y FD(v)s FG(\()p Fv(x)p FD(;)14 b Fv(p)p FG(\))38 b(^)-56 b FD(')q FG(\()p Fv(p)p FG(\))23 b FD(d)p Fv(p)p FD(:)386 5125 y FG(In)h(this)h(represen)n(tation,)e(the)i(de\014ning)g (equations)e(for)h(the)h(transformation)d(op)r(erators)h(turn)386 5224 y(in)n(to)18 b(algebraic)f(equations)g(for)h(their)g(sym)n(b)r (ols,)i(whic)n(h)e(are)g(readily)f(solv)-5 b(able.)33 b(In)19 b(our)e(w)n(ork)g(w)n(e)386 5324 y(will)28 b(call)g(the)g (general)f(transformation)f(sc)n(heme)i(b)n(y)f(means)h(of)g(unitary)f (pseudo)r(di\013eren)n(tial)386 5423 y(op)r(erators)f(the)i('Sob)r (olev)f(transformation)f(sc)n(heme'.)486 5523 y(W)-7 b(olf,)27 b(Reiher)g(and)g(Hess)g(\(2002,)e(2004\))h(ha)n(v)n(e)g(ev)-5 b(aluated)27 b(the)g(transformed)f(Dirac)h(op)r(er-)386 5623 y(ator)f(up)h(to)g(\014fth)h(order)e(in)h FD(\015)k FG(for)c(its)g(use)g(in)g(quan)n(tum)g(c)n(hemical)f(v)-5 b(ariational)26 b(calculations.)p eop %%Page: 7 13 7 12 bop 3342 259 a FC(7)386 459 y FG(A)26 b(thorough)f(mathematical)g (analysis)f(has,)i(ho)n(w)n(ev)n(er,)e(only)h(b)r(een)i(p)r(erformed)e (for)g(the)h(\014rst-)386 558 y(order)e(term)h(whic)n(h)g(agrees)e (with)j(the)f(Bro)n(wn-Ra)n(v)n(enhall)e(op)r(erator)g(de\014ned)i(ab)r (o)n(v)n(e)f(\(Ev)-5 b(ans,)386 658 y(P)n(erry)26 b(and)h(Sieden)n(top) g(1996,)f(Balinksy)h(and)g(Ev)-5 b(ans)27 b(1998,)f(1999\).)486 758 y(F)-7 b(or)19 b(the)i(second-order)d(pseudo-relativistic)h(op)r (erator,)i(in)n(tro)r(duced)f(b)n(y)g(Jansen)g(and)g(Hess)386 857 y(\(1989\),)35 b(only)g(the)g(b)r(oundedness)g(from)f(b)r(elo)n(w)g (for)h(sub)r(critical)f(p)r(oten)n(tial)h(strength)f(\()p FD(\015)40 b FA(\024)386 957 y FG(1)p FD(:)p FG(006\))48 b(is)27 b(kno)n(wn.)36 b(Also,)26 b(p)r(ositivit)n(y)h(of)f(the)i (Jansen-Hess)d(op)r(erator)g(in)i(the)g(\014ctitious)g(case)386 1056 y(of)35 b(a)f(massless)g(particle)h(has)f(b)r(een)h(sho)n(wn)g (\(Brummelh)n(uis,)h(Sieden)n(top)f(and)g(Sto)r(c)n(kmey)n(er)386 1156 y(2002\).)h(The)27 b(v)-5 b(alue)28 b(1.006)e(for)h(the)h (critical)f(p)r(oten)n(tial)h(strength)f(is)h(v)n(ery)f(close)g(to)g (the)h(exact)386 1256 y(v)-5 b(alue)27 b(1)h(where)f(the)h (ground-state)e(energy)g FD(E)1864 1268 y FC(0)1929 1256 y FG(b)r(ecomes)i(zero.)486 1355 y(In)e(the)i(case)e(of)g(more)g(than)h (one)f(particle)h(in)g(a)f(cen)n(tral)g(\014eld,)h(the)g(in)n(tuitiv)n (e)g(w)n(a)n(y)f(of)h(con-)386 1467 y(structing)f(an)g(op)r(erator)f (whic)n(h)h(is)h(the)g(sum)f(of)h(one-particle)e(Dirac)h(op)r(erators)e FD(D)3025 1423 y FC(\()p FB(k)q FC(\))3023 1489 y(0)3134 1467 y FG(+)16 b FD(V)3282 1436 y FC(\()p FB(k)q FC(\))386 1566 y FG(plus)27 b(the)g(t)n(w)n(o-particle)d(in)n(teraction)i(terms,) h(do)r(es)f(not)g(lead)h(to)f(correct)f(results.)36 b(Indeed,)27 b(as)386 1666 y(w)n(as)j(p)r(oin)n(ted)i(out)g(b)n(y)f(Bro)n(wn)f(and)h (Ra)n(v)n(enhall)f(\(1951\),)i(one)f(do)r(es)g(not)g(get)h(stable)f(b)r (ound-)386 1765 y(state)c(solutions.)36 b(Instead,)27 b(a)f(consisten)n(t)h(form)n(ulation)f(within)i(quan)n(tum)f(electro)r (dynamics)386 1865 y(is)36 b(required.)63 b(An)38 b(op)r(erator)c (acting)i(on)h(4-spinors)e(can)h(b)r(e)h(deriv)n(ed)f(from)g(the)h (full)g(QED)386 1965 y(Hamiltonian.)k(It)29 b(can)g(b)r(e)h(split)f(in) n(to)g(a)f(part)h FD(H)1937 1977 y FB(no)p Fz(\000)p FB(pair)2222 1965 y FG(whic)n(h)g(describ)r(es)g(stationary)e(elec-)386 2064 y(tronic)j(states)h(\(conserving)e(the)j(n)n(um)n(b)r(er)f(of)g (electrons\),)g(a)f(second)h(part)f FD(H)2866 2076 y FB(pair)3028 2064 y FG(whic)n(h)h(ac-)386 2164 y(coun)n(ts)36 b(for)h(pair)f(creation,)j(and)d(remaining)h(parts)f(whic)n(h)h (additionally)f(in)n(v)n(olv)n(e)g(the)h(ra-)386 2264 y(diation)g(\014eld.)67 b(The)38 b(mere)f(consideration)e(of)j FD(H)2004 2276 y FB(no)p Fz(\000)p FB(pair)2298 2264 y FG(in)f(this)h(op)r(erator)e(is)h(called)g(the)386 2363 y('no-pair')22 b(appro)n(ximation,)g(where)g(one)h(disregards)e (pair)h(creation)g(and)h(the)h(coupling)e(to)h(the)386 2463 y(photon)j(\014eld.)36 b(F)-7 b(or)26 b(the)g(t)n(w)n(o-particle)e (case,)i FD(N)32 b FG(=)22 b(2,)k(Suc)n(her)g(\(1958\))e(deriv)n(ed)h (the)i(follo)n(wing)386 2562 y(op)r(erator)f(\(whic)n(h)i(b)r(elo)n(w)f (will)h(b)r(e)g(called)f(Coulom)n(b-Dirac)f(op)r(erator\))812 2816 y FD(H)881 2828 y FB(no)p Fz(\000)p FB(pair)1183 2816 y FG(=)1338 2713 y FC(2)1294 2738 y Fu(X)1294 2916 y FB(k)q FC(=1)1415 2816 y FG(\()p FD(D)1518 2773 y FC(\()p FB(k)q FC(\))1516 2839 y(0)1643 2816 y FG(+)32 b FD(V)1807 2782 y FC(\()p FB(k)q FC(\))1899 2816 y FG(\))42 b(+)f FD(P)2144 2773 y FC(\(1\))2132 2837 y(+)2247 2816 y FD(P)2312 2773 y FC(\(2\))2300 2837 y(+)2424 2816 y FD(V)2491 2782 y FC(\(12\))2627 2816 y FD(P)2692 2773 y FC(\(1\))2680 2837 y(+)2795 2816 y FD(P)2860 2773 y FC(\(2\))2848 2837 y(+)386 3071 y FG(from)23 b(the)h(Bethe-Salp)r(eter)e(equation)h(of)h (quan)n(tum)f(electro)r(dynamics)f(\(Bethe)i(and)f(Salp)r(eter)386 3186 y(1957\).)69 b(The)38 b FD(P)924 3143 y FC(\()p FB(k)q FC(\))912 3207 y(+)1056 3186 y FG(are)g(the)h(exact)f(pro)5 b(jectors)37 b(de\014ned)i(ab)r(o)n(v)n(e,)h(relating)e(to)h(the)g (single-)386 3309 y(particle)c(op)r(erator)e FD(D)1107 3266 y FC(\()p FB(k)q FC(\))1105 3331 y(0)1223 3309 y FG(+)23 b FD(V)1378 3279 y FC(\()p FB(k)q FC(\))1506 3309 y FG(of)35 b(particle)g FD(k)s FG(,)i(and)e FD(V)2258 3279 y FC(\(12\))2415 3309 y FG(is)g(the)h(in)n(teraction)e(b)r(et)n(w) n(een)386 3409 y(paericles)h(1)h(and)g(2.)62 b(A)36 b(nice)g(accoun)n (t)g(of)g(the)g(deriv)-5 b(ation)36 b(of)g(this)g(op)r(erator)f(is)h (giv)n(en)f(b)n(y)386 3508 y(Douglas)20 b(and)g(Kroll)g(\(1974\))f(who) h(call)h(the)g(eigen)n(v)-5 b(alue)20 b(equation)g(of)g FD(H)2636 3520 y FB(no)p Fz(\000)p FB(pair)2913 3508 y FG(the)i(Coulom)n(b)386 3608 y(ladder)27 b(equation.)486 3719 y(Alternativ)n(ely)-7 b(,)22 b(it)h(w)n(as)e(later)g(suggested)g (\(Mittleman)i(1981\))d(to)i(replace)f FD(P)2881 3676 y FC(\()p FB(k)q FC(\))2869 3740 y(+)2996 3719 y FG(b)n(y)h(the)g(free) 386 3879 y(pro)5 b(jectors)23 b(\003)832 3836 y FC(\()p FB(k)q FC(\))832 3900 y(+)924 3879 y FG(,)i(but)h(to)e(pro)5 b(ject)24 b(the)h(complete)g(op)r(erator)2364 3801 y FC(2)2337 3817 y Fu(P)2321 3954 y FB(k)q FC(=1)2441 3879 y FG(\()p FD(D)2544 3836 y FC(\()p FB(k)q FC(\))2542 3902 y(0)2650 3879 y FG(+)13 b FD(V)2794 3849 y FC(\()p FB(k)q FC(\))2887 3879 y FG(\))g(+)g FD(V)3076 3849 y FC(\(12\))3198 3879 y FD(;)48 b FG(i.e.)386 4029 y(also)26 b(the)i(single-particle)e(con)n(tribution.)36 b(This)27 b(is,)g(lik)n(e)g(the)h(Bro)n(wn-Ra)n(v)n(enhall)c(op)r(erator)i(in)386 4128 y(the)i(single-particle)e(case,)h(only)g(an)g(appro)n(ximation)f (linear)h(in)h(the)g(p)r(oten)n(tials.)486 4228 y(One)h(aim)h(of)g(the) h(presen)n(t)e(w)n(ork)g(is)h(to)g(get)f(some)h(additional)f (information)h(on)f FD(D)3141 4240 y FC(0)3178 4228 y FG(-form)386 4328 y(b)r(oundedness)36 b(and)h(the)f(sp)r(ectral)g(prop) r(erties)g(of)g(the)h(\(single-particle\))e(Jansen-Hess)g(op-)386 4427 y(erator)i(for)g(massiv)n(e)g(particles.)69 b(The)38 b(ma)5 b(jor)37 b(goal)g(is,)k(ho)n(w)n(ev)n(er,)e(to)g(apply)f(the)g (Sob)r(olev)386 4527 y(transformation)c(sc)n(heme)i(to)g FD(N)9 b FG(-electron)35 b(atoms.)62 b(F)-7 b(or)35 b FD(N)46 b FG(=)37 b(2)p FD(;)f FG(the)g(pseudo-relativistic)386 4626 y(op)r(erator)26 b(has)i(b)r(een)g(deriv)n(ed)f(b)n(y)h(Douglas)f (and)h(Kroll)f(\(1974\),)g(and)g(they)i(also)e(pro)n(vide)f(the)386 4726 y(second-order)18 b(terms)j(of)g(the)g(transformed)f (electron-electron)f(Coulom)n(b)h(in)n(teraction.)34 b(Ho)n(w-)386 4826 y(ev)n(er,)21 b(within)h(the)g(Douglas-Kroll)d (transformation)g(sc)n(heme,)k(these)e(second-order)e(terms)i(are)386 4925 y(so)26 b(complicated)h(that)g(they)g(ha)n(v)n(e)e(b)r(een)j (neglected)e(in)h(an)n(y)f(n)n(umerical)g(computation)h(\(Hess)386 5025 y(1986\))21 b(assuming)h(that)h(they)g(are)f(small)h(an)n(yw)n(a)n (y)e(\(W)-7 b(olf,)24 b(Reiher)f(and)g(Hess)f(2004\).)34 b(Applica-)386 5125 y(tion)29 b(of)f(the)i(Sob)r(olev)e(transformation) f(sc)n(heme)h(pro)n(vides)g(a)g(breakthrough)f(in)i(the)g(resp)r(ect) 386 5224 y(that)23 b(the)g(resulting)g(transformed)f(second-order)e(op) r(erators)h(ha)n(v)n(e)h(a)g(v)n(ery)g(simple)h(structure.)386 5324 y(Th)n(us)k(a)g(detailed)h(mathematical)f(analysis)g(b)r(ecomes)g (feasible.)486 5423 y(The)k(la)n(y-out)f(of)h(the)h(presen)n(t)f(w)n (ork)f(is)h(as)g(follo)n(ws.)47 b(After)32 b(an)f(o)n(v)n(erview)e(of)j (some)e(basic)386 5523 y(auxiliary)i(theorems)h(and)g(of)g(the)h (pseudo)r(di\013eren)n(tial)f(op)r(erator)f(tec)n(hnique)h(\(section)h (I.1\),)386 5623 y(the)40 b(single-particle)e(Sob)r(olev)i (transformation)e(sc)n(heme)h(is)h(describ)r(ed)f(for)g(the)i(Coulom)n (b)p eop %%Page: 8 14 8 13 bop 386 259 a FC(8)386 459 y FG(p)r(oten)n(tial)38 b(\(section)h(I.2\),)i(and)d(its)h(con)n(v)n(ergence)d(is)j(sho)n(wn)e (not)i(for)f(the)h(Coulom)n(b)e(\014eld,)386 558 y(but)e(for)f(the)h (sligh)n(tly)f(less)g(singular)f(p)r(oten)n(tial)h FD(V)54 b FG(=)34 b FA(\000)p FD(\015)5 b(=x)2368 528 y FC(1)p Fz(\000)p FB(\017)2553 558 y FG(\(0)34 b FD(<)g(\017)g FA(\034)h FG(1\))p FD(:)69 b FG(Section)386 658 y(I.3)32 b(furnishes)f(the)i(equiv)-5 b(alence)32 b(to)g(the)g(Douglas-Kroll)d (transformation)i(sc)n(heme,)i(as)e(w)n(ell)386 758 y(as)38 b(the)h(explicit)g(form)f(of)h(the)g(transformed)e(op)r(erator)g(to)i (second)f(order)f(in)i FD(\015)k FG(in)c(either)386 857 y(represen)n(tation.)47 b(Subsequen)n(tly)-7 b(,)33 b FD(D)1569 869 y FC(0)1606 857 y FG(-form)e(b)r(oundedness)g(of)h(the)g (p)r(oten)n(tial)f(terms)h(of)f(the)386 957 y(Jansen-Hess)j(op)r (erator)g(is)h(sho)n(wn)g(as)g(w)n(ell)g(as)g(sub)r(ordinacy)f(of)i (the)g(second-order)d(terms)386 1056 y(with)28 b(resp)r(ect)f(to)g(the) h(\014rst-order)d(terms)i(for)g(su\016cien)n(tly)g(small)g(p)r(oten)n (tial)h(strength.)36 b(Also,)386 1156 y(p)r(ositivit)n(y)28 b(is)g(pro)n(v)n(en,)f(although)h(w)n(e)g(did)h(not)f(succeed)g(in)h (sho)n(wing)e(it)i(for)f(the)g(same)g(range)386 1256 y(of)e FD(\015)32 b FG(as)25 b(in)i(the)g(massless)e(case)h(\(section)g (I.4\).)37 b(The)26 b(single-particle)f(in)n(v)n(estigations)g(are)g (ter-)386 1355 y(minated)f(with)g(the)g(lo)r(calisation)e(of)i(the)g (essen)n(tial)e(sp)r(ectrum)i(of)g(the)f(Jansen-Hess)g(op)r(erator)386 1455 y(at)30 b([)p FD(m;)14 b FA(1)p FG(\))p FD(;)59 b FG(with)31 b(the)g(pro)r(of)f(of)h(the)g(absence)f(of)g(singular)g (con)n(tin)n(uous)f(sp)r(ectrum)i(as)f(w)n(ell)386 1555 y(as)j(the)h(absence)e(of)i(em)n(b)r(edded)g(eigen)n(v)-5 b(alues)32 b(in)i([)p FD(m;)14 b FA(1)p FG(\))33 b(up)h(to)g(certain)e (critical)h(coupling)386 1654 y(strengths,)26 b(and)f(\014nally)h(with) h(sho)n(wing)d(the)j(absolute)e(con)n(tin)n(uit)n(y)h(of)f(the)i(sp)r (ectrum)f(for)f(the)386 1754 y(\014ctitious)33 b(massless)f(particle.) 53 b(These)33 b(results)g(are)f(a)h(generalisation)e(of)i(those)g(kno)n (wn)f(for)386 1853 y(the)c(Bro)n(wn-Ra)n(v)n(enhall)c(op)r(erator.)486 1953 y(P)n(art)32 b(I)r(I)j(deals)f(with)h(the)f(t)n(w)n(o-particle)f (Coulom)n(b-Dirac)g(op)r(erator.)55 b(In)35 b(this)f(case,)h(the)386 2053 y(second)27 b(unitary)g(transformation)g(con)n(tains)f(a)i (correlated)e(t)n(w)n(o-particle)g(con)n(tribution)h(\(the)386 2152 y(b)r(oundedness)38 b(of)h(whic)n(h)f(is)g(pro)n(v)n(en)f(in)i (section)f(I)r(I.3\))h(whic)n(h,)i(ho)n(w)n(ev)n(er,)e(do)r(es)f(not)g (en)n(ter)386 2252 y(in)n(to)26 b(the)i(transformed)d(second-order)f (op)r(erator)h(\(section)i(I)r(I.4\).)37 b(Again,)26 b(the)i(relativ)n(e)d(form)386 2352 y(b)r(oundedness)j(of)g(the)h (transformed)e(p)r(oten)n(tial)h(terms)g(as)f(w)n(ell)h(as)g(p)r (ositivit)n(y)g(of)g(the)g(result-)386 2451 y(ing)36 b(pseudo-relativistic)e(op)r(erator)g(is)i(pro)n(v)n(en)e(for)h(sub)r (critical)h(p)r(oten)n(tial)g(strength.)61 b(The)386 2551 y(maxim)n(um)24 b(p)r(ossible)h(p)r(oten)n(tial)f(strength)h(for)f (these)h(prop)r(erties)e(to)i(hold,)g(deriv)n(ed)f(from)g(the)386 2651 y(presen)n(t)35 b(t)n(yp)r(e)g(of)h(estimates,)h(is)e(sligh)n(tly) g(b)r(elo)n(w)g(the)h(one)f(for)f(the)i(single-particle)e(op)r(era-)386 2750 y(tor.)j(P)n(art)27 b(I)r(I)h(is)g(closed)g(b)n(y)f(lo)r(calising) g(the)i(essen)n(tial)e(sp)r(ectrum)h(of)g(the)g(pseudo-relativistic)386 2850 y(t)n(w)n(o-particle)23 b(op)r(erator.)35 b(With)25 b(the)h(help)f(of)g(its)h(b)r(eha)n(viour)d(under)i(translations)f(it)i (is)e(found)386 2949 y(that)29 b(the)g(essen)n(tial)f(sp)r(ectrum)h(of) g(the)g(free)g(t)n(w)n(o-particle)e(op)r(erator,)h([2)p FD(m;)14 b FA(1)p FG(\))p FD(;)54 b FG(is)28 b(a)h(subset)386 3049 y(of)h(the)g(essen)n(tial)f(sp)r(ectrum)i(of)f(the)g(transformed)f (t)n(w)n(o-particle)f(op)r(erator)h(with)h(all)g(p)r(oten-)386 3149 y(tial)36 b(terms)g(included.)63 b(The)36 b(essen)n(tial)f(sp)r (ectrum)h(of)h(the)f(latter)g(do)r(es)g(not)g(c)n(hange)f(when)386 3248 y(the)d(transformed)f(t)n(w)n(o-particle)f(second-order)f(p)r (oten)n(tial)j(is)f(dropp)r(ed.)50 b(The)31 b(pro)r(of)h(of)f(the)386 3348 y(conjecture)24 b(that)i(the)f(in\014m)n(um)g(of)g(the)h(essen)n (tial)e(sp)r(ectrum)h(of)g(a)f(t)n(w)n(o-electron)f(ion)i(is)g(giv)n (en)386 3448 y(b)n(y)h(the)h(ground-state)f(energy)f(of)i(the)g (corresp)r(onding)e(one-electron)g(ion,)h(increased)g(b)n(y)g(the)386 3547 y(rest)39 b(energy)f(of)h(the)h(second)e(electron)h(\(the)h (relativistic)e(v)n(ersion)g(of)h(the)h(HVZ)f(theorem)386 3647 y(\(Reed-Simon)28 b(1978,)d(Theorem)i(XI)r(I)r(I.17\)\))h(is)g (left)g(to)g(future)g(in)n(v)n(estigations.)486 3746 y(W)-7 b(e)32 b(close)f(our)g(w)n(ork)f(b)n(y)i(deriving)f(the)h (second-order)d(pseudo-relativistic)h(op)r(erator)h(in)386 3846 y(the)i FD(N)9 b FG(-particle)32 b(case,)i FD(N)41 b(>)32 b FG(2)64 b(\(part)33 b(I)r(I)r(I\).)h(T)-7 b(o)33 b(this)g(order)f(it)i(turns)f(out)g(to)g(b)r(e)g(a)g(simple)386 3946 y(generalisation)24 b(of)j(the)f(op)r(erator)f(deriv)n(ed)h(for)f FD(N)32 b FG(=)23 b(2)p FD(;)j FG(and)g(hence)h(its)f(p)r(ositivit)n(y) g(is)h(readily)386 4045 y(sho)n(wn.)43 b(In)30 b(the)h(case)e(of)h (neutral)f(atoms)h(\()p FD(N)36 b FG(=)26 b FD(Z)6 b FG(\))30 b(the)g(corresp)r(onding)f(critical)g(p)r(oten)n(tial)386 4145 y(strength)e(is)g(found)g(to)g(b)r(e)g(considerably)f(smaller)g (than)h(for)g(one-)f(or)g(t)n(w)n(o-electron)f(ions.)37 b(Its)386 4245 y(reduction)e(for)h FD(N)45 b FG(=)37 b FD(Z)k FG(is)36 b(a)f(consequence)g(of)h(the)g(additional)g(presence) f(of)g(the)i(second-)386 4344 y(order)27 b(t)n(w)n(o-particle)g(in)n (teraction)h(terms)h(whic)n(h)f(scale)g(with)i FD(N)2419 4314 y FC(2)2456 4344 y FD(:)54 b FG(Although)29 b(con)n(trolled)e(b)n (y)386 4444 y(the)36 b(\014rst-order)d(electron-electron)h(in)n (teraction,)i(these)f(terms)g(are)g(conjectured)g(to)g(b)r(e)h(of)386 4543 y(opp)r(osite)j(sign.)72 b(Hence)39 b(their)g(neglect)g(in)h(quan) 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FD(')37 b FA(2)23 b FD(L)2328 2079 y FC(2)2365 2067 y FG(\()p Fx(R)2451 2032 y FC(3)2495 2067 y FG(\))p FA(g)647 b FG(\(I.2\))386 2215 y(with)28 b(the)g(scalar)e(pro)r(duct)885 2401 y(\()p FD(';)14 b( )s FG(\))1097 2413 y FB(\033)1165 2401 y FG(:=)46 b(\()p FD(f)1381 2367 y FB(\033)1439 2401 y FD(';)14 b(f)1580 2367 y FB(\033)1639 2401 y FD( )s FG(\))46 b(=)1885 2288 y Fu(Z)1931 2477 y Fy(R)1978 2460 y Ft(3)2010 2401 y FG(\(1)18 b(+)g FD(p)2227 2367 y FC(2)2264 2401 y FG(\))2296 2367 y FB(\033)p 2364 2329 172 4 v 2377 2401 a FG(^)-55 b FD(')p FG(\()p Fv(p)p FG(\))2576 2379 y(^)2559 2401 y FD( )s FG(\()p Fv(p)p FG(\))24 b FD(d)p Fv(p)p FD(:)340 b FG(\(I.3\))386 2601 y(These)34 b(Sob)r(olev)h(spaces)f(are)g(Hilb)r (ert)h(spaces)f(and)h(are)e(dense)i(subspaces)f(of)h FD(L)3007 2613 y FC(2)3044 2601 y FG(\()p Fx(R)3130 2571 y FC(3)3173 2601 y FG(\))p FD(:)71 b FG(In)386 2701 y(turn,)32 b FA(S)37 b FG(is)31 b(dense)g(in)g FD(H)1171 2713 y FB(\033)1216 2701 y FG(\()p Fx(R)1302 2671 y FC(3)1345 2701 y FG(\))60 b(\(W)-7 b(erner)30 b(1995,)g FA(x)p FG(V.2;)j(F)-7 b(olland)30 b(1995,)g(p.192\).)46 b(Our)30 b(cases)386 2800 y(of)d(in)n(terest)g(are)f FD(\033)g FG(=)d(1)j(and)h FD(\033)g FG(=)22 b(1)p FD(=)p FG(2)p FD(:)k FG(The)h(space)f FD(H)2102 2815 y FC(1)p FB(=)p FC(2)2207 2800 y FG(\()p Fx(R)2293 2770 y FC(3)2336 2800 y FG(\))18 b FA(\002)f Fx(C)2522 2770 y FC(4)2592 2800 y FG(is)27 b(the)g(form)g(domain)g(of)386 2900 y FD(D)455 2912 y FC(0)492 2900 y FG(.)486 3000 y(The)19 b(one-electron)f(Dirac)h (op)r(erator)f(is)i(de\014ned)g(b)n(y)f FD(H)30 b FG(=)23 b FD(D)2367 3012 y FC(0)2406 3000 y FG(+)r FD(V)39 b FG(with)20 b FD(V)42 b FG(:=)22 b FA(\000)p FD(\015)5 b(=x)19 b FG(where)386 3104 y FD(x)35 b FG(:=)g FA(j)p Fv(x)p FA(j)p FD(:)70 b FG(F)-7 b(or)34 b(sub)r(critical)g(p)r(oten)n (tial)h(strengths)e FD(\015)40 b(<)2254 3035 y FA(p)p 2323 3035 42 4 v 69 x FG(3)p FD(=)p FG(2)34 b(one)g(has)g FA(D)r FG(\()p FD(H)7 b FG(\))35 b(=)g FA(D)r FG(\()p FD(D)3305 3116 y FC(0)3342 3104 y FG(\))386 3204 y(with)28 b FD(H)35 b FG(b)r(eing)28 b(self-adjoin)n(t)g(on)f FA(D)r FG(\()p FD(D)1613 3216 y FC(0)1651 3204 y FG(\))h(and)g(for)g FD(\015)g(<)23 b FG(1)p FD(;)61 b(H)35 b FG(has)27 b(a)h(self-adjoin)n (t)f(extension)386 3303 y(\(Thaller)g(1992,)f(p.114\).)386 3599 y(I.1.)41 b Fv(Preliminaries.)486 3798 y FG(This)35 b(section)h(con)n(tains)f(a)g(compilation)h(of)g(some)f(auxiliary)f (theorems)h(whic)n(h)h(will)g(b)r(e)386 3897 y(frequen)n(tly)28 b(used)g(in)h(the)f(follo)n(wing.)38 b(They)28 b(concern)g(estimates)g (of)g(essen)n(tially)f(self-adjoin)n(t)386 3997 y(op)r(erators)19 b(from)i(ab)r(o)n(v)n(e,)g(in)h(the)g(form)f(as)f(w)n(ell)h(as)g(in)g (the)h(norm)f(sense.)34 b(They)21 b(are)f(form)n(ulated)386 4097 y(for)30 b(single-particle)e(op)r(erators,)h(but)i(they)f(are)g (readily)f(generalised)g(to)h(the)g(m)n(ulti-particle)386 4196 y(case)24 b(\(see)h(part)f(I)r(I\).)i(Also,)f(the)g(pseudo)r 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Fv(p)2113 5511 y Fz(0)2137 5545 y FG(\))2214 5489 y FD(f)9 b FG(\()p FD(p)p FG(\))p 2202 5526 180 4 v 2202 5602 a FD(f)g FG(\()p FD(p)2326 5578 y Fz(0)2349 5602 y FG(\))2391 5545 y FD(:)737 b FG(\(I.1.1\))p eop %%Page: 10 16 10 15 bop 386 259 a FC(10)486 459 y FG(The)26 b(lemma)h(is)g(easily)f (deriv)n(ed)g(from)g(the)h(Sc)n(h)n(ur)f(test)h(for)f(the)i(b)r (oundedness)e(of)h(in)n(tegral)386 558 y(op)r(erators)d(\(see)i(e.g.)36 b(Halmos)25 b(and)h(Sunder)h(1978,)d(p.22\).)36 b(It)27 b(can)e(b)r(e)i(pro)n(v)n(ed)d(with)j(the)g(help)386 658 y(of)g(the)h(Sc)n(h)n(w)n(arz)e(inequalit)n(y)h(\(Lieb)h(and)g(Y)-7 b(au)27 b(1988,)f(EPS)h(1996\).)386 788 y Fv(Lemma)i(I.2)j FG(\(Lieb)c(and)f(Y)-7 b(au)28 b(form)n(ula)e(for)h(arbitrary)f(k)n (ernels\))p Fv(.)386 888 y Fn(L)l(et)j FD(A)h Fn(b)l(e)g(an)g(inte)l (gr)l(al)g(op)l(er)l(ator)h(de\014ne)l(d)f(by)g(the)g(kernel)g FD(k)s FG(\()p Fv(p)p FD(;)14 b Fv(p)2459 858 y Fz(0)2482 888 y FG(\))p Fn(,)1274 1091 y FG(\()p FD(A')p FG(\)\()p Fv(p)p 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y FA(j)p FD(k)s FG(\()p Fv(p)p FD(;)g Fv(p)3261 3187 y Fz(0)3284 3217 y FG(\))p FA(j)3349 3165 y Ft(1)p 3350 3174 V 3350 3207 a(2)386 3330 y FA(j)p FD( )s FG(\()p Fv(p)551 3300 y Fz(0)575 3330 y FG(\))p FA(j)p FD(:)47 b FG(In)n(tro)r(ducing)23 b(the)i(factor)e(unit)n(y)h(=)e(\()p FD(f)9 b FG(\()p FD(p)p FG(\))p FD(=f)g FG(\()p FD(p)2174 3300 y Fz(0)2197 3330 y FG(\)\))2271 3278 y Ft(1)p 2272 3287 V 2272 3320 a(2)2314 3330 y FG(\()p FD(f)g FG(\()p FD(p)2470 3300 y Fz(0)2493 3330 y FG(\))p FD(=f)g FG(\()p FD(p)p FG(\)\))2765 3278 y Ft(1)p 2765 3287 V 2765 3320 a(2)2855 3330 y FG(as)23 b(in)h(the)g(pro)r(of)386 3430 y(of)j(Lemma)h(I.1,)f(\(I.1.3\))g (results)h(from)f(the)h(Sc)n(h)n(w)n(arz)e(inequalit)n(y)-7 b(.)486 3529 y(Symmetric)34 b(op)r(erators)e(ful\014lling)48 b(\()p FD(';)14 b(A')p FG(\))49 b(=)34 b(\()p FD(A';)14 b(')p FG(\))69 b(can)34 b(readily)f(b)r(e)i(sho)n(wn)e(from)386 3629 y(\(I.1.2\))24 b(to)h(ha)n(v)n(e)e(k)n(ernels)h(with)h FD(k)s FG(\()p Fv(p)p FD(;)14 b Fv(p)1603 3599 y Fz(0)1626 3629 y FG(\))24 b(=)e FD(k)1815 3599 y Fz(\003)1853 3629 y FG(\()p Fv(p)1938 3599 y Fz(0)1962 3629 y FD(;)14 b Fv(p)p FG(\))p FD(:)48 b FG(Both)24 b FA(j)p FD(k)s FG(\()p Fv(p)p FD(;)14 b Fv(p)2602 3599 y Fz(0)2626 3629 y FG(\))p FA(j)25 b FG(and)f FA(j)p FD(k)2933 3599 y Fz(\003)2972 3629 y FG(\()p Fv(p)p FD(;)14 b Fv(p)3147 3599 y Fz(0)3170 3629 y FG(\))p FA(j)25 b FG(can)386 3735 y(b)r(e)j(estimated)f(b)n(y)g FD(c)1028 3747 y FC(0)1067 3713 y FG(~)1065 3735 y FD(k)s FG(\()p Fv(p)p FD(;)14 b Fv(p)1286 3705 y Fz(0)1309 3735 y FG(\))51 b(where)1634 3713 y(~)1632 3735 y FD(k)s FG(\()p Fv(p)p FD(;)14 b Fv(p)1853 3705 y Fz(0)1876 3735 y FG(\))28 b(is)f(a)g(nonnegativ)n(e)f(function)h(c)n(haracterising)386 3835 y(the)33 b(sym)n(b)r(ol)g(class)f(of)h FD(A)65 b FG(\(to)33 b(b)r(e)h(explained)e(in)i(subsection)e(b\).)54 b(Hence,)34 b(b)r(oth)g FD(I)3064 3847 y FC(1)3102 3835 y FG(\()p FD(p)p FG(\))f(and)386 3934 y FD(I)422 3946 y FC(2)460 3934 y FG(\()p FD(p)p FG(\))28 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b(that)g(the)h(tr)l(ansforme) l(d)f(Dir)l(ac)h(op)l(er)l(ator)g(c)l(an)f(b)l(e)g(written)f(in)i(the) 386 4641 y(fol)t(lowing)g(way)1036 4789 y FG(\()p FD(U)1125 4801 y FC(1)1176 4789 y FA(\001)14 b(\001)g(\001)f FD(U)1343 4801 y FB(n)1388 4789 y FG(\))1420 4755 y Fz(\000)p FC(1)1524 4789 y FD(H)29 b(U)1679 4801 y FC(1)1730 4789 y FA(\001)14 b(\001)g(\001)g FD(U)1898 4801 y FB(n)1989 4789 y FG(=)45 b FD(H)2175 4755 y FC(\()p FB(n)p FC(\))2314 4789 y FG(+)c FD(R)q FG(\()p FD(\015)2564 4755 y FB(n)p FC(+1)2693 4789 y FG(\))426 b(\(I.2.1\))963 5021 y FD(H)1039 4987 y FC(\()p FB(n)p FC(\))1182 5021 y FG(=)46 b(\003)1351 5033 y FC(+)1443 4879 y Fu( )1548 4918 y FB(n)1509 4942 y Fu(X)1508 5121 y FB(k)q FC(=0)1643 5021 y FD(H)1712 5033 y FB(k)1753 4879 y Fu(!)1846 5021 y FG(\003)1904 5033 y FC(+)2000 5021 y FG(+)41 b(\003)2164 5033 y Fz(\000)2257 4879 y Fu( )2363 4918 y FB(n)2323 4942 y Fu(X)2323 5121 y FB(k)q FC(=0)2457 5021 y FD(H)2526 5033 y FB(k)2567 4879 y Fu(!)2661 5021 y FG(\003)2719 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y(op)r(erator)h(analysed)g(b)n(y)i(EPS)e(\(1996\).) 486 1671 y(Let)i(us)g(no)n(w)g(consider)f FD(n)d FG(=)f(2)p FD(:)52 b FG(F)-7 b(rom)28 b(\(I.2.11\))f(it)i(follo)n(ws)e(that)i(the) f(transformed)f(Dirac)386 1771 y(op)r(erator)38 b(in)h(second)g(order)f (is)h(determined)h(b)n(y)f(the)g(\014rst)g(transformation,)i FD(U)3016 1783 y FC(1)3096 1771 y FG(=)h FD(e)3242 1740 y FB(iB)3315 1748 y Ft(1)3352 1771 y FD(;)386 1870 y FG(only)-7 b(.)53 b(Ho)n(w)n(ev)n(er,)33 b(the)g(existence)g(of)g(the)h (second)e(transformation,)h FD(U)2657 1882 y FC(2)2727 1870 y FG(=)f FD(e)2863 1840 y FB(iB)2936 1848 y Ft(2)2972 1870 y FD(;)i FG(has)e(to)h(b)r(e)386 1970 y(established)23 b(to)g(sho)n(w)f(that)i FD(H)1357 1940 y FC(\(2\))1469 1970 y FG(is)f(indeed)h(the)g(transformed)e(op)r(erator,)g(with)i(a)f (remainder)386 2069 y(of)k(order)g FD(\015)746 2039 y FC(3)783 2069 y FD(:)50 b FG(W)-7 b(e)28 b(ha)n(v)n(e)429 2254 y FD(U)495 2219 y Fz(\000)p FC(1)486 2276 y(1)598 2254 y FD(H)20 b(U)744 2266 y 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2524 29 3 v 1623 2557 a(2)1690 2384 y Fu(Z)1786 2497 y FD(d)p Fv(p)24 b FA(j)13 b FG(^)-55 b FD(')p FG(\()p Fv(p)p FG(\))p FA(j)2123 2463 y FC(2)2203 2497 y FA(\001)18 b FD(I)2280 2509 y FC(1)2318 2497 y FG(\()p FD(p)p FG(\))686 b(\(I.2.28\))1011 2765 y FD(I)1047 2777 y FC(1)1084 2765 y FG(\()p FD(p)p FG(\))24 b(:=)1348 2652 y Fu(Z)1444 2765 y FD(d)p Fv(q)1676 2709 y FG(1)p 1571 2746 252 4 v 1571 2822 a FA(j)p Fv(q)19 b FA(\000)f Fv(p)p FA(j)2171 2709 y FG(1)p 1865 2746 653 4 v 1865 2822 a(\()p FA(j)p Fv(q)h FA(\000)f Fv(p)p FA(j)33 b FG(+)18 b FD(p)g FG(+)g(1\))2481 2798 y FC(2)2561 2709 y FD(f)9 b FG(\()p FD(p)p FG(\))p 2561 2746 156 4 v 2562 2822 a FD(f)g FG(\()p FD(q)s FG(\))2727 2765 y FD(:)386 2948 y FG(It)29 b(remains)f(to)h(pro)n(v)n(e)e(that)i FD(I)1333 2960 y FC(1)1371 2948 y FG(\()p FD(p)p FG(\))g(is)g(b)r (ounded)g(for)f FD(p)d FA(2)g Fx(R)2260 2960 y FC(+)2321 2948 y FD(:)54 b FG(F)-7 b(or)29 b(the)g(con)n(v)n(ergence)d(gener-)386 3047 y(ating)32 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b(w)n(e)c(ha)n(v)n(e)g(to)386 960 y(pro)n(v)n(e)26 b(the)i(follo)n(wing)e(prop)r(osition.)386 1087 y Fv(Prop)s(osition)k (I.1)h FG(\(Existence)d(of)f(Sob)r(olev)g(transformations\))p Fv(.)386 1187 y Fn(L)l(et)k FD(U)588 1199 y FB(n)659 1187 y FG(=)25 b FD(e)788 1157 y FB(iB)861 1165 y Fp(n)907 1187 y FD(;)65 b(n)26 b FA(\025)g FG(1)p FD(;)57 b Fn(b)l(e)31 b(the)h(unitary)f(tr)l(ansformations)h(fr)l(om)g(The)l(or)l(em)h(I.1.) 44 b(L)l(et)31 b FD(\036)3329 1199 y FB(n)386 1287 y Fn(b)l(e)j(the)h(symb)l(ol)g(of)g FD(B)1074 1299 y FB(n)1154 1287 y Fn(and)g FD(W)1398 1299 y FB(n)1477 1287 y Fn(the)g(p)l (otential)g(in)g(the)f(de\014ning)h(e)l(quation)f(for)i FD(\036)3054 1299 y FB(n)3099 1287 y FD(:)66 b Fn(Then)386 1386 y FD(W)464 1398 y FB(n)539 1386 y Fn(is)30 b FD(p)p Fn(-form)g(b)l(ounde)l(d)g(on)g FD(H)1404 1401 y FC(1)p FB(=)p FC(2)1509 1386 y FG(\()p Fx(R)1595 1356 y FC(3)1638 1386 y FG(\))19 b FA(\002)f Fx(C)1826 1356 y FC(4)1899 1386 y Fn(by)30 b(me)l(ans)g(of)1450 1542 y FA(j)p 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FB(k)2458 3046 y FD(;)60 b(k)26 b FA(\024)c FD(n)2747 3016 y Fz(0)2786 3046 y FA(\000)15 b FG(1)p FD(;)25 b FG(whic)n(h)h(en)n(ter)386 3146 y(in)n(to)h(a)g(giv)n(en)g(comm)n (utator)f(con)n(tributing)h(to)g FD(W)1960 3158 y FB(n)2001 3141 y Fl(0)2056 3146 y FG(m)n(ust)g(add)g(to)h FD(n)2574 3116 y Fz(0)2615 3146 y FA(\000)18 b FG(1,)27 b(the)g(last)h(factor)e FD(\015)386 3245 y FG(b)r(eing)i(supplied)g(b)n(y)f(the)i(linearit)n(y) d(in)i FD(V)19 b FG(.)38 b(Hence,)28 b(the)g(induction)g(h)n(yp)r (othesis)g(implies)g(that)386 3345 y(all)34 b(comm)n(utators)f(of)h (smaller)g(order)f(than)i FD(\015)1884 3315 y FB(n)1963 3345 y FG(are)e FD(p)p FG(-form)h(b)r(ounded.)58 b(In)35 b(the)f(induction)386 3445 y(step)28 b(one)f(has)h(to)g(sho)n(w)f(that) h([)p FD(V)5 b(;)14 b(B)1527 3457 y FB(n)p Fz(\000)p FC(1)1657 3445 y FG(])28 b(and)g([[)p FA(\001)p FG(])p FD(;)14 b(B)2062 3457 y FB(k)2103 3445 y FG(])p FD(;)61 b(k)26 b(<)e(n)18 b FA(\000)g FG(1)p FD(;)51 b FG(are)27 b FD(p)p 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y(class.)40 b(With)30 b(the)f(help)h(of)f(the)g(Lieb)g(and)g(Y)-7 b(au)29 b(form)n(ula)f(\(I.1.1\),)h(the)h FD(p)p FG(-form)e(b)r (oundedness)386 4103 y(of)492 4077 y Fu(b)480 4103 y FG([)p FA(\001)p FG(])g(can)g(b)r(e)g(expressed)e(in)i(the)g(follo)n (wing)e(w)n(a)n(y)401 4306 y FA(j)p FG(\()p FD(';)14 b FG([)p FA(\001)p FG(])g FD(')p FG(\))p FA(j)38 b(\024)998 4250 y FG(1)p 888 4287 261 4 v 888 4365 a(\(2)p FD(\031)s FG(\))1044 4341 y FC(3)p FB(=)p FC(2)1173 4193 y Fu(Z)1270 4306 y FD(d)p Fv(p)27 b FA(j)14 b FG(^)-56 b FD(')q FG(\()p Fv(p)p FG(\))p FA(j)1611 4264 y FC(2)1676 4193 y Fu(Z)1773 4306 y FD(d)p Fv(q)1895 4214 y Fu(\020)1958 4211 y(\014)1958 4261 y(\014)1958 4310 y(\014)1997 4280 y(b)1986 4306 y FG([)p FA(\001)p FG(]\()p Fv(q)19 b FA(\000)f Fv(p)p FD(;)c Fv(p)p FG(\))2414 4211 y Fu(\014)2414 4261 y(\014)2414 4310 y(\014)2474 4306 y FG(+)2571 4211 y Fu(\014)2571 4261 y(\014)2571 4310 y(\014)2610 4280 y(b)2599 4306 y FG([)p FA(\001)p FG(])2668 4240 y Fz(\003)2706 4306 y 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y(prop)r(erties)27 b(hold)h(also)f(for)h(the)h(sym)n (b)r(ol)e FD(w)1744 5424 y FB(k)1814 5412 y FG(of)h(the)h(self-adjoin)n (t)e(op)r(erator)g FD(W)2894 5424 y FB(k)2935 5412 y FD(:)52 b FG(Inequalit)n(y)386 5523 y(\(I.2.23\))30 b(is)h(therefore)f (also)f(v)-5 b(alid)31 b(for)47 b(^)-59 b FD(w)1683 5535 y FB(k)1756 5523 y FG(in)31 b(place)f(of)2180 5497 y Fu(b)2169 5523 y FG([)p FA(\001)p FG(])p FD(;)h FG(if)g FD(k)g(<)d(n)60 b FG(\(when)31 b FD(W)2978 5535 y FB(k)3050 5523 y FG(is)g FD(p)p FG(-form)386 5623 y(b)r(ounded\).)p eop %%Page: 18 24 18 23 bop 386 259 a FC(18)486 459 y FG(W)-7 b(e)24 b(presen)n(t)f(the)h (pro)r(of)g(of)f FD(p)p FG(-form)g(b)r(oundedness)h(of)g([[)p FA(\001)p FG(])p FD(;)14 b(B)2382 471 y FB(k)2423 459 y FG(];)47 b(the)24 b(corresp)r(onding)e(pro)r(of)386 558 y(for)27 b([)p FD(V)5 b(;)14 b(B)689 570 y FB(n)p Fz(\000)p FC(1)819 558 y FG(])28 b(can)f(b)r(e)h(carried)e(out)i(along) e(the)i(same)f(lines.)486 658 y(First)h(w)n(e)h(estimate)f(the)i(sym)n (b)r(ol)e 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y FD(';)2541 1666 y Fu(d)2539 1693 y FG([)p FA(\001)p FG(])p FD(')q FG(\))p FA(j)662 1897 y(\024)836 1841 y FG(1)p 760 1878 194 4 v 760 1954 a(\(2)p FD(\031)s FG(\))916 1930 y FC(3)977 1784 y Fu(Z)1074 1897 y FD(d)p Fv(p)1170 1863 y Fz(0)1194 1897 y FD(d)p Fv(p)14 b FD(d)p Fv(q)37 b FA(j)14 b FG(^)-56 b FD(')q FG(\()p Fv(p)p FG(\))p FA(j)1666 1805 y Fu(n)1722 1801 y(\014)1722 1851 y(\014)1722 1901 y(\014)1761 1870 y(b)1749 1897 y FG([)p FA(\001)p FG(])1818 1831 y Fz(\003)1857 1897 y FG(\()p Fv(p)1942 1863 y Fz(0)1984 1897 y FA(\000)18 b Fv(p)p FD(;)c Fv(p)p FG(\))2242 1801 y Fu(\014)2242 1851 y(\014)2242 1901 y(\014)2307 1897 y FA(j)2341 1875 y FG(^)2330 1897 y FD(\036)2379 1909 y FB(k)2420 1897 y FG(\()p Fv(p)2505 1863 y Fz(0)2547 1897 y FA(\000)k Fv(q)p FD(;)c Fv(q)p FG(\))p FA(j)288 b FG(\(I.2.35\))1151 2123 y(+)23 b FA(j)1273 2101 y FG(^)1262 2123 y FD(\036)1311 2089 y Fz(\003)1311 2143 y FB(k)1352 2123 y FG(\()p Fv(p)1437 2089 y Fz(0)1479 2123 y FA(\000)c Fv(p)p FD(;)14 b Fv(p)p 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1693 y Fu(\014)1101 1743 y(\014)1101 1793 y(\014)1140 1762 y(b)1129 1788 y FG([)p FA(\001)p FG(])1198 1813 y FB(n)1243 1788 y FG(\()p Fv(p)h FA(\000)f Fv(q)p FD(;)c Fv(q)p FG(\))1599 1693 y Fu(\014)1599 1743 y(\014)1599 1793 y(\014)1674 1732 y FD(p)1716 1702 y FC(2)p 1674 1769 80 4 v 1675 1845 a FD(q)1715 1821 y FC(2)1782 1788 y FA(\001)k FD(q)1863 1754 y FC(1)p Fz(\000)p FB(\016)2031 1788 y FA(\024)46 b FD(p)2184 1754 y Fz(\000)p FB(\016)2285 1675 y Fu(Z)2382 1788 y FD(d)p Fv(q)2513 1693 y Fu(\014)2513 1743 y(\014)2513 1793 y(\014)2552 1762 y(b)2540 1788 y FG([)p FA(\001)p FG(])2610 1813 y FB(n)2655 1788 y FG(\()p Fv(p)19 b FA(\000)f Fv(q)p FD(;)c Fv(q)p FG(\))3011 1693 y Fu(\014)3011 1743 y(\014)3011 1793 y(\014)3086 1732 y FD(p)3128 1702 y FC(2+)p FB(\016)p 3086 1769 163 4 v 3087 1845 a FD(q)3127 1821 y FC(2+)p FB(\016)1654 2005 y FA(\024)45 b FD(c)23 b(p)1865 1970 y FC(1)p Fz(\000)p FB(n\017)p Fz(\000)p FB(\016)3110 2005 y FG(\(I.2.67\))386 2137 y(if)28 b(2)18 b(+)g FD(\016)40 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3710 y FA(j)p FG(\()p FD(';)14 b(V)1115 3722 y FC(1)1167 3710 y FD(')p FG(\))p FA(j)47 b(\024)e FD(c)1469 3722 y FC(0)1530 3710 y FG(\()p FD(';)14 b(p)1695 3675 y FC(1)p Fz(\000)p FB(\017)1826 3710 y FD(')p FG(\))46 b FA(\024)g FD(c)23 b FG(\()p FD(';)14 b(p)g(')p FG(\))42 b(+)f FD(C)30 b FG(\()p FD(';)14 b(')p FG(\))271 b(\(I.2.70\))386 3857 y(with)34 b FD(c)g(<)e FG(1)p FD(:)67 b FG(F)-7 b(or)34 b FD(k)i(>)d FG(1)p FD(;)g FG(w)n(e)h(assume)f(that)h FD(V)1967 3869 y FB(k)2042 3857 y FG(is)g FD(p)p FG(-form)e(b)r(ounded) j(with)f(form)f(b)r(ound)386 3956 y FD(c)422 3968 y FC(1)482 3956 y FD(<)23 b FG(1)p FD(:)50 b FG(Then)28 b(w)n(e)f(ha)n(v)n(e)g (from)g(\(ii\))423 4104 y FA(j)p FG(\()p FD(';)14 b(V)617 4116 y FB(k)q FC(+1)757 4104 y FD(')p FG(\))p FA(j)38 b(\024)e FD(\016)17 b FA(j)p FG(\()p FD(';)d(V)1253 4116 y FB(k)1295 4104 y FD(')p FG(\))p FA(j)33 b FG(+)f FD(C)20 b FG(\()p FD(';)14 b(')p FG(\))38 b FA(\024)e FD(\016)31 b FG(\()p FD(c)2097 4116 y FC(1)2148 4104 y FG(\()p 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1030 y Fz(0)2454 1054 y FA(j)2477 1030 y FC(1)p Fz(\000)p FB(\017)2650 997 y FA(\024)45 b FD(c)2796 963 y Fz(0)3110 997 y FG(\(I.2.73\))386 1162 y(since)28 b(the)i(in)n(tegral)d(is)i(\014nite)g(for)f FD(p)d FA(!)g FG(0)j(and)h(for)f FD(p)d FA(!)g(1)j FG(and)h(the)g (singularit)n(y)e(of)i(the)g(last)386 1270 y(factor)f(at)h Fv(p)d FG(=)f FA(\000)p Fv(q)1012 1240 y Fz(0)1064 1270 y FG(is)k(in)n(tegrable.)40 b(The)30 b(con)n(v)n(ergence)c(generating)i (function)i FD(f)9 b FG(\()p FD(p)p FG(\))25 b(=)g FD(p)3262 1214 y Ft(1)p Fl(\000)p Fp(\017)p 3262 1227 99 3 v 3297 1260 a Ft(2)386 1369 y FG(w)n(as)g(c)n(hosen)h(to)g(allo)n(w)f(for)h(a) g(\(I.2.45\)-t)n(yp)r(e)f(estimate)h(when)h(sho)n(wing)e(that)i(the)f (presence)g(of)386 1469 y FD(U)443 1481 y FB(k)511 1469 y FG(pla)n(ys)h(no)g(role)g(\(but)h(to)g(pro)n(v)n(e)e(b)r(oundedness)h (of)h FD(I)2129 1481 y FB(B)2179 1489 y Ft(1)2216 1469 y FG(,)g(one)f(can)g(also)g(tak)n(e)f FD(f)9 b FG(\()p FD(p)p FG(\))23 b(=)g(1\).)486 1578 y(F)-7 b(or)21 b FD(B)692 1590 y FC(2)729 1578 y FG(,)i(w)n(e)f(use)g(the)g(estimate)g (\(I.2.34\))f(of)1881 1556 y(^)1870 1578 y FD(\036)1919 1590 y FC(2)1979 1578 y FG(b)n(y)38 b(^)-59 b FD(w)2147 1590 y FC(2)2207 1578 y FG(and)22 b(recall)f(that)h FD(W)2831 1590 y FC(2)2891 1578 y FG(is)g(determined)386 1677 y(from)27 b(the)h(comm)n(utator)f([)p FD(V)5 b(;)14 b(B)1370 1689 y FC(1)1407 1677 y FG(])p FD(:)51 b FG(Consider)26 b(the)i(sym)n(b)r (ol)f(of)h FD(V)19 b(B)2502 1689 y FC(1)2567 1677 y FG(via)27 b(\(I.1.17\),)639 1856 y FA(j)667 1834 y Fu(c)662 1856 y FD(v)s(\036)754 1876 y FC(1)792 1856 y FG(\()p Fv(q)p FD(;)14 b Fv(p)p FG(\))p FA(j)47 b(\024)1299 1800 y FD(c)p 1186 1837 261 4 v 1186 1914 a FG(\(2)p FD(\031)s FG(\))1342 1890 y FC(3)p FB(=)p FC(2)1471 1743 y Fu(Z)1568 1856 y FD(d)p Fv(p)1664 1822 y Fz(0)1895 1800 y FG(1)p 1720 1837 391 4 v 1720 1913 a FA(j)p Fv(q)19 b FA(\000)f Fv(p)1948 1889 y Fz(0)1972 1913 y FA(j)1995 1889 y FC(2+)p FB(\017)2139 1856 y FA(\001)2486 1800 y FG(1)p 2190 1837 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2523 y Fu(c)604 2545 y FD(v)s(\036)697 2565 y FC(1)734 2545 y FG(\()p Fv(q)p FD(;)14 b Fv(p)p FG(\))p FA(j)48 b(\024)1133 2489 y FG(~)-44 b FD(c)p 1129 2526 41 4 v 1129 2602 a(q)1193 2432 y Fu(Z)1276 2452 y Fz(1)1239 2620 y FC(0)1370 2489 y FD(dp)1455 2458 y Fz(0)p 1370 2526 109 4 v 1376 2602 a FD(p)1418 2561 y Fl(0)1441 2578 y FB(\017)1727 2489 y FG(1)p 1521 2526 454 4 v 1521 2602 a(\()p FD(p)1595 2578 y Fz(0)1637 2602 y FG(+)18 b FD(p)g FG(+)h(1\))1938 2578 y FC(2)1998 2428 y Fu(\022)2191 2489 y FG(1)p 2069 2526 285 4 v 2069 2602 a FA(j)p FD(q)j FA(\000)c FD(p)2276 2578 y Fz(0)2299 2602 y FA(j)2322 2578 y FB(\017)2405 2545 y FA(\000)2643 2489 y FG(1)p 2521 2526 V 2521 2602 a FA(j)p FD(q)k FG(+)c FD(p)2728 2578 y Fz(0)2751 2602 y FA(j)2774 2578 y FB(\017)2816 2428 y Fu(\023)2891 2545 y FD(:)196 b FG(\(I.2.75\))386 2731 y(Since)24 b(the)g(div)n(ergence)e(at)h FD(q)j FG(=)d(0)g(results) g(from)g(the)h(b)r(eha)n(viour)f(of)g(the)h(in)n(tegral)f(near)f FD(p)3176 2701 y Fz(0)3223 2731 y FG(=)g(0,)386 2831 y(it)41 b(is)f(su\016cien)n(t)h(to)f(reduce)h(the)g(in)n(tegration)e (region)g(to)i([0)p FD(;)14 b FG(1])39 b(and)i(estimate)f(\()p FD(p)3099 2800 y Fz(0)3150 2831 y FG(+)26 b FD(p)h FG(+)386 2930 y(1\))460 2900 y Fz(\000)p FC(2)582 2930 y FA(\024)33 b FG(1)p FD(:)66 b FG(The)34 b(resulting)f(in)n(tegral)f(can)h(b)r(e)h (p)r(erformed)f(analytically)g(with)h(the)g(help)g(of)386 3030 y(h)n(yp)r(ergeometric)j(functions)h(\(Gradsh)n(teyn)g(and)g (Ryzhik)g(1965,)i(p.284\),)g(and)e(it)h(b)r(eha)n(v)n(es)386 3129 y FA(\030)23 b FD(q)514 3099 y FC(1)p Fz(\000)p FC(2)p FB(\017)691 3129 y FG(for)k FD(q)f FA(!)d FG(0)p FD(:)69 b(B)1184 3141 y FC(1)1221 3129 y FD(V)47 b FG(is)27 b(in)h(the)g(same)f(op)r(erator)f(class)h(suc)n(h)g(that)h(w)n(e)f (obtain)1311 3302 y FA(j)17 b FG(^)-59 b FD(w)1393 3314 y FC(2)1430 3302 y FG(\()p Fv(q)p FD(;)14 b Fv(p)p FG(\))p FA(j)47 b(\024)f FD(c)2043 3246 y FG(1)18 b(+)g FD(q)2226 3216 y FB(\017)p 1884 3283 534 4 v 1884 3359 a FD(q)1924 3335 y FC(2)p FB(\017)1989 3359 y FG(\()p FD(q)k FG(+)c FD(p)g FG(+)g(1\))2380 3335 y FC(2)2427 3302 y FD(:)660 b FG(\(I.2.76\))386 3487 y(By)38 b(induction,)k(one)c(can)g(sho)n(w)f (that)i(for)f FD(k)44 b(>)d FG(2,)f(one)f(has)e FA(j)17 b FG(^)-59 b FD(w)2506 3499 y FB(k)2548 3487 y FG(\()p Fv(q)p FD(;)14 b Fv(p)p FG(\))p FA(j)56 b(\024)3109 3454 y FB(c)p 2946 3468 355 4 v 2946 3515 a FC(\()p FB(q)r FC(+)p FB(p)p FC(+1\))3199 3499 y Ft(2+)p Fp(\017)3311 3487 y FD(:)386 3596 y FG(Th)n(us)27 b(one)h(obtains)f(regularisation)e (up)r(on)j(increasing)e FD(k)s FG(,)i(resulting)f(in)h(b)r(ounded)g(op) r(erators)386 3696 y FD(B)449 3708 y FB(k)490 3696 y FD(;)60 b(k)26 b(>)c FG(1)p FD(:)2488 b Fm(\004)486 3842 y FG(Prop)r(osition)25 b(I.2)h(pro)n(vides)g(justi\014cation)h(for)f (represen)n(ting)f(the)i(transformed)f(Dirac)g(op-)386 3942 y(erator)e(in)i(terms)f(of)g(a)g(series)g(expansion)g(in)g(the)h (p)r(oten)n(tial)g(strength.)36 b(Note,)26 b(ho)n(w)n(ev)n(er,)e(that) 386 4050 y(the)36 b(limit)h FD(\017)f FA(!)h FG(0)f(cannot)f(b)r(e)i (carried)d(out)i(since)g(in)g(\(I.2.55\),)2608 4017 y FB(\017)p 2545 4031 154 4 v 2545 4079 a FC(1)p Fz(\000)p FB(n\017)2723 4050 y FD(\016)2763 4020 y Fz(\000)2825 3994 y Ft(1)p Fl(\000)p Fp(n\017)p 2825 4007 136 3 v 2879 4040 a(\017)3025 4050 y FA(!)h(1)72 b FG(as)386 4156 y FD(\017)29 b FA(!)g FG(0)p FD(;)i FG(whic)n(h)h(implies)f FD(C)36 b FA(!)29 b(1)j FG(in)f(\(I.2.51\).)48 b(Therefore,)31 b(this)h(limit)g(cannot)f(b)r(e)h(used)f(to)386 4256 y(pro)n(v)n(e)f FD(p)p FG(-form)i(b)r(oundedness)g(of)g FD(V)1524 4268 y FB(k)1565 4256 y FD(;)75 b FG(1)30 b FA(\024)g FD(k)j FA(\024)d FD(n)63 b FG(with)33 b(form)e(b)r(ound)i (less)e(than)i(one,)g(in)386 4355 y(the)28 b(case)f(of)g(the)h(Coulom)n (b)f(p)r(oten)n(tial.)386 4626 y(I.3.)41 b Fv(Relation)31 b(to)g(the)h(Douglas-Kroll)e(transformation)h(sc)m(heme.)486 4826 y FG(The)38 b(Douglas-Kroll)d(transformation)i(sc)n(heme)h(for)f (the)i(Dirac)f(op)r(erator)e(is)i(based)g(on)386 4925 y(the)30 b(F)-7 b(oldy-W)g(outh)n(uysen)29 b(\(1950\))g (transformation,)g(aimed)g(at)h(casting)f(the)h(free)g(Dirac)f(op-)386 5025 y(erator)35 b(in)n(to)h(an)g(op)r(erator)f(whic)n(h)h(do)r(es)g (not)h(couple)f(the)h(upp)r(er)f(and)g(lo)n(w)n(er)f(comp)r(onen)n(ts) 386 5125 y(of)g(the)g(relativistic)g(w)n(a)n(v)n(efunction.)58 b(In)35 b(this)g(section,)i(the)e(F)-7 b(oldy-W)g(outh)n(uysen)34 b(transfor-)386 5224 y(mation,)f(generalised)e(b)n(y)h(Douglas)f(and)h (Kroll)f(\(1974\))g(to)i(Dirac)e(op)r(erators)g(including)h(an)386 5324 y(electrostatic)f(p)r(oten)n(tial,)i(will)f(b)r(e)g(describ)r(ed.) 49 b(Subsequen)n(tly)-7 b(,)33 b(it)g(will)f(b)r(e)g(sho)n(wn)f(that)h (the)386 5423 y(Douglas-Kroll)e(transformation)h(sc)n(heme)h(is)g (equiv)-5 b(alen)n(t)32 b(to)h(the)g(Sob)r(olev)e(transformation)386 5523 y(sc)n(heme.)48 b(Finally)-7 b(,)32 b(the)g(adv)-5 b(an)n(tages)30 b(and)h(dra)n(wbac)n(ks)e(of)j(the)f(resulting)g(op)r (erators)f(will)h(b)r(e)386 5623 y(discussed.)p eop %%Page: 25 31 25 30 bop 3309 259 a FC(25)386 459 y Fo(a\))33 b(The)g(Douglas-Kr)-5 b(ol)5 b(l)34 b(tr)-5 b(ansformation)36 b(scheme)486 606 y FG(Lik)n(e)17 b(the)i(Sob)r(olev)f(transformation)f(sc)n(heme,)i (the)g(Douglas-Kroll)d(transformation)h(sc)n(heme)386 705 y(consists)25 b(of)h(a)f(series)g(of)h(unitary)f(op)r(erators)f FD(U)1887 675 y Fz(0)1878 727 y FB(j)1913 705 y FD(:)49 b FG(The)26 b(zeroth-order)d(transformation)i(op)r(er-)386 805 y(ator)h(is)i(diagonal)e(in)i(momen)n(tum)g(space)f(and)g(is)g(giv) n(en)g(b)n(y)598 1013 y FD(U)664 979 y Fz(0)655 1034 y FC(0)715 1013 y FG(:=)46 b FD(A)p FG(\()p FD(p)p FG(\))1055 896 y Fu(\022)1116 1013 y FG(1)41 b(+)g FD(\014)1459 957 y Fw(\013)p Fv(p)p 1380 994 275 4 v 1380 1070 a FD(E)1441 1082 y FB(p)1498 1070 y FG(+)18 b FD(m)1664 896 y Fu(\023)1739 1013 y FD(;)346 b(A)p FG(\()p FD(p)p FG(\))23 b(:=)2433 896 y Fu(\022)2504 957 y FD(E)2565 969 y FB(p)2622 957 y FG(+)18 b FD(m)p 2504 994 V 2571 1070 a FG(2)p FD(E)2674 1082 y FB(p)2788 896 y Fu(\023)2859 891 y Ft(1)p 2859 900 29 3 v 2859 933 a(2)2916 1013 y FD(;)212 b FG(\(I.3.1\))386 1230 y(and)18 b(for)g(the)h(transformed)e(Dirac)h(op)r(erator)e(one)i (obtains)g(with)h FD(U)2457 1170 y Fl(0)2479 1195 y Fz(\000)p FC(1)2448 1253 y(0)2592 1230 y FG(=)j FD(A)p FG(\()p FD(p)p FG(\))14 b(\(1)p FA(\000)p FD(\014)3140 1174 y Fw(\013)p Fv(p)p 3061 1211 275 4 v 3061 1287 a FD(E)3122 1299 y FB(p)3179 1287 y FG(+)k FD(m)3345 1230 y FG(\))1313 1438 y FD(U)1379 1404 y Fz(0)1370 1459 y FC(0)1407 1438 y FD(H)7 b(U)1549 1378 y Fl(0)1571 1403 y Fz(\000)p FC(1)1540 1460 y(0)1706 1438 y FG(=)46 b FD(\014)t(E)1929 1450 y FB(p)2009 1438 y FG(+)c FA(E)2160 1450 y FC(1)2238 1438 y FG(+)f FA(O)2410 1450 y FC(1)1049 1626 y FA(E)1093 1638 y FC(1)1153 1626 y FG(:=)k FD(A)p FG(\()p FD(p)p FG(\))1468 1509 y Fu(\022)1530 1626 y FD(V)60 b FG(+)1833 1570 y Fw(\013)q Fv(p)p 1754 1607 V 1754 1683 a FD(E)1815 1695 y FB(p)1873 1683 y FG(+)18 b FD(m)2062 1626 y(V)2240 1570 y Fw(\013)p Fv(p)p 2161 1607 V 2161 1683 a FD(E)2222 1695 y FB(p)2280 1683 y FG(+)g FD(m)2445 1509 y Fu(\023)2520 1626 y FD(A)p FG(\()p FD(p)p FG(\))p FD(;)440 b FG(\(I.3.2\))1012 1853 y FA(O)1078 1865 y FC(1)1138 1853 y FG(:=)46 b FD(\014)t(A)p FG(\()p FD(p)p FG(\))1505 1736 y Fu(\022)1656 1797 y Fw(\013)p Fv(p)p 1577 1834 V 1577 1910 a FD(E)1638 1922 y FB(p)1695 1910 y FG(+)18 b FD(m)1884 1853 y(V)60 b FA(\000)41 b FD(V)2277 1797 y Fw(\013)p Fv(p)p 2198 1834 V 2198 1910 a FD(E)2259 1922 y FB(p)2316 1910 y FG(+)18 b FD(m)2482 1736 y Fu(\023)2557 1853 y FD(A)p FG(\()p FD(p)p FG(\))p FD(:)386 2017 y FG(When)36 b(a)g(p)r(oten)n(tial)g(is)f (presen)n(t)h(the)g(op)r(erator)e FA(E)2000 2029 y FC(1)2037 2017 y FD(;)i FG(called)g(an)f(ev)n(en)h(op)r(erator)e(b)r(ecause)h(it) 386 2117 y(comm)n(utes)30 b(with)i FD(\014)t FG(,)g(is)f(also)f (diagonal.)45 b(But)31 b(there)g(is)f(an)h(additional)f(term,)i(the)f (so-called)386 2217 y(o)r(dd)19 b(term)g FA(O)797 2229 y FC(1)853 2217 y FG(whic)n(h)f(an)n(ticomm)n(utes)g(with)i FD(\014)j FG(and)c(whic)n(h)f(do)r(es)h(couple)f(the)h(t)n(w)n(o)f (comp)r(onen)n(ts)386 2316 y(of)31 b FD( )s FG(.)47 b(Therefore,)31 b(the)g(next)h(transformation,)e FD(U)2011 2286 y Fz(0)2002 2337 y FC(1)2039 2316 y FG(,)i(is)f(aimed)g(at)g(eliminating)g FA(O)3037 2328 y FC(1)3074 2316 y FD(:)g FG(As)g(has)386 2416 y(b)r(een)20 b(sho)n(wn)f(b)n(y)g(W)-7 b(olf,)22 b(Reiher)d(and)h(Hess)f(\(2002\),)h(the)g(c)n(hoice)f(of)g(this)h (transformation)e(is)i(not)386 2524 y(unique.)36 b(Historically)-7 b(,)24 b(a)h(square-ro)r(ot)d(form)i(w)n(as)g(c)n(hosen,)g FD(U)2349 2494 y Fz(0)2340 2546 y FB(j)2398 2524 y FG(=)e(\(1)13 b(+)g FD(W)2740 2494 y FC(2)2728 2546 y FB(j)2776 2524 y FG(\))2818 2472 y Ft(1)p 2818 2481 29 3 v 2818 2514 a(2)2873 2524 y FG(+)g FD(W)3029 2536 y FB(j)3064 2524 y FD(;)50 b(j)28 b FA(\025)23 b FG(1)p FD(;)386 2628 y FG(with)30 b(an)n(tisymmetric)e(op)r(erators)g FD(W)1571 2640 y FB(j)1606 2628 y FG(,)i(i.e.)42 b(\()p FD(';)14 b(W)2008 2640 y FB(j)2044 2628 y FD(')p FG(\))40 b(=)25 b FA(\000)p FG(\()p FD(W)2435 2640 y FB(j)2471 2628 y FD(';)14 b(')p FG(\))p FD(:)56 b FG(Ho)n(w)n(ev)n(er,)28 b(in)h(order)386 2727 y(to)c(establish)f(the)h(equiv)-5 b(alence)24 b(with)i(the)f(Sob)r(olev)f(transformations,)f(an)i(exp)r (onen)n(tial)f(form)386 2827 y(has)j(to)g(b)r(e)h(tak)n(en,)1411 2927 y FD(U)1477 2892 y Fz(0)1468 2947 y FB(j)1526 2927 y FG(:=)45 b FD(e)1698 2892 y Fz(\000)p FB(iS)1814 2900 y Fp(j)1849 2927 y FD(;)263 b(j)28 b FA(\025)23 b FG(1)p FD(;)801 b FG(\(I.3.3\))386 3044 y(with)32 b(essen)n(tially)e (self-adjoin)n(t)h(op)r(erators)f FD(S)1834 3056 y FB(j)1869 3044 y FD(:)i FG(F)-7 b(or)30 b(this)i(exp)r(onen)n(tial)f(form,)i(the) e(transfor-)386 3144 y(mation)c(sc)n(heme)g(\(I.2.6\))h(from)f(the)h (previous)e(section)i(can)f(b)r(e)h(used,)g(suc)n(h)f(that)598 3291 y FD(U)664 3257 y Fz(0)655 3312 y FC(1)692 3291 y FD(U)758 3257 y Fz(0)749 3312 y FC(0)800 3291 y FD(H)20 b(U)955 3231 y Fl(0)977 3256 y Fz(\000)p FC(1)946 3313 y(0)1066 3291 y FD(U)1132 3231 y Fl(0)1155 3256 y Fz(\000)p FC(1)1123 3313 y(1)1290 3291 y FG(=)46 b FD(\014)t(E)1513 3303 y FB(p)1593 3291 y FG(+)41 b FA(E)1743 3303 y FC(1)1822 3291 y FG(+)g FA(O)1994 3303 y FC(1)2073 3291 y FG(+)g FD(i)p FG([)p FD(\014)t(E)2343 3303 y FB(p)2381 3291 y FD(;)14 b(S)2469 3303 y FC(1)2506 3291 y FG(])42 b(+)f FD(i)p FG([)p FA(E)2773 3303 y FC(1)2828 3291 y FG(+)18 b FA(O)2977 3303 y FC(1)3015 3291 y FD(;)c(S)3103 3303 y FC(1)3140 3291 y FG(])1353 3468 y FA(\000)1428 3412 y FG(1)p 1428 3449 42 4 v 1428 3525 a(2)1502 3468 y([)g([)p FD(\014)t(E)1674 3480 y FB(p)1713 3468 y FD(;)g(S)1801 3480 y FC(1)1838 3468 y FG(])p FD(;)28 b(S)1963 3480 y FC(1)2000 3468 y FG(])42 b(+)f FD(R)q FG(\()p FD(\015)2315 3434 y FC(3)2352 3468 y FG(\))p FD(:)744 b FG(\(I.3.4\))386 3609 y FD(S)437 3621 y FC(1)502 3609 y FG(is)27 b(c)n(hosen)g(as)g(an)g (o)r(dd)h(op)r(erator)e(whic)n(h)h(eliminates)h FA(O)2263 3621 y FC(1)2328 3609 y FG(according)e(to)1253 3743 y FD(i)d FG([)p FD(\014)t(E)1440 3755 y FB(p)1479 3743 y FD(;)14 b(S)1567 3755 y FB(j)1602 3743 y FG(])46 b(=)g FA(\000O)1913 3755 y FB(j)1947 3743 y FD(;)346 b(j)28 b FG(=)23 b(1)643 b(\(I.3.5\))386 3878 y(hence)32 b FD(S)672 3890 y FC(1)741 3878 y FG(is)g(of)f(\014rst)h(order)f(in)h(the)g(p)r (oten)n(tial)g(lik)n(e)f FA(O)2150 3890 y FC(1)2188 3878 y FD(;)h FG(and)f(the)h(terms)g(whic)n(h)g(are)f(disre-)386 3978 y(garded)e(in)i(\(I.3.4\))f(are)f(in)i(fact)g(at)f(least)g(of)g (third)h(order)e(in)i(the)g(p)r(oten)n(tial.)45 b(The)30 b(transfor-)386 4077 y(mation)25 b(sc)n(heme)g(is)g(con)n(tin)n(ued)f (in)i(the)f(sense)g(that)h(after)e FD(k)k FG(transformations,)c(the)i (p)r(oten)n(tial)386 4177 y(term)31 b(whic)n(h)h(is)f(of)h FD(k)24 b FG(+)c(1st)31 b(order)f(in)i(the)g(p)r(oten)n(tial)g (strength,)g(is)f(decomp)r(osed)g(in)n(to)g(ev)n(en)386 4277 y(\()p FA(E)462 4289 y FB(k)q FC(+1)587 4277 y FG(\))25 b(and)f(o)r(dd)h(\()p FA(O)1061 4289 y FB(k)q FC(+1)1186 4277 y FG(\))g(con)n(tributions)f(\(corresp)r(onding)f(to)i(their)f(b)r (eha)n(viour)g(when)g(com-)386 4376 y(m)n(uted)32 b(with)f FD(\014)t FG(\),)j(and)d(the)g(successiv)n(e)f(transformation)g FD(U)2313 4346 y Fz(0)2304 4400 y FB(k)q FC(+1)2472 4376 y FG(=)f FD(e)2605 4346 y Fz(\000)p FB(iS)2721 4355 y Fp(k)q Ft(+1)2863 4376 y FG(eliminates)i(the)386 4477 y(o)r(dd)26 b(term)h(b)n(y)f(means)f(of)i(the)f(c)n(hoice)g(\(I.3.5\))g (for)g FD(j)i FG(=)22 b FD(k)d FG(+)c(1)p FD(:)49 b FG(After)27 b FD(n)15 b FG(+)h(1)26 b(transformations)386 4577 y(one)h(arriv)n(es)f (at)1351 4677 y FD(H)1427 4642 y Fz(0)1420 4697 y FB(n)1489 4677 y FG(:=)45 b FD(\014)t(E)1734 4689 y FB(p)1815 4677 y FG(+)c FA(E)1965 4689 y FC(1)2044 4677 y FG(+)18 b FD(:::)g FG(+)41 b FA(E)2364 4689 y FB(n)3151 4677 y FG(\(I.3.6\))386 4794 y(whic)n(h)28 b(only)g(consists)g(of)g(ev)n(en)g (terms)g(and)g(agrees)e(with)j(the)g(transformed)e(Dirac)h(op)r(erator) 386 4894 y(to)f(the)h(order)f(of)g FD(\015)990 4864 y FB(n)p FC(+1)1119 4894 y FD(:)486 4993 y FG(With)20 b FD(H)768 4963 y Fz(0)761 5014 y FB(n)826 4993 y FG(of)f(this)h(form,)h (one)e(can)h(easily)e(eliminate)i(the)g(lo)n(w)n(er)e(comp)r(onen)n(ts) h(of)h FD( )i FG(in)e(order)386 5093 y(to)i(obtain)g(e.g.)35 b(the)22 b(nonrelativistic)g(limit.)36 b(This)22 b(is)g(either)g(done)g (b)n(y)g(c)n(ho)r(osing)f FD( )26 b FG(:=)3117 5026 y Fu(\000)3155 5056 y FB(u)3158 5122 y FC(0)3194 5026 y Fu(\001)3232 5093 y FD(;)46 b FG(or)386 5193 y(equiv)-5 b(alen)n(tly)e(,)28 b(b)n(y)f(pro)5 b(jecting)27 b(the)h(blo)r(c)n (k-diagonalised)d(matrix-v)-5 b(alued)27 b(op)r(erator)f FD(H)3149 5163 y Fz(0)3142 5213 y FB(n)3216 5193 y FG(on)n(to)386 5299 y(the)h(upp)r(er)g(blo)r(c)n(k)f(\(b)n(y)h(means)g(of)f(forming) 1794 5262 y FC(1+)p FB(\014)p 1794 5280 125 4 v 1839 5328 a FC(2)1942 5299 y FD(H)2018 5269 y Fz(0)2011 5319 y FB(n)2080 5262 y FC(1+)p FB(\014)p 2080 5280 V 2126 5328 a FC(2)2215 5299 y FG(\))p FD(:)50 b FG(With)28 b(this)f(pro)r(cedure,)f(one)g(has)386 5399 y(a)h(reduction)g(from)h (the)g(4-dimensional)e(space)h(of)g FD( )k FG(to)c(the)h(2-dimensional) f(space)g(of)g FD(u)p FG(.)486 5498 y(Let)c FD( )j FG(:=)821 5431 y Fu(\000)859 5461 y FB(u)862 5527 y FC(0)898 5431 y Fu(\001)983 5498 y FG(with)e FD(u)e FA(2)i FD(H)1386 5513 y FC(1)p FB(=)p FC(2)1490 5498 y FG(\()p Fx(R)1576 5468 y FC(3)1620 5498 y FG(\))10 b FA(\002)g Fx(C)1791 5468 y FC(2)1881 5498 y FG(an)23 b(arbitrary)e(function.)36 b(Then)24 b(one)f(can)g(form)386 5623 y(the)29 b(exp)r(ectation)f(v)-5 b(alue)28 b(of)h FD(H)1365 5593 y Fz(0)1358 5643 y FB(n)1431 5623 y FG(and)g(in)f(this)h(w)n(a)n(y)e(de\014ne)i(an)f(op)r(erator)f FD(b)2752 5580 y FC(\()p FB(n)p FC(\))2752 5632 y FB(m)2877 5623 y FG(whic)n(h)h(acts)g(on)p eop %%Page: 26 32 26 31 bop 386 259 a FC(26)386 459 y FD(H)455 474 y FC(1)p FB(=)p FC(2)559 459 y FG(\()p Fx(R)646 429 y FC(3)689 459 y FG(\))18 b FA(\002)g Fx(C)876 429 y FC(2)970 459 y FG(\(instead)27 b(of)h FD(H)1452 474 y FC(1)p FB(=)p FC(2)1556 459 y FG(\()p Fx(R)1642 429 y FC(3)1686 459 y FG(\))18 b FA(\002)g Fx(C)1873 429 y FC(4)1916 459 y FG(\))51 b(b)n(y)27 b(means)g(of)g(\(Douglas)g(and)g(Kroll)g(1974,) 386 559 y(EPS)g(1996\))1449 679 y(\()p FD( )s(;)14 b(H)1651 645 y Fz(0)1644 699 y FB(n)1713 679 y FD( )s FG(\))46 b(=)g(\()p FD(u;)14 b(b)2112 645 y FC(\()p FB(n)p FC(\))2112 699 y FB(m)2231 679 y FD(u)p FG(\))840 b(\(I.3.7\))1262 822 y(=)46 b(\()p FD(u;)14 b FG(\()p FD(b)1558 834 y FC(0)p FB(m)1686 822 y FG(+)32 b FD(b)1819 834 y FC(1)p FB(m)1947 822 y FG(+)18 b FD(:::)h FG(+)f FD(b)2237 834 y FB(nm)2340 822 y FG(\))24 b FD(u)p FG(\))p FD(:)386 952 y FG(The)j(index)g FD(m)g FG(refers)f(to)h(the)g(particle)g(mass)f (while)h(the)g(other)g(index)g(denotes)g(the)g(order)f(in)386 1052 y(the)i(p)r(oten)n(tial)f(strength.)386 1270 y Fo(b\))32 b(Equivalenc)-5 b(e)33 b(of)f(Sob)-5 b(olev)34 b(and)g(Douglas-Kr)-5 b(ol)5 b(l)33 b(tr)-5 b(ansformation)37 b(scheme)486 1417 y FG(If)24 b(the)g(Douglas-Kroll)d(unitary)j(transformations)d (are)i(c)n(hosen)g(of)h(exp)r(onen)n(tial)f(t)n(yp)r(e,)i(one)386 1516 y(has)f(a)g(term)n(wise)g(equiv)-5 b(alence)24 b(of)h(the)f (transformed)g(Dirac)g(op)r(erator)f(to)h(an)n(y)g(order)f FD(n)i FG(in)f(the)386 1616 y(p)r(oten)n(tial)30 b(strength.)46 b(F)-7 b(or)30 b(non-exp)r(onen)n(tial)f(unitary)h(transformations,)g (the)h(transformed)386 1715 y(op)r(erator)e(will)h(di\013er)h(from)f (the)g(Sob)r(olev)g(transformed)f(op)r(erator)g(for)h(su\016cien)n(tly) g(high)g FD(n)386 1815 y FG(\(e.g.)77 b FD(n)45 b(>)g FG(4)c(for)f(a)h(square-ro)r(ot)d(t)n(yp)r(e\).)78 b(Ho)n(w)n(ev)n(er,) 42 b(the)g(equiv)-5 b(alence)40 b(p)r(ersists)h(if)g(the)386 1915 y(same)30 b(t)n(yp)r(e)g(of)h(unitary)f(transformation)e(\(not)j (necessarily)e(exp)r(onen)n(tial\))h(is)g(used)h(in)f(b)r(oth)386 2014 y(transformation)c(sc)n(hemes.)486 2114 y(W)-7 b(e)23 b(consider)g(exp)r(ectation)g(v)-5 b(alues)23 b(of)h(the)f(Sob)r(olev)g (transformed)g(Dirac)g(op)r(erator)e(tak)n(en)386 2214 y(with)28 b(a)g(4-spinor)e FD(')j FG(in)f(the)g(p)r(ositiv)n(e)g(sp)r (ectral)g(subspace)f FA(H)2325 2226 y FC(+)p FB(;)p FC(1)2457 2214 y FG(:=)c(\003)2626 2226 y FC(+)2681 2214 y FG(\()p FD(H)2782 2229 y FC(1)p FB(=)p FC(2)2886 2214 y FG(\()p Fx(R)2973 2183 y FC(3)3016 2214 y FG(\))c FA(\002)f Fx(C)3204 2183 y FC(4)3247 2214 y FG(\))29 b(of)386 2313 y(the)e(free)g(Dirac)f (op)r(erator)f FD(D)1314 2325 y FC(0)1351 2313 y FD(:)i FG(Suc)n(h)g(a)f(spinor)g(can)g(in)i(momen)n(tum)f(space)f(b)r(e)h (expressed)f(in)386 2413 y(terms)h(of)h(P)n(auli)e(spinors)h FD(u)c FA(2)g FD(H)1428 2428 y FC(1)p FB(=)p FC(2)1532 2413 y FG(\()p Fx(R)1619 2383 y FC(3)1662 2413 y FG(\))c FA(\002)f Fx(C)1849 2383 y FC(2)1943 2413 y FG(\(Rose)28 b(1961,)d(EPS)i(1996\),)1088 2619 y(^)-55 b FD(')p FG(\()p Fv(p)p FG(\))38 b(=)1672 2563 y(1)p 1404 2600 577 4 v 1404 2617 a Fu(p)p 1487 2617 494 4 v 69 x FG(2)p FD(E)1590 2698 y FB(p)1642 2686 y FG(\()p FD(E)1735 2698 y FB(p)1793 2686 y FG(+)18 b FD(m)p FG(\))2014 2502 y Fu(\022)2075 2563 y FG(\()p FD(E)2168 2575 y FB(p)2225 2563 y FG(+)g FD(m)p FG(\))29 b(^)-47 b FD(u)o FG(\()p Fv(p)p FG(\))2188 2676 y Fv(p)p Fw(\033)31 b FG(^)-47 b FD(u)o FG(\()p Fv(p)p FG(\))2602 2502 y Fu(\023)2663 2619 y FD(;)465 b FG(\(I.3.8\))386 2827 y(where)34 b Fw(\033)j FG(is)d(the)h(v)n(ector) e(of)h(the)h(three)f(P)n(auli)f(matrices)h(giv)n(en)f(in)i(the)g(in)n (tro)r(duction.)56 b(W)-7 b(e)386 2926 y(ha)n(v)n(e)26 b(the)i(follo)n(wing)f(theorem.)386 3050 y Fv(Theorem)j(I.2.)386 3150 y Fn(L)l(et)37 b FD(')h FA(2)g FG(\003)780 3162 y FC(+)835 3150 y FG(\()p FA(S)6 b FG(\()p Fx(R)1010 3120 y FC(3)1053 3150 y FG(\))25 b FA(\002)f Fx(C)1253 3120 y FC(4)1296 3150 y FG(\))76 b Fn(b)l(e)37 b(a)i(4-spinor)f(in)g (the)g(p)l(ositive)h(sp)l(e)l(ctr)l(al)f(subsp)l(ac)l(e)h(of)f FD(D)3312 3162 y FC(0)3349 3150 y Fn(,)386 3249 y(which)e(de\014nes)e (a)g(Pauli)h(spinor)g FD(u)e Fn(ac)l(c)l(or)l(ding)j(to)e(\(I.3.8\).)53 b(L)l(et)33 b FD(H)2555 3219 y Fz(0)2548 3270 y FB(n)2628 3249 y Fn(b)l(e)h(the)g(Douglas-Kr)l(ol)t(l)386 3349 y(tr)l(ansforme)l(d)24 b(Dir)l(ac)g(op)l(er)l(ator)h(to)e FD(n)p Fn(-th)g(or)l(der)i(in)e(the)h(p)l(otential)g(str)l(ength,)h (using)e(exp)l(onential)386 3449 y(unitary)30 b(op)l(er)l(ators)h FD(U)1105 3418 y Fz(0)1096 3470 y FB(j)1131 3449 y Fn(.)40 b(L)l(et)30 b FD(H)1416 3418 y FC(\()p FB(n)p FC(\))1543 3449 y Fn(b)l(e)h(the)f(Sob)l(olev-tr)l(ansforme)l(d)i(op)l(er)l(ator)f (fr)l(om)g(The)l(or)l(em)386 3548 y(I.1.)39 b(Then)31 b(their)f(exp)l(e)l(ctation)g(values)g(agr)l(e)l(e)h(to)e(any)h(or)l (der)h FD(n)p Fn(,)493 3766 y FG(\()p FD(';)14 b(H)692 3732 y FC(\()p FB(n)p FC(\))813 3766 y FD(')p FG(\))46 b(=)1056 3624 y Fu( )1122 3766 y FD(';)1253 3662 y FB(n)1213 3687 y Fu(X)1213 3866 y FB(k)q FC(=0)1348 3766 y FD(H)1417 3778 y FB(k)1471 3766 y FD(')1525 3624 y Fu(!)1628 3766 y FG(=)1739 3649 y Fu(\022\022)1861 3710 y FD(u)1864 3823 y FG(0)1909 3649 y Fu(\023)1970 3766 y FD(;)14 b(H)2083 3732 y Fz(0)2076 3786 y FB(n)2121 3649 y Fu(\022)2182 3710 y FD(u)2185 3823 y FG(0)2230 3649 y Fu(\023\023)2620 3766 y FD(n)23 b FG(=)g(1)p FD(;)14 b FG(2)p FD(;)g(:::)g(:)106 b FG(\(I.3.9\))486 3988 y(The)27 b(pro)r(of)g(is)h(p)r(erformed)f(with) h(the)g(help)g(of)f(a)h(lemma.)386 4112 y Fv(Lemma)h(I.7)j FG(\(Relation)27 b(b)r(et)n(w)n(een)h(transformed)e(p)r(oten)n(tials\)) p Fv(.)386 4211 y Fn(L)l(et)37 b FD(H)57 b FG(=)37 b FD(D)834 4223 y FC(0)895 4211 y FG(+)24 b FD(V)93 b Fn(and)38 b FD(U)1351 4223 y FB(j)1436 4211 y FG(=)f FD(e)1577 4181 y FB(iB)1650 4189 y Fp(j)1685 4211 y FD(;)88 b(j)42 b FG(=)37 b(1)p FD(;)14 b(:::;)g(n)73 b Fn(b)l(e)38 b(the)f(Sob)l(olev) i(tr)l(ansformation)386 4311 y(scheme,)h(wher)l(e)e(the)f(p)l(otential) g(term)g(of)h FD(k)s Fn(-th)e(or)l(der)i(in)f(the)g(p)l(otential)g(str) l(ength)g FD(\015)k Fn(is)c(de-)386 4411 y(c)l(omp)l(ose)l(d)f(into)e FD(V)980 4423 y FB(k)1043 4411 y FG(+)22 b FD(W)1208 4423 y FB(k)1249 4411 y FD(:)66 b Fn(L)l(et)35 b FD(U)1553 4380 y Fz(0)1544 4431 y FC(0)1580 4411 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y FG(~)2183 1876 y FD(D)2252 1888 y FC(0)2289 1876 y FG(\()p Fv(p)2374 1842 y Fz(00)2416 1876 y FG(\)\)\()2673 1820 y(1)p 2522 1857 342 4 v 2522 1933 a FD(E)2583 1945 y FB(p)2617 1929 y Fl(00)2682 1933 y FG(+)18 b FD(E)2826 1945 y FB(p)2884 1876 y FG(+)3130 1820 y(1)p 2968 1857 365 4 v 2968 1933 a FD(E)3029 1945 y FB(p)3063 1929 y Fl(00)3127 1933 y FG(+)g FD(E)3271 1945 y FB(p)3305 1929 y Fl(0)3342 1876 y FG(\))3110 2027 y(\(I.3.27\))386 2133 y(where)645 2112 y(~)626 2133 y FD(D)695 2145 y FC(0)732 2133 y FG(\()p Fv(p)p FG(\))28 b(^)-56 b FD(')q FG(\()p Fv(p)p FG(\))37 b(=)f(^)-55 b FD(')p FG(\()p Fv(p)p FG(\))52 b(has)27 b(b)r(een)h(used)f(whenev)n(er)2300 2112 y(~)2281 2133 y FD(D)2350 2145 y FC(0)2414 2133 y FG(is)h(acting)f(on)g FD(':)486 2232 y FG(F)-7 b(rom)24 b(this)g(k)n(ernel,)g(one)h(can)f (deriv)n(e)f(the)i(co)r(ordinate-space)d(represen)n(tation)h(of)h FD(B)3125 2244 y FC(2)p FB(m)3221 2232 y FG(.)36 b(T)-7 b(o)386 2332 y(this)28 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2316 732 a FB(x)2377 703 y FD(')p FG(\))38 b FA(\025)22 b FG(0)p FD(:)386 926 y Fo(d\))33 b(Choic)-5 b(e)33 b(of)g(tr)-5 b(ansformation)37 b(scheme)486 1073 y FG(F)-7 b(rom)34 b(a)g(comparison)f(of)h(the)h (Jansen-Hess)e(op)r(erator)g(in)h(its)h(represen)n(tations)e FD(b)3144 1085 y FB(m)3241 1073 y FG(and)386 1184 y FD(B)453 1141 y FC(\(2\))449 1194 y FB(m)566 1184 y FG(in)25 b(co)r(ordinate)e (space,)h(\(I.3.20\))f(resp)r(ectiv)n(e)g(\(I.3.24\))h(with)g (\(I.3.30\),)g(it)h(is)f(ob)n(vious)f(that)386 1300 y FD(B)453 1257 y FC(\(2\))449 1310 y FB(m)542 1300 y FG(,)i(obtained)g (with)g(the)g(Sob)r(olev)f(transformation)f(sc)n(heme,)h(has)g(a)g(m)n (uc)n(h)h(simpler)f(shap)r(e.)486 1400 y(Although)i(w)n(e)f(did)h(not)g (\014nd)h(it)f(p)r(ossible)g(to)f(sho)n(w)g(p)r(ositivit)n(y)h(of)g (the)g(second-order)e(term)386 1499 y FD(B)449 1511 y FC(2)p FB(m)585 1499 y FG(from)39 b(the)g(represen)n(tation)f (\(I.3.33\),)k(it)e(is)f(eviden)n(t)h(from)f(\(I.3.27\))f(that)i(its)g (k)n(ernel)386 1599 y(has)29 b(a)h(p)r(ositiv)n(e)f(real)g(part.)44 b(There)29 b(is)h(a)f(pro)r(of)h(b)n(y)f(Ian)n(tc)n(henk)n(o)g(\(see)g (IJA)h(2003\),)f(in)n(v)n(olving)386 1699 y(the)38 b(partial-w)n(a)n(v) n(e)c(represen)n(tation)i(of)i(the)g(op)r(erator)e FD(b)2211 1711 y FC(2)p FB(m)2344 1699 y FG(sho)n(wn)h(in)h(App)r(endix)g(B,)g (that)386 1798 y(the)33 b(resp)r(ectiv)n(e)f(\(partial-w)n(a)n(v)n(e\)) e(k)n(ernels)h(of)i(the)g(second-order)d(term)j(of)f(the)h(Jansen-Hess) 386 1898 y(op)r(erator)26 b(are)g(indeed)i(p)r(ositiv)n(e.)486 1998 y(The)21 b(breakthrough)e(pro)n(vided)i(b)n(y)g(the)h(Sob)r(olev)e (transformation)g(sc)n(heme)h(b)r(ecomes)g(clear)386 2097 y(when)27 b(one)f(pro)r(ceeds)g(from)g(the)h(one-particle)f(Dirac) g(op)r(erator)f(to)h(the)h(m)n(ulti-particle)f(Cou-)386 2197 y(lom)n(b-Dirac)g(op)r(erator)g(to)h(b)r(e)h(discussed)e(in)i (parts)f(I)r(I)g(and)h(I)r(I)r(I.)g(F)-7 b(or)26 b(example,)h(compare)f (the)386 2296 y(Douglas-Kroll)i(transformed)h(t)n(w)n(o-particle)g (second-order)g(op)r(erator)g(\(Douglas)g(and)i(Kroll)386 2396 y(1974,)25 b(eqs.)36 b(\(4.22\))26 b({)h(\(4.24\)\),)g(\014lling)f (nearly)g(a)h(whole)f(page,)h(with)g(the)g(resp)r(ectiv)n(e)f(Sob)r (olev)386 2496 y(transformed)g(op)r(erator)g(\(I)r(I.4.4\))i(whic)n(h)f (op)r(ens)h(the)g(w)n(a)n(y)e(to)i(mathematical)f(analysis.)486 2595 y(The)36 b(adv)-5 b(an)n(tage)35 b(of)h(the)h(Douglas-Kroll)c (transformed)j(op)r(erators)e(lies)i(in)h(their)f(ready)386 2695 y(access)e(to)i(n)n(umerical)f(calculations)f(in)i(quan)n(tum)g(c) n(hemistry)f(\(Hess)h(1986,)g(W)-7 b(olf,)38 b(Reiher)386 2795 y(and)21 b(Hess)h(2002,2004\).)31 b(If)22 b(atomic)f(binding)h (energies)e(and)i(energy)e(shifts)i(due)g(to)g(relativistic)386 2894 y(pro)r(cesses)37 b(are)g(of)h(in)n(terest,)i(v)-5 b(ariational)37 b(calculations)g(are)g(a)g(v)n(ery)g(e\016cien)n(t)i (to)r(ol.)68 b(Suc)n(h)386 2994 y(computations)25 b(are)g(m)n(uc)n(h)h (simpler)f(if)h(carried)f(out)h(in)g Fn(two)p FG(-dimensional)f(space,) g(p)r(ermitting)386 3093 y(an)h Fn(arbitr)l(ary)i FG(c)n(hoice)d(of)i (w)n(a)n(v)n(efunction)e FD(u)h FG(\(without)h(ha)n(ving)e(to)i(care)e (for)h(the)h(restriction)e(to)386 3193 y(the)j(p)r(ositiv)n(e)f(sp)r (ectral)g(subspace\).)386 3571 y(I.4.)41 b Fv(Boundedness)23 b(prop)s(erties)h(and)h(p)s(ositivit)m(y)e(of)i(the)f(Jansen-Hess)g(op) s(erator.)486 3770 y FG(F)-7 b(rom)26 b(Theorem)h(I.2,)g(one)g(has)f (the)i(relation)e(b)r(et)n(w)n(een)i(the)f(Douglas-Kroll)e(transformed) 386 3881 y(op)r(erator)h FD(b)757 3838 y FC(\()p FB(n)p FC(\))757 3891 y FB(m)881 3881 y FG(and)i(the)g(Sob)r(olev-transformed) d(op)r(erator)h FD(B)2361 3838 y FC(\()p FB(n)p FC(\))2357 3891 y FB(m)2458 3881 y FD(;)963 4007 y Fu(\022)1066 4082 y FD(b)1102 4039 y FC(\()p FB(n)p FC(\))1102 4092 y FB(m)1282 4082 y FG(0)1112 4182 y(0)128 b(0)1365 4007 y Fu(\023)1472 4124 y FG(=)1593 4068 y(1)p 1593 4105 42 4 v 1593 4181 a(2)1667 4124 y(\(1)18 b(+)g FD(\014)t FG(\))24 b FD(U)2015 4090 y Fz(0)2006 4145 y FC(0)2066 4124 y FD(B)2133 4090 y FC(\()p FB(n)p FC(\))2129 4145 y FB(m)2253 4124 y FD(U)2319 4063 y Fl(0)2342 4089 y Fz(\000)p FC(1)2310 4146 y(0)2464 4068 y FG(1)p 2464 4105 V 2464 4181 a(2)2538 4124 y(\(1)19 b(+)f FD(\014)t FG(\))354 b(\(I.4.1\))386 4372 y(whic)n(h)29 b(di\013er)f(only)h(b)n(y) f(the)h(b)r(ounded)g(op)r(erators)e FD(\014)33 b FG(and)c FD(U)2302 4342 y Fz(0)2293 4393 y FC(0)2330 4372 y FD(:)53 b FG(Therefore,)28 b(in)h(most)g(cases)e(it)386 4488 y(pla)n(ys)e(no)h(role)f(whether)h(the)h(transformed)e(op)r(erator)g (is)h(c)n(hosen)f(in)i(the)f(form)g FD(b)2941 4444 y FC(\()p FB(n)p FC(\))2941 4497 y FB(m)3064 4488 y FG(or)f FD(B)3231 4444 y FC(\()p FB(n)p FC(\))3227 4497 y FB(m)3329 4488 y FD(:)386 4587 y FG(In)j(the)g(follo)n(wing)e(analysis,)h(b)r (oth)h(represen)n(tations)d(will)j(b)r(e)g(used.)486 4687 y(The)g(b)r(oundedness)g(prop)r(erties)g(of)g(an)g(essen)n(tially) f(self-adjoin)n(t)h(op)r(erator)f FD(A)h FG(de\014ned)h(on)386 4786 y(a)k(dense)f(subset)i(of)f(a)f(Hilb)r(ert)i(space)e FA(H)i FG(are)e(crucial)g(for)h(its)g(extension)g(to)g(a)f(self-adjoin) n(t)386 4886 y(op)r(erator.)37 b(Let)29 b(the)g(op)r(erator)d FD(A)j FG(consist)f(of)g(the)h(kinetic)f(energy)g(op)r(erator)e FD(D)2929 4898 y FC(0)2995 4886 y FG(and)i(a)g(sum)386 4986 y(of)g(p)r(oten)n(tial)g(terms)f FD(V)1111 4998 y FC(1)1149 4986 y FD(;)14 b(:::;)g(V)1340 4998 y FB(k)1381 4986 y FD(:)52 b FG(Assume)28 b FD(V)1812 4998 y FC(1)1868 4986 y FG(+)18 b FD(:::)h FG(+)f FD(V)2170 4998 y FB(k)2263 4986 y FG(is)28 b FD(D)2416 4998 y FC(0)2453 4986 y FG(-form)f(b)r (ounded)i(with)f(form)386 5085 y(b)r(ound)k FD(c)g FG(less)f(than)h(1,) g(suc)n(h)f(that)h(the)g(form)g(domain)f(of)h FD(A)g FG(is)f(the)h(same)f(as)g(that)h(of)g FD(D)3285 5097 y FC(0)3322 5085 y FD(;)386 5185 y FG(implying)g(that)g FD(A)g FG(is)f(w)n(ell-de\014ned)h(in)g(the)g(form)f(sense.)49 b(Then)32 b FD(A)g FG(is)g(form-b)r(ounded)f(from)386 5285 y(b)r(elo)n(w,)46 b(\()p FD(';)14 b(A)g(')p FG(\))45 b(=)30 b(\()p FD(';)14 b FG(\()p FD(D)1312 5297 y FC(0)1371 5285 y FG(+)21 b FD(V)1505 5297 y FC(1)1564 5285 y FG(+)g FD(:::)g FG(+)g FD(V)1874 5297 y FB(k)1915 5285 y FG(\))14 b FD(')p FG(\))45 b FA(\025)30 b FG(\(1)21 b FA(\000)g FD(c)p FG(\))14 b(\()p FD(';)g(D)2642 5297 y FC(0)2679 5285 y FD(')p FG(\))45 b FD(>)29 b FG(0)p FD(;)62 b FG(suc)n(h)32 b(that)386 5384 y(there)f(exists)h(the)g(self-adjoin)n(t)f(F)-7 b(riedric)n(hs)31 b(extension)g(of)g FD(A)h FG(on)g FA(H)h FG(\(P)n(earson)c(1988,)i(p.104;)386 5484 y(EPS)c(1996\).)p eop %%Page: 32 38 32 37 bop 386 259 a FC(32)386 459 y Fo(a\))33 b(Col)5 b(le)-5 b(ction)34 b(of)f(liter)-5 b(atur)g(e)34 b(r)-5 b(esults)486 624 y FG(The)34 b(prop)r(erties)g(giv)n(en)f(b)r(elo)n(w)i (concern)e(the)i(Bro)n(wn-Ra)n(v)n(enhall)d(op)r(erator)h FD(B)3055 581 y FC(\(1\))3051 633 y FB(m)3144 624 y FD(;)48 b(b)3251 581 y FC(\(1\))3251 633 y FB(m)386 723 y FG(\(EPS)37 b(1996,)i(Tix)f(1997,)h(1998,)g(Burenk)n(o)n(v)d(and)i(Ev)-5 b(ans)37 b(1998\))g(as)g(w)n(ell)h(as)g(the)g(massless)386 826 y(Jansen-Hess)i(op)r(erator)f FD(b)46 b FG(:=)g FD(b)1466 796 y FC(\(2\))1555 826 y FG(;)87 b(a)41 b(few)h(results)f(are)f(also)h (kno)n(wn)f(for)h(the)h(massiv)n(e)386 937 y(Jansen-Hess)26 b(op)r(erator)g FD(b)1223 894 y FC(\(2\))1223 947 y FB(m)1339 937 y FG(\(Sto)r(c)n(kmey)n(er)h(2002,)f(BSS)h(2002\).)386 1063 y Fv(Lemma)i(I.8)j FG(\(Bro)n(wn-Ra)n(v)n(enhall)25 b(op)r(erator\))p Fv(.)386 1162 y Fn(L)l(et)k FD(u)23 b FA(2)g(S)6 b FG(\()p Fx(R)821 1132 y FC(3)864 1162 y FG(\))19 b FA(\002)f Fx(C)1052 1132 y FC(2)1148 1162 y Fn(and)30 b FD(\015)1352 1174 y FB(B)s(R)1483 1162 y FG(=)22 b(2)p FD(=)p FG(\()1696 1130 y FB(\031)p 1696 1144 41 4 v 1700 1191 a FC(2)1765 1162 y FG(+)1862 1130 y FC(2)p 1858 1144 V 1858 1191 a FB(\031)1909 1162 y FG(\))p FD(:)546 1312 y FG(\(i\))42 b Fn(L)l(et)28 b FD(b)853 1269 y FC(\(1\))853 1322 y FB(m)965 1312 y FG(=)22 b FD(b)1088 1324 y FC(0)p FB(m)1200 1312 y FG(+)16 b FD(b)1317 1324 y FC(1)p FB(m)1413 1312 y FD(:)52 b Fn(Then)29 b FD(b)1739 1324 y FC(1)p FB(m)1864 1312 y Fn(is)g FD(b)1988 1324 y FC(0)p FB(m)2084 1312 y Fn(-form)g(b)l(ounde)l(d)g(with)g(form)h (b)l(ound)f FD(<)22 b FG(1)675 1411 y Fn(for)30 b FD(\015)e(<)23 b(\015)1009 1423 y FB(B)s(R)1116 1411 y FD(:)522 1511 y FG(\(ii\))43 b Fn(L)l(et)29 b FD(b)854 1481 y FC(\(1\))966 1511 y FG(:=)22 b FD(b)1112 1523 y FC(0)1168 1511 y FG(+)c FD(b)1287 1523 y FC(1)1353 1511 y Fn(b)l(e)30 b(the)g(massless)g(Br)l (own-R)l(avenhal)t(l)i(op)l(er)l(ator.)39 b(Then)1241 1701 y FA(j)p FG(\()p FD(u;)28 b FG(\()p FD(b)1463 1713 y FC(1)p FB(m)1591 1701 y FA(\000)k FD(b)1724 1713 y FC(1)1761 1701 y FG(\))14 b FD(u)p FG(\))p FA(j)46 b(\024)2077 1645 y FG(3)p 2077 1682 42 4 v 2077 1758 a(2)2142 1701 y FD(m)14 b(\015)28 b FG(\()p FD(u;)14 b(u)p FG(\))p FD(:)631 b FG(\(I.4.2\))1104 1945 y FA(j)p FG(\()p FD(u;)27 b FG(\()p FD(b)1325 1910 y FC(\(1\))1325 1965 y FB(m)1447 1945 y FA(\000)k FD(b)1579 1910 y FC(\(1\))1668 1945 y FG(\))14 b FD(u)p FG(\))p FA(j)46 b(\024)g FD(m)23 b FG(\(1)18 b(+)2269 1889 y(3)p 2269 1926 V 2269 2002 a(2)2334 1945 y FD(\015)5 b FG(\))23 b(\()p FD(u;)14 b(u)p FG(\))p FD(:)499 2131 y FG(\(iii\))43 b Fn(F)-6 b(or)30 b FD(\015)d FA(\024)c FD(\015)1031 2143 y FB(B)s(R)1138 2131 y FD(;)60 b(b)1257 2088 y FC(\(1\))1257 2141 y FB(m)1376 2131 y Fn(is)30 b(p)l(ositive.)40 b(Explicitly,)1337 2289 y FG(\()p FD(u;)14 b(b)1490 2255 y FC(\(1\))1490 2310 y FB(m)1593 2289 y FD(u)p FG(\))46 b FA(\025)f FD(m)23 b FG(\(1)c FA(\000)f FD(\015)5 b FG(\))23 b(\()p FD(u;)14 b(u)p FG(\))p FD(:)727 b FG(\(I.4.3\))534 2457 y(iv\))42 b Fn(In)27 b(the)g(p)l(artial-wave)j(r)l(epr)l(esentation)e(of)g FD(b)2050 2414 y FC(\(1\))2050 2467 y FB(m)2139 2457 y FD(;)50 b Fn(the)28 b(gr)l(ound-state)f(c)l(on\014gur)l(ation)g(is) 675 2557 y FD(l)d FG(=)f(0)p FD(;)36 b(s)23 b FG(=)1073 2524 y FC(1)p 1073 2538 34 4 v 1073 2585 a(2)1116 2557 y FD(:)525 2680 y FG(\(v\))42 b Fn(If)30 b FD(\015)d(>)c(\015)963 2692 y FB(B)s(R)1071 2680 y FD(;)60 b(b)1190 2637 y FC(\(1\))1190 2690 y FB(m)1308 2680 y Fn(is)30 b(unb)l(ounde)l(d)g(fr)l(om)g(b)l (elow.)486 2806 y FG(The)24 b(\014rst)h(item)g(of)f(\(ii\))i(w)n(as)d (pro)n(v)n(ed)g(b)n(y)i(Tix)f(\(1997\),)g(but)i(no)e(explicit)h(b)r (ound)g(w)n(as)f(giv)n(en.)386 2905 y(This)40 b(w)n(as)f(pro)n(vided)g (later)g(b)n(y)h(Sto)r(c)n(kmey)n(er)e(\(2002\).)73 b(In)40 b(order)f(to)h(calculate)f(the)h(form)386 3005 y(b)r(ound)c(of)g(the)g (massiv)n(e)e(Bro)n(wn-Ra)n(v)n(enhall)f(op)r(erator)h(relativ)n(e)g (to)h(the)i(massless)d(one)h(w)n(e)386 3104 y(use)j(the)g(mean)g(v)-5 b(alue)37 b(theorem)h(of)f(di\013eren)n(tial)h(calculus)f(to)h(deduce)g (0)i FA(\024)53 b FD(b)2972 3116 y FC(0)p FB(m)3093 3104 y FA(\000)25 b FD(b)3219 3116 y FC(0)3310 3104 y FG(=)386 3132 y Fu(p)p 469 3132 291 4 v 73 x FD(p)511 3181 y FC(2)566 3205 y FG(+)18 b FD(m)722 3181 y FC(2)771 3205 y FA(\000)11 b FD(p)36 b FG(=)22 b FD(m)14 b FG(\()1141 3171 y FB(db)1205 3179 y Ft(0)p Fp(m)p 1141 3186 148 4 v 1168 3233 a FB(dm)1298 3205 y FG(\))1330 3217 y FB(m)p FC(=)p FB(\026)1522 3205 y FG(=)23 b FD(m)11 b FA(\001)1847 3168 y FB(\026)p 1738 3186 260 4 v 1738 3196 a FA(p)p 1807 3196 191 4 v 57 x FB(p)1841 3237 y Ft(2)1873 3253 y FC(+)p FB(\026)1964 3237 y Ft(2)2067 3205 y FA(\024)23 b FD(m)47 b FG(for)23 b(an)n(y)g(n)n(um)n(b)r(er)h FD(\026)f FA(2)g FG(\(0)p FD(;)14 b(m)p FG(\))24 b(and)386 3354 y(all)j FD(p)c FA(\025)g FG(0)p FD(:)50 b FG(Adding)28 b(the)g(b)r(ound)1471 3321 y FC(3)p 1471 3335 34 4 v 1471 3382 a(2)1514 3354 y FD(m\015)k FG(of)c(the)g(\014rst-order)d(terms)i(completes)h(the)g (pro)r(of.)486 3453 y(P)n(ositivit)n(y)38 b(w)n(as)h(pro)n(v)n(ed)f(b)n (y)i(Tix)f(\(1998\),)j(and)e(items)g(\(iv\))g(and)g(\(v\))g(w)n(ere)f (sho)n(wn)g(b)n(y)386 3553 y(EPS\(1996\).)486 3653 y(In)g(order)f(for)g FD(D)1037 3665 y FC(0)1100 3653 y FG(+)26 b FD(V)58 b FG(to)39 b(b)r(e)g(a)g(w)n(ell-de\014ned)g(op)r(erator)e(in)j(the)f (form)g(sense,)i FD(V)58 b FG(has)386 3752 y(to)32 b(b)r(e)g FD(D)678 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y(massiv)n(e)26 b(case.)36 b(Instead,)28 b(one)f(has)p eop %%Page: 33 39 33 38 bop 3309 259 a FC(33)386 459 y Fv(Lemma)29 b(I.10)i FG(\(Bounds)d(for)f(the)h(massiv)n(e)e(Jansen-Hess)g(op)r(erator\))p Fv(.)386 574 y Fn(L)l(et)j FD(u)23 b FA(2)g(S)6 b FG(\()p Fx(R)821 544 y FC(3)864 574 y FG(\))19 b FA(\002)f Fx(C)1052 544 y FC(2)1095 574 y FD(;)53 b Fn(and)30 b(let)g FD(b)1485 586 y FB(m)1571 574 y FG(:=)22 b FD(b)1717 531 y FC(\(2\))1717 584 y FB(m)1843 574 y FG(=)h FD(b)1967 586 y FC(0)p FB(m)2081 574 y FG(+)18 b FD(b)2200 586 y FC(1)p FB(m)2314 574 y FG(+)g FD(b)2433 586 y FC(2)p FB(m)2529 574 y FD(:)546 693 y FG(\(i\))1246 797 y FA(j)p FG(\()p FD(u;)27 b FG(\()p FD(b)1467 809 y FC(2)p FB(m)1582 797 y FA(\000)18 b FD(b)1701 809 y FC(2)1738 797 y FG(\))c FD(u)p FG(\))p FA(j)46 b(\024)f FD(m)14 b(\015)2178 763 y FC(2)2215 797 y FD(d)2258 809 y FC(0)2319 797 y FG(\()p FD(u;)g(u)p FG(\))635 b(\(I.4.5\))675 923 y Fn(with)30 b FD(d)898 935 y FC(0)959 923 y FG(:=)22 b(8)c(+)g(12)1296 854 y 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b(the)h(estimate)g(of)g(the)h(energy)e(in)h(a)g(partial-w)n(a)n(v)n(e)d (state)i FD(a)2549 570 y FB(\027)2591 558 y FG(,)i(de\014ned)f(b)n(y)g (\(B.1\))g(and)386 658 y(\(B.3\),)767 782 y(\()p FD(a)843 794 y FB(\027)884 782 y FD(;)14 b(b)957 794 y FB(lsm)1087 782 y FD(a)1131 794 y FB(\027)1172 782 y FG(\))46 b FA(\025)g FG(\()p FD(a)1437 794 y FB(\027)1478 782 y FD(;)14 b(b)1551 794 y FC(0)p FB(m)1661 782 y FD(a)1705 794 y FB(\027)1746 782 y FG(\))42 b(+)f(\()p FD(a)2002 794 y FB(\027)2043 782 y FD(;)14 b(b)2116 739 y FC(\(1\))2116 807 y FB(lsm)2245 782 y FD(a)2289 794 y FB(\027)2331 782 y FG(\))41 b FA(\000)h(j)p FG(\()p FD(a)2610 794 y FB(\027)2651 782 y FD(;)14 b(b)2724 739 y FC(\(2\))2724 807 y FB(lsm)2853 782 y FD(a)2897 794 y FB(\027)2938 782 y FG(\))p FA(j)754 936 y(\025)45 b FG(\()p FD(a)940 948 y FB(\027)982 936 y FD(;)14 b(b)1055 948 y FC(0)p FB(m)1164 936 y FD(a)1208 948 y FB(\027)1250 936 y FG(\))41 b FA(\000)g FG(\()p FA(j)p FD(a)1528 948 y FB(\027)1570 936 y FA(j)p FD(;)28 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FD(dp)761 2455 y Fz(0)798 2372 y Fu(\024)842 2489 y FD(q)879 2501 y FB(l)905 2489 y FG(\()959 2433 y FD(p)p 947 2470 V 947 2546 a(p)989 2522 y Fz(0)1022 2489 y FG(\))42 b(+)18 b FD(h)1227 2455 y FC(2)1264 2489 y FG(\()p FD(p)p FG(\))23 b FD(q)1430 2501 y FB(l)p FC(+1)1540 2489 y FG(\()1594 2433 y FD(p)p 1582 2470 V 1582 2546 a(p)1624 2522 y Fz(0)1657 2489 y FG(\))1689 2372 y Fu(\025)1769 2433 y FD(p)p 1757 2470 V 1757 2546 a(p)1799 2522 y Fz(0)1878 2489 y FA(\024)1989 2376 y Fu(Z)2072 2396 y Fz(1)2035 2565 y FC(0)2156 2489 y FD(dp)2241 2455 y Fz(0)2278 2372 y Fu(\024)2322 2489 y FD(q)2359 2501 y FC(0)2396 2489 y FG(\()2450 2433 y FD(p)p 2438 2470 V 2438 2546 a(p)2480 2522 y Fz(0)2514 2489 y FG(\))41 b(+)19 b FD(h)2719 2455 y FC(2)2756 2489 y FG(\()p FD(p)p FG(\))k FD(q)2922 2501 y FC(1)2959 2489 y FG(\()3013 2433 y FD(p)p 3001 2470 V 3001 2546 a(p)3043 2522 y Fz(0)3077 2489 y FG(\))3109 2372 y Fu(\025)3188 2433 y FD(p)p 3176 2470 V 3176 2546 a(p)3218 2522 y Fz(0)1455 2751 y FG(=)46 b 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1642 34 4 v 545 1689 a(2)589 1661 y FG(,)k(pro)n(vided)g(b)n(y)g(Tix\(1998\),)f(is) i(used,)964 1851 y(\()p FD(a)1040 1863 y FB(\027)1081 1851 y FD(;)14 b FG(\()p FD(b)1186 1863 y FC(0)p FB(m)1300 1851 y FG(+)k FD(b)1419 1808 y FC(\(1\))1419 1876 y FB(lsm)1535 1851 y FG(\))c FD(a)1625 1863 y FB(\027)1666 1851 y FG(\))47 b FA(\025)1832 1738 y Fu(Z)1915 1758 y Fz(1)1878 1927 y FC(0)1999 1851 y FD(dp)23 b FA(j)p FD(a)2174 1863 y FB(\027)2216 1851 y FG(\()p FD(p)p FG(\))p FA(j)2345 1817 y FC(2)2405 1851 y FD(E)2466 1863 y FB(p)2528 1851 y FD(T)2577 1871 y FC(0)2620 1849 y Ft(1)p 2620 1858 29 3 v 2620 1891 a(2)2662 1851 y FG(\()p FD(x)p FG(\))p FD(;)314 b FG(\(I.4.52\))448 2117 y FD(T)497 2138 y FC(0)539 2115 y Ft(1)p 539 2124 V 539 2158 a(2)581 2117 y FG(\()p FD(x)p FG(\))25 b(:=)45 b(1)c FA(\000)1026 2061 y FD(\015)p 1026 2098 48 4 v 1029 2174 a FG(2)1098 1975 y Fu(\()1165 2117 y FG(\()1197 2036 y Fu(p)p 1280 2036 228 4 v 81 x FD(x)1327 2093 y FC(2)1383 2117 y FG(+)18 b(1)g(+)g(1\))1716 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a(2)2612 2521 y FG(\()p FD(x)p FG(\))g(and)f FD(G)2993 2541 y FC(1)p Fz(\000)3088 2519 y Ft(1)p 3088 2528 V 3088 2561 a(2)3130 2521 y FG(\()p FD(x)p FG(\))i(re-)386 2646 y(placed)e(b)n(y)h(new)f(functions)1339 2625 y(~)1320 2646 y FD(G)1385 2667 y FC(0)1428 2644 y Ft(1)p 1428 2653 V 1428 2687 a(2)1471 2646 y FG(\()p FD(x)p FG(\))i(and)1807 2625 y(~)1788 2646 y FD(G)1853 2667 y FC(1)p Fz(\000)1948 2644 y Ft(1)p 1948 2653 V 1948 2687 a(2)1991 2646 y FG(\()p FD(x)p FG(\))p FD(;)73 b FG(resp)r(ectiv)n(ely)-7 b(.)60 b(One)35 b(can)g(sho)n(w)g(n)n(u-)386 2771 y(merically)e(that)955 2750 y(~)936 2771 y FD(G)1001 2792 y FC(0)1045 2769 y Ft(1)p 1045 2778 V 1045 2812 a(2)1087 2771 y FG(\()p FD(x)p FG(\))i(is)f(monotonically)f(decreasing)g(in)h FD(x)p FG(,)i(attaining)e(its)g(in\014m)n(um)h(at)386 2871 y FD(x)23 b FA(!)h(1)p FD(;)811 3069 y FG(inf)776 3122 y FB(x)p Fz(2)p Fy(R)906 3130 y Ft(+)980 3048 y FG(~)961 3069 y FD(G)1026 3089 y FC(0)1069 3067 y Ft(1)p 1069 3076 V 1069 3109 a(2)1112 3069 y 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b(n)n(umerically)386 3449 y(without)j(an)n(y)f(further)h(appro)n (ximation,)f(and)h(the)g(factor)f FD(h)p FG(\()p FD(p)p FG(\))p FD(h)p FG(\()p FD(p)2550 3419 y Fz(0)2573 3449 y FG(\))i(is)e(k)n(ept)h(in)g(the)h(k)n(ernel)386 3548 y(when)39 b(applying)f(the)h(Lieb)g(and)f(Y)-7 b(au)39 b(form)n(ula.)69 b(Then)39 b(it)g(is)f(found)h(\(n)n(umerically\))g (that)405 3633 y(~)386 3654 y FD(G)451 3674 y FC(1)p Fz(\000)546 3652 y Ft(1)p 546 3661 29 3 v 546 3694 a(2)589 3654 y FG(\()p FD(x)p FG(\))c(is)f(monotonically)f(decreasing)g(with)i (its)g(in\014m)n(um)g(at)f FD(x)h FG(=)f FA(1)g FG(again)f(giv)n(en)h (b)n(y)386 3779 y(\(I.4.53\).)63 b(Moreo)n(v)n(er,)36 b(one)g(alw)n(a)n(ys)f(has)1749 3758 y(~)1730 3779 y FD(G)1795 3799 y FC(1)p Fz(\000)1890 3777 y Ft(1)p 1890 3786 V 1890 3819 a(2)1933 3779 y FG(\()p FD(x)p FG(\))53 b FD(>)2218 3758 y FG(~)2199 3779 y FD(G)2264 3799 y FC(0)2308 3777 y Ft(1)p 2308 3786 V 2308 3819 a(2)2350 3779 y FG(\()p FD(x)p FG(\))p FD(:)e FG(Th)n(us)2774 3758 y(~)2755 3779 y FD(G)2820 3799 y FC(1)p Fz(\000)2915 3777 y Ft(1)p 2915 3786 V 2915 3819 a(2)2957 3779 y FG(\()p FD(x)p FG(\))39 b FD(>)f FG(0)74 b(if)386 3883 y FD(\015)28 b(<)d FG(~)-45 b FD(\015)587 3895 y FB(c)621 3883 y FD(:)386 3983 y FG(Collecting)27 b(results,)g(w)n(e)g(ha)n(v)n(e)g FD(b)1415 3995 y FB(lsm)1553 3983 y FD(>)c FG(0)50 b(for)27 b FD(s)c FG(=)g FA(\006)2085 3950 y FC(1)p 2084 3964 34 4 v 2084 4011 a(2)2155 3983 y FG(and)k FD(\015)h(<)e FG(~)-45 b FD(\015)2518 3995 y FB(c)2552 3983 y FG(.)707 b Fm(\004)486 4130 y FG(The)24 b(presen)n(t)h(pro)r(of)f(of)h(p)r (ositivit)n(y)g(b)n(y)f(means)g(of)h(the)h(Lieb)e(and)h(Y)-7 b(au)25 b(form)n(ula)f(cannot)g(b)r(e)386 4229 y(extended)30 b(to)f(pro)n(vide)f(critical)h(p)r(oten)n(tial)h(strengths)e(b)r(ey)n (ond)33 b(~)-45 b FD(\015)2469 4241 y FB(c)2502 4229 y FG(.)43 b(This)30 b(is)f(lo)n(w)n(er)f(than)h(the)386 4329 y(Bro)n(wn-Ra)n(v)n(enhall)e(v)-5 b(alue,)30 b FD(\015)1322 4341 y FB(B)s(R)1455 4329 y FG(=)c(0)p FD(:)p 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4855 y FG(where)27 b FD(b)1861 4812 y FC(\(2\))1861 4880 y FB(lsm)1976 4855 y FG(\()p FD(p;)14 b(p)2129 4825 y Fz(0)2152 4855 y FG(\))28 b(is)g(dominating.)386 5125 y(I.5.)41 b Fv(Sp)s(ectral)32 b(prop)s(erties)f(of)h(the)f(Jansen-Hess)g(op)s(erator.)486 5324 y FG(The)i(sp)r(ectrum)h FD(\033)k FG(of)33 b(a)h(self-adjoin)n(t) f(op)r(erator)f(\(whic)n(h)i(in)g(our)f(case)g(is)h(the)g(F)-7 b(riedric)n(hs)386 5423 y(extension)23 b(on)g(the)g(Hilb)r(ert)h(space) f FD(L)1555 5435 y FC(2)1592 5423 y FG(\()p Fx(R)1678 5393 y FC(3)1721 5423 y FG(\))10 b FA(\002)g Fx(C)1892 5393 y FC(2)1935 5423 y FG(\))47 b(consists)22 b(of)h(the)h(essen)n (tial)e(sp)r(ectrum)i FD(\033)3276 5435 y FB(ess)386 5523 y FG(and)33 b(the)h(eigen)n(v)-5 b(alues)33 b(of)g(\014nite)i(m)n (ultiplicit)n(y)-7 b(.)55 b(The)34 b(essen)n(tial)e(sp)r(ectrum,)k(in)e (turn,)h(is)e(the)386 5623 y(union)e(of)h(the)f(absolute)g(con)n(tin)n (uous)f(sp)r(ectrum)i FD(\033)2030 5635 y FB(ac)2100 5623 y FG(,)h(the)e(singular)f(con)n(tin)n(uous)h(sp)r(ectrum)p eop %%Page: 41 47 41 46 bop 3309 259 a FC(41)386 459 y FD(\033)433 471 y FB(sc)499 459 y FG(,)34 b(the)g(eigen)n(v)-5 b(alues)32 b(of)h(in\014nite)h(m)n(ultiplicit)n(y)-7 b(,)35 b(and)e(the)h(limit)g (p)r(oin)n(ts)f(of)g FD(\033)2917 471 y FB(p)2956 459 y FG(,)i(where)d(the)386 558 y(p)r(oin)n(t)c(sp)r(ectrum)f FD(\033)1012 570 y FB(p)1079 558 y FG(is)g(the)h(set)g(of)g(eigen)n(v) -5 b(alues)26 b(\(Reed-Simon)i(1980,)d(p.231,236\).)486 658 y(This)36 b(section)g(concerns)f(the)i(essen)n(tial)f(sp)r(ectrum)g (of)h(the)f(Jansen-Hess)f(op)r(erator)g FD(b)3289 670 y FB(m)3352 658 y FG(,)386 758 y(b)r(oth)j(for)f FD(m)j FG(=)f(0)f(and)f FD(m)j FA(6)p FG(=)g(0)p FD(:)77 b FG(In)38 b(particular,)h(the)f(absence)f(of)g(singular)g(con)n(tin)n(uous)386 857 y(sp)r(ectrum)g(and)g(em)n(b)r(edded)h(eigen)n(v)-5 b(alues)36 b(will)h(b)r(e)g(sho)n(wn.)65 b(The)37 b(strategy)f(of)h (pro)r(of)g(is)g(in)386 957 y(man)n(y)h(cases)f(the)i(same)f(as)g (applied)h(for)f(the)h(corresp)r(onding)d(theorems)i(concerning)f(the) 386 1056 y(Bro)n(wn-Ra)n(v)n(enhall)28 b(op)r(erator)h(\(EPS)i(1996,)f (Balinsky)g(and)h(Ev)-5 b(ans)31 b(1998\).)45 b(In)32 b(these)f(cases,)386 1156 y(the)d(pro)r(ofs)f(will)g(only)h(b)r(e)g (outlined.)486 1256 y(Let)f(us)h(start)f(b)n(y)g(recalling)f(the)i(kno) n(wn)f(results)g(for)h(the)f(Bro)n(wn-Ra)n(v)n(enhall)e(op)r(erator.) 386 1376 y Fv(Lemma)k(I.12)i FG(\(Sp)r(ectrum)e(of)e(Bro)n(wn-Ra)n(v)n (enhall)e(op)r(erator\))p Fv(.)386 1492 y Fn(L)l(et)k FD(b)565 1449 y FC(\(1\))565 1502 y FB(m)677 1492 y FG(=)23 b FD(b)801 1504 y FC(0)p FB(m)915 1492 y FG(+)18 b FD(b)1034 1504 y FC(1)p FB(m)1130 1492 y FD(;)53 b Fn(and)30 b(assume)f FD(\015)f(<)23 b(\015)1859 1504 y FB(B)s(R)1989 1492 y FG(=)g(2)p FD(=)p FG(\()2203 1460 y FB(\031)p 2202 1474 41 4 v 2206 1521 a FC(2)2271 1492 y FG(+)2368 1460 y FC(2)p 2364 1474 V 2364 1521 a FB(\031)2415 1492 y FG(\))p FD(:)53 b Fn(Then)1510 1649 y FD(\033)1557 1661 y FB(ess)1656 1649 y FG(\()p FD(b)1724 1615 y FC(\(1\))1724 1670 y FB(m)1813 1649 y FG(\))46 b(=)g([)p FD(m;)14 b FA(1)p FG(\))1619 1806 y FD(\033)1666 1818 y FB(sc)1731 1806 y FG(\()p FD(b)1799 1772 y FC(\(1\))1799 1826 y FB(m)1888 1806 y FG(\))47 b(=)e FA(;)p FD(:)1139 1945 y Fn(If)30 b FD(m)23 b FG(=)g(0)29 b Fn(and)h FD(\015)e FA(\024)22 b FD(\015)1843 1957 y FB(B)s(R)1951 1945 y FD(;)183 b(\033)2204 1957 y FB(p)2243 1945 y FG(\()p FD(b)2311 1911 y FC(\(1\))2400 1945 y FG(\))38 b(=)22 b FA(;)p FD(;)386 2089 y Fn(i.e.)39 b FD(b)574 2046 y FC(\(1\))574 2098 y FB(m)689 2089 y Fn(has)28 b(no)f(singular)g(c)l (ontinuous)g(sp)l(e)l(ctrum)f(and)h FD(b)2221 2059 y FC(\(1\))2333 2089 y FG(=)c FD(b)2457 2101 y FC(0)2506 2089 y FG(+)12 b FD(b)2619 2101 y FC(1)2682 2089 y Fn(has)28 b(no)f(eigenvalues,)386 2188 y(such)j(that)f FD(\033)s FG(\()p FD(b)862 2158 y FC(\(1\))952 2188 y FG(\))h Fn(is)g(absolutely) h(c)l(ontinuous.)486 2309 y FG(The)g(absence)g(of)h(em)n(b)r(edded)g (eigen)n(v)-5 b(alues)31 b(in)h(the)g(essen)n(tial)f(sp)r(ectrum)h(for) f FD(m)f FA(6)p FG(=)g(0)h(can)386 2409 y(b)r(e)d(sho)n(wn)f(with)h (the)g(help)g(of)f(the)h(virial)f(theorem.)386 2529 y Fv(Lemma)i(I.13)i FG(\(Absence)d(of)g(em)n(b)r(edded)g(eigen)n(v)-5 b(alues)26 b(for)h(Bro)n(wn-Ra)n(v)n(enhall)e(op)r(erator\))p Fv(.)386 2645 y Fn(L)l(et)k FD(\015)f FA(\024)23 b FD(\015)731 2657 y FC(0)768 2645 y FD(:)53 b Fn(Then)30 b FD(b)1096 2602 y FC(\(1\))1096 2655 y FB(m)1214 2645 y Fn(has)h(no)f(eigenvalues) h(in)f FG([)p FD(m;)14 b FA(1)p FG(\))p FD(:)486 2766 y FG(This)29 b(lemma)g(w)n(as)f(pro)n(v)n(en)f(b)n(y)i(Balinsky)f(and)h (Ev)-5 b(ans)29 b(\(1998\))f(for)g FD(\015)2677 2778 y FC(0)2740 2766 y FG(=)2840 2733 y FC(3)p 2840 2747 34 4 v 2840 2795 a(4)2883 2766 y FD(:)55 b FG(By)29 b(impro)n(v-)386 2866 y(ing)k(their)g(estimates)g(w)n(e)g(ha)n(v)n(e)f(obtained)g FD(\015)1819 2878 y FC(0)1889 2866 y FG(=)g FD(\015)2029 2878 y FB(B)s(R)2136 2866 y FD(;)i FG(the)f(maxim)n(um)g(p)r(ossible)g (p)r(oten)n(tial)386 2977 y(strength)e(for)h(stabilit)n(y)g(of)g FD(b)1313 2934 y FC(\(1\))1313 2987 y FB(m)1401 2977 y FD(:)63 b FG(The)32 b(pro)r(of)f(of)h(this)g(new)g(result)g(will)g(b) r(e)h(giv)n(en)e(after)g(the)386 3076 y(pro)r(of)c(of)g(Theorem)g(I.4.) 486 3223 y(Let)21 b(us)g(no)n(w)g(turn)h(to)f(the)h(Jansen-Hess)d(op)r (erator)h(and)h(state)g(our)g(corresp)r(onding)e(results.)386 3344 y Fv(Theorem)30 b(I.3.)386 3444 y Fn(L)l(et)c FD(b)562 3456 y FB(m)648 3444 y FG(=)d FD(b)772 3456 y FC(0)p FB(m)880 3444 y FG(+)12 b FD(b)993 3456 y FC(1)p FB(m)1100 3444 y FG(+)g FD(b)1213 3456 y FC(2)p FB(m)1335 3444 y Fn(b)l(e)27 b(the)g(Jansen-Hess)f(op)l(er)l(ator)i(and)f(assume)g FD(\015)g(<)c(\015)3016 3456 y FB(J)3089 3444 y Fn(with)28 b(the)386 3543 y(critic)l(al)j(p)l(otential)f(str)l(ength)f FD(\015)1362 3555 y FB(J)1432 3543 y FG(=)22 b(1)p FD(:)p FG(006)28 b Fn(as)i(in)g(L)l(emma)g(I.9.)40 b(Then)546 3664 y FG(\(i\))i FD(\033)722 3676 y FB(ess)820 3664 y FG(\()p FD(b)888 3676 y FB(m)951 3664 y FG(\))47 b(=)e FD(\033)1187 3676 y FB(ess)1286 3664 y FG(\()p FD(b)1354 3676 y FC(0)p FB(m)1450 3664 y FG(\))i(=)e([)p FD(m;)14 b FA(1)p FG(\))522 3764 y(\(ii\))43 b FD(\033)722 3776 y FB(sc)787 3764 y FG(\()p FD(b)855 3776 y FB(m)918 3764 y FG(\))k(=)e FA(;)p FD(:)499 3863 y FG(\(iii\))e Fn(If)53 b FD(m)23 b FG(=)f(0)p FD(;)60 b(\033)1140 3875 y FB(p)1179 3863 y FG(\()p FD(b)p FG(\))23 b(=)g FA(;)p FD(;)52 b Fn(i.e.)40 b(the)30 b(sp)l(e)l(ctrum)f(of)h FD(b)g Fn(is)g(absolutely)h (c)l(ontinuous.)486 3984 y FG(F)-7 b(or)39 b(the)h(pro)r(of,)j(one)d (needs)g(the)g(b)r(eha)n(viour)f(of)h FD(b)2158 3996 y FB(m)2261 3984 y FG(under)g(complex)g(dilations.)73 b(F)-7 b(or)386 4084 y FD(\022)25 b FG(:=)e FD(e)600 4054 y FB(\030)659 4084 y FA(2)g Fx(R)791 4096 y FC(+)853 4084 y FD(;)48 b FG(the)25 b(unitary)f(group)g(of)h(dilation)f(op)r (erators)f FD(d)2388 4096 y FB(\022)2451 4084 y FG(is)h(in)n(tro)r (duced)h(b)n(y)f(means)h(of)1263 4232 y FD(d)1306 4244 y FB(\022)1363 4232 y FG(^)-48 b FD(u)p FG(\()p Fv(p)p FG(\))46 b(=)g FD(\022)1720 4197 y Fz(\000)p FC(3)p FB(=)p FC(2)1896 4232 y FG(^)-47 b FD(u)o FG(\()p Fv(p)p FD(=\022)r FG(\))47 b(=:)28 b(^)-48 b FD(u)2343 4244 y FB(\022)2380 4232 y FG(\()p Fv(p)p FG(\))654 b(\(I.5.1\))386 4373 y(with)34 b FD(u)f FA(2)g FD(L)807 4385 y FC(2)844 4373 y FG(\()p Fx(R)930 4343 y FC(3)974 4373 y FG(\))22 b FA(\002)g Fx(C)1169 4343 y FC(2)1213 4373 y FD(:)67 b FG(Then)33 b(for)h FA(j)p FD(\030)t FA(j)47 b FD(<)33 b(\030)1926 4385 y FC(0)2030 4373 y FG(with)h(a)f(suitably)h(c)n(hosen) f FD(\030)2931 4385 y FC(0)3001 4373 y FD(>)g FG(0)p FD(;)80 b(\022)36 b FG(is)386 4473 y(extended)30 b(to)g(the)h(complex)f (domain)f FD(D)1685 4485 y FB(\030)1715 4493 y Ft(0)1779 4473 y FG(:=)e FA(f)p FD(\022)i FG(=)e FD(e)2135 4443 y FB(\030)2199 4473 y FG(:)54 b FD(\030)32 b FA(2)c Fx(C)14 b FD(;)48 b FA(j)p FD(\030)t FA(j)27 b FD(<)g(\030)2792 4485 y FC(0)2830 4473 y FA(g)p FD(:)57 b FG(One)30 b(de\014nes)386 4573 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FG(\))p FD(;)263 b(b)1545 1823 y FB(k)q(;\022)1639 1811 y FG(\()p Fv(p)p FD(;)14 b Fv(p)1814 1776 y Fz(0)1837 1811 y FG(\))47 b(=)2036 1754 y(1)p 2036 1791 V 2036 1867 a FD(\022)2111 1811 y(b)2147 1823 y FB(k)2187 1811 y FG(\()p Fv(p)p FD(;)14 b Fv(p)2362 1776 y Fz(0)2386 1811 y FG(\))p FD(;)226 b(k)26 b FG(=)d(1)p FD(;)14 b FG(2)p FD(:)183 b FG(\(I.5.6\))386 2055 y Fn(Pr)l(o)l(of)56 b(outline)30 b(of)h(\(i\).)386 2155 y FG(F)-7 b(rom)24 b(Lemma)h(I.12)f(w)n(e)g(kno)n(w)g(that)h(the)h(essen)n(tial)d(sp)r (ectrum)i(of)g FD(b)2522 2167 y FC(0)p FB(m)2631 2155 y FG(+)13 b FD(b)2745 2167 y FC(1)p FB(m)2865 2155 y FG(coincides)24 b(with)386 2254 y(that)k(of)f FD(b)696 2266 y FC(0)p FB(m)792 2254 y FD(;)h FG(so)f(it)h(remains)f(to)g(sho)n (w)g(that)h(adding)f FD(b)2133 2266 y FC(2)p FB(m)2256 2254 y FG(leads)g(to)h(no)f(c)n(hanges.)486 2354 y(It)35 b(is)h(kno)n(wn)f(that)h(a)f(compact)g(op)r(erator)f(do)r(es)h(not)h(c) n(hange)e(the)i(essen)n(tial)f(sp)r(ectrum,)386 2453 y(ho)n(w)n(ev)n(er)26 b FD(b)741 2465 y FC(2)p FB(m)865 2453 y FG(is)i(not)g(b)r(ounded)g(from)g(ab)r(o)n(v)n(e)f(and)h(hence)g (is)g(not)g(compact.)37 b(Therefore,)27 b(the)386 2553 y(strategy)f(of)i(W)-7 b(eyl's)27 b(essen)n(tial)g(sp)r(ectral)g (theorem)g(\(Reed-Simon)h(1978,)e(p.122\))g(is)i(used:)386 2653 y(F)-7 b(rom)27 b(the)h(compactness)f(of)g(the)h(resolv)n(en)n(t)e (di\013erence)1187 2798 y FD(R)1250 2810 y FB(m)1313 2798 y FG(\()p FD(\026)p FG(\))e(:=)46 b(\()p FD(b)1653 2810 y FB(m)1734 2798 y FG(+)18 b FD(\026)p FG(\))1899 2764 y Fz(\000)p FC(1)2030 2798 y FA(\000)41 b FG(\()p FD(b)2204 2810 y FC(0)p FB(m)2319 2798 y FG(+)18 b FD(\026)p FG(\))2484 2764 y Fz(\000)p FC(1)3151 2798 y FG(\(I.5.7\))386 2944 y(with)33 b FD(\026)e FA(\025)g FG(1)h(a)h(constan)n(t)e(suc)n(h)i (that)f(the)h(resolv)n(en)n(ts)e(are)h(b)r(ounded)h(\(note)f(that)h FD(b)3087 2956 y FC(0)p FB(m)3214 2944 y FA(\025)e FG(0)p FD(;)386 3043 y FG(and)24 b FD(b)580 3055 y FB(m)668 3043 y FG(is)g(also)g(b)r(ounded)h(from)f(b)r(elo)n(w)h(for)f FD(\015)j FA(\024)c FD(\015)1999 3055 y FB(J)2070 3043 y FG(according)g(to)i(Lemma)f(I.10\),)h(it)g(follo)n(ws)386 3143 y(that)j(the)g(essen)n(tial)e(sp)r(ectra)h(of)h FD(b)1459 3155 y FB(m)1549 3143 y FG(and)g FD(b)1747 3155 y FC(0)p FB(m)1870 3143 y FG(coincide.)486 3243 y(With)g(the)g(help)g(of)f(the)h(second)f(resolv)n(en)n(t)f(iden)n(tit) n(y)-7 b(,)28 b FD(A)2262 3213 y Fz(\000)p FC(1)2388 3243 y FG(=)23 b FD(B)2543 3213 y Fz(\000)p FC(1)2651 3243 y FA(\000)18 b FD(B)2801 3213 y Fz(\000)p FC(1)2890 3243 y FG(\()p FD(A)h FA(\000)f FD(B)t FG(\))p FD(A)3247 3213 y Fz(\000)p FC(1)3337 3243 y FD(;)386 3342 y(R)449 3354 y FB(m)512 3342 y FG(\()p FD(\026)p FG(\))28 b(is)g(decomp)r (osed,)991 3489 y FD(R)1054 3501 y FB(m)1117 3489 y FG(\()p FD(\026)p FG(\))47 b(=)e FA(\000)p FG(\()p FD(b)1521 3501 y FC(0)p FB(m)1636 3489 y FG(+)18 b FD(\026)p FG(\))1801 3455 y Fz(\000)p FC(1)1913 3489 y FG(\()p FD(b)1981 3501 y FC(1)p FB(m)2096 3489 y FG(+)g FD(b)2215 3501 y FC(2)p FB(m)2310 3489 y FG(\))24 b(\()p FD(b)2434 3501 y FB(m)2515 3489 y FG(+)18 b FD(\026)p FG(\))2680 3455 y Fz(\000)p FC(1)3151 3489 y FG(\(I.5.8\))518 3671 y(=)46 b FA(\000)708 3578 y Fu(n)762 3671 y FG(\()p FD(b)830 3683 y FC(0)p FB(m)945 3671 y FG(+)18 b FD(\026)p FG(\))1110 3636 y Fz(\000)p FC(1)1222 3671 y FG(\()p FD(b)1290 3683 y FC(1)p FB(m)1405 3671 y FG(+)g FD(b)1524 3683 y FC(2)p FB(m)1620 3671 y FG(\))23 b(\()p FD(b)1743 3683 y FC(0)p FB(m)1858 3671 y FG(+)18 b FD(\026)p FG(\))2023 3636 y Fz(\000)p FC(1)p FB(=)p FC(2)2179 3578 y Fu(o)37 b(h)2311 3671 y FG(\()p FD(b)2379 3683 y FC(0)p FB(m)2493 3671 y FG(+)18 b FD(\026)p FG(\))2658 3636 y FC(1)p FB(=)p FC(2)2786 3671 y FG(\()p FD(b)2854 3683 y FB(m)2935 3671 y FG(+)g FD(\026)p FG(\))3100 3636 y Fz(\000)p FC(1)3204 3578 y Fu(i)386 3824 y FG(and)34 b(it)g(is)g(sho)n(wn)f(that)h FD(R)1240 3836 y FB(m)1303 3824 y FG(\()p FD(\026)p FG(\))h(is)f (compact)f(b)n(y)h(means)g(of)f(compactness)g(of)h(the)h(term)f(in)386 3923 y(curly)27 b(brac)n(k)n(ets)f(and)h(b)r(oundedness)h(of)f(the)h (second)f(factor.)486 4023 y(T)-7 b(o)33 b(sho)n(w)f(b)r(oundedness)h (of)g(the)h(term)f(in)h(square)e(brac)n(k)n(ets)g(w)n(e)h(split)g (o\013)h(the)f(b)r(ounded)386 4123 y(op)r(erator)24 b(\()p FD(b)787 4135 y FB(m)865 4123 y FG(+)15 b FD(\026)p FG(\))1027 4092 y Fz(\000)p FC(1)p FB(=)p FC(2)1183 4123 y FD(:)49 b FG(Boundedness)25 b(of)h(\()p FD(b)1912 4135 y FC(0)p FB(m)2023 4123 y FG(+)15 b FD(\026)p FG(\))2185 4092 y FC(1)p FB(=)p FC(2)2289 4123 y FG(\()p FD(b)2357 4135 y FB(m)2435 4123 y FG(+)g FD(\026)p FG(\))2597 4092 y Fz(\000)p FC(1)p FB(=)p FC(2)2803 4123 y FG(results)25 b(from)g(the)386 4222 y FD(b)422 4234 y FC(0)p FB(m)518 4222 y FG(-form)i(b)r(oundedness)g(of)h FD(b)1360 4234 y FB(m)1473 4222 y FG(\(Prop)r(osition)e(I.4\))i(expressed)e(as)883 4368 y(\()p FD( )s(;)i FG(\()p FD(b)1091 4380 y FC(1)p FB(m)1206 4368 y FG(+)18 b FD(b)1325 4380 y FC(2)p FB(m)1421 4368 y FG(\))c FD( )s FG(\))46 b FA(\025)g(\000)p FG(\(1)18 b FA(\000)g FD(\017)p FG(\))23 b(\()p FD( )s(;)14 b(b)2204 4380 y FC(0)p FB(m)2314 4368 y FD( )s FG(\))41 b FA(\000)h FD(C)29 b FG(\()p FD( )s(;)14 b( )s FG(\))p FD(:)274 b FG(\(I.5.9\))386 4517 y(F)-7 b(rom)25 b(this)h(w)n(e)f(obtain)g(with) h FD( )g FG(:=)d(\()p FD(b)1584 4529 y FB(m)1661 4517 y FG(+)14 b FD(\026)p FG(\))1822 4486 y Fz(\000)p FC(1)p FB(=)p FC(2)1992 4517 y FD(u)48 b FG(the)26 b(required)f(b)r (oundedness)g(condition)386 4669 y FA(k)p FG(\()p FD(b)496 4681 y FC(0)p FB(m)596 4669 y FG(+)5 b FD(\026)p FG(\))748 4635 y FC(1)p FB(=)p FC(2)866 4669 y FG(\()p FD(b)934 4681 y FB(m)1002 4669 y FG(+)g FD(\026)p FG(\))1154 4635 y Fz(\000)p FC(1)p FB(=)p FC(2)1323 4669 y FD(u)p FA(k)1413 4635 y FC(2)1496 4669 y FG(=)46 b(\()p FD( )s(;)28 b FG(\()p FD(b)1815 4681 y FC(0)p FB(m)1915 4669 y FG(+)5 b FD(\026)p FG(\))14 b FD( )s FG(\))46 b FA(\024)g FD(c)2363 4681 y FC(0)2423 4669 y FA(k)p FD(u)p 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2221 5157 a Fz(p)p 2275 5157 38 3 v 2275 5199 a FB(x)2372 5168 y FG(is)26 b(compact.)36 b(F)-7 b(or)25 b(the)i(second-)386 5283 y(order)h(con)n(tribution,)h FD(b)1139 5295 y FC(2)p FB(m)1235 5283 y FD(;)g FG(w)n(e)g(use)g(the)h (Sob)r(olev)e(represen)n(tation)g(\(I.3.24\))g(and)h(\(I.3.30\))g(of) 386 5383 y(op)r(erators)d(and)h(in)n(tro)r(duce)g(the)h(follo)n(wing)f (factorisation)665 5577 y(\()p FD(D)766 5589 y FC(0)836 5577 y FG(+)32 b FD(\026)p FG(\))1015 5543 y Fz(\000)p FC(1)1127 5577 y FD(B)1190 5589 y FC(2)p FB(m)1309 5577 y FG(\()p FD(D)1410 5589 y FC(0)1480 5577 y FG(+)g FD(\026)p FG(\))1659 5543 y Fz(\000)p FC(1)p FB(=)p FC(2)1861 5577 y FG(=)2004 5521 y FD(\015)2052 5491 y FC(2)p 1982 5558 130 4 v 1982 5634 a FG(8)p FD(\031)2074 5610 y FC(2)2158 5460 y Fu(\032)2220 5577 y FG(\()p FD(D)2321 5589 y FC(0)2377 5577 y FG(+)18 b FD(\026)p FG(\))2542 5543 y Fz(\000)p FC(1)2679 5521 y FG(1)p 2641 5558 117 4 v 2641 5575 a FA(p)p 2711 5575 48 4 v 2711 5634 a FD(x)2768 5460 y Fu(\033)3110 5577 y FG(\(I.5.11\))p eop %%Page: 43 49 43 48 bop 3309 259 a FC(43)667 496 y FA(\001)704 379 y Fu(\024)796 440 y FG(1)p 758 477 117 4 v 758 494 a FA(p)p 827 494 48 4 v 59 x FD(x)908 496 y FG(\(1)18 b FA(\000)1102 475 y FG(~)1083 496 y FD(D)1152 508 y FC(0)1189 496 y FG(\()p Fv(p)p FG(\)\)\()p FD(E)1431 508 y FB(p)1489 496 y FG(+)g FD(\026)p FG(\))1654 462 y Fz(\000)p FC(1)p FB(=)p FC(2)1811 379 y Fu(\025)1891 404 y(h)1931 496 y FG(\()p FD(E)2024 508 y FB(p)2081 496 y FG(+)g FD(\026)p FG(\))2246 462 y FC(1)p FB(=)p FC(2)2365 496 y FD(V)2413 508 y FC(10)p FB(;m)2576 496 y FG(\()p FD(D)2677 508 y FC(0)2733 496 y FG(+)g FD(\026)p FG(\))2898 462 y Fz(\000)p FC(1)p FB(=)p FC(2)3054 404 y Fu(i)402 725 y FG(+)499 669 y FD(\015)547 639 y FC(2)p 477 706 130 4 v 477 782 a FG(8)p FD(\031)569 758 y FC(2)616 633 y Fu(h)655 725 y FG(\()p FD(D)756 737 y FC(0)812 725 y FG(+)5 b FD(\026)p FG(\))964 691 y Fz(\000)p FC(1)1052 725 y FD(V)1100 737 y FC(10)p FB(;m)1250 725 y FG(\(1)18 b FA(\000)1431 704 y FG(~)1412 725 y FD(D)1481 737 y FC(0)1517 725 y FG(\()p Fv(p)p FG(\)\)\()p FD(E)1759 737 y FB(p)1818 725 y FG(+)5 b FD(\026)p FG(\))1970 633 y Fu(i)1994 608 y(\032)2057 725 y FG(\()p FD(E)2150 737 y FB(p)2207 725 y FG(+)g FD(\026)p FG(\))2359 691 y Fz(\000)p FC(1)2495 669 y FG(1)p 2458 706 117 4 v 2458 723 a FA(p)p 2527 723 48 4 v 60 x FD(x)2584 608 y Fu(\033)-13 b(\024)2724 669 y FG(1)p 2687 706 117 4 v 2687 723 a FA(p)p 2756 723 48 4 v 60 x FD(x)2827 725 y FG(\()p FD(D)2928 737 y FC(0)2984 725 y FG(+)5 b FD(\026)p FG(\))3136 691 y Fz(\000)p FC(1)p FB(=)p FC(2)3291 608 y Fu(\025)3335 725 y FD(:)386 888 y FG(Since)35 b(these)h(op)r(erators)d(act)i(on)g(the)h(p)r(ositiv)n(e) f(sp)r(ectral)g(subspace)f(of)h FD(D)2786 900 y FC(0)2823 888 y FD(;)h FG(the)g(lemma)f(of)386 987 y(Herbst)27 b(\(1977\))f(assures)g(that)i(the)g(op)r(erators)d(in)j(curly)f(brac)n (k)n(ets)e(in)j(\(I.5.11\))f(are)f(compact.)386 1087 y(The)32 b(op)r(erators)f(in)i(square)e(brac)n(k)n(ets)f(are)i(readily) f(sho)n(wn)h(to)h(b)r(e)f(b)r(ounded)h(b)n(y)f(using)h(that)405 1167 y(~)386 1188 y FD(D)455 1200 y FC(0)492 1188 y FG(\()p Fv(p)p FG(\))24 b(is)e(b)r(ounded)i(as)e(is)h FD(V)1270 1200 y FC(10)p FB(;m)1466 1188 y FG(\(see)g(\(I.4.22\)\),)g(and)g(b)n (y)g(applying)f(Kato's)g(\(1966\))f(inequal-)386 1294 y(it)n(y)-7 b(,)46 b(1)p FD(=x)28 b FA(\024)807 1261 y FB(\031)p 807 1275 41 4 v 811 1322 a FC(2)872 1294 y FD(p;)60 b FG(to)31 b(the)g(terms)g(in)n(v)n(olving)1881 1261 y FC(1)p 1852 1275 93 4 v 1852 1284 a Fz(p)p 1906 1284 38 3 v 1906 1326 a FB(x)1968 1294 y FG(\()p FD(D)2069 1306 y FC(0)2127 1294 y FG(+)20 b FD(\026)p FG(\))2294 1264 y Fz(\000)p FC(1)p FB(=)p FC(2)2451 1294 y FD(:)60 b FG(F)-7 b(or)30 b(example,)i(using)e(the)386 1409 y(Lieb)e(and)f(Y)-7 b(au)28 b(form)n(ula)e(\(Lemma)i(I.1\),)g(one)f(can)g(estimate)g(for)h FD(')23 b FA(2)g(H)2704 1421 y FC(+)p FB(;)p FC(1)1211 1555 y FA(k)p FG(\()p FD(E)1346 1567 y FB(p)1402 1555 y FG(+)18 b FD(\026)p FG(\))1567 1520 y FC(1)p FB(=)p FC(2)1695 1555 y FD(V)1743 1567 y FC(10)p FB(;m)1916 1555 y FG(\()p FD(D)2017 1567 y FC(0)2072 1555 y FG(+)g FD(\026)p FG(\))2237 1520 y Fz(\000)p FC(1)p FB(=)p FC(2)2417 1555 y FD(')p FA(k)2513 1520 y FC(2)3110 1555 y FG(\(I.5.12\))620 1760 y(=)731 1647 y Fu(Z)828 1760 y FD(d)p Fv(p)938 1639 y Fu(\014)938 1689 y(\014)938 1739 y(\014)938 1789 y(\014)966 1760 y FG(\()p FD(E)1059 1772 y FB(p)1116 1760 y FG(+)g FD(\026)p FG(\))1281 1725 y FC(1)p FB(=)p FC(2)1400 1647 y Fu(Z)1497 1760 y FD(d)p Fv(p)1593 1725 y Fz(0)1785 1704 y FG(1)p 1649 1741 315 4 v 1649 1817 a FA(j)p Fv(p)h FA(\000)f Fv(p)1880 1793 y Fz(0)1903 1817 y FA(j)1926 1793 y FC(2)2147 1704 y FG(1)p 2006 1741 324 4 v 2006 1817 a FD(E)2067 1829 y FB(p)2124 1817 y FG(+)g FD(E)2268 1829 y FB(p)2302 1812 y Fl(0)2363 1760 y FG(\()p FD(E)2456 1772 y FB(p)2490 1756 y Fl(0)2535 1760 y FG(+)g FD(\026)p FG(\))2700 1725 y Fz(\000)p FC(1)p FB(=)p FC(2)2893 1760 y FG(^)-55 b FD(')p FG(\()p Fv(p)3019 1725 y Fz(0)3043 1760 y FG(\))3075 1639 y Fu(\014)3075 1689 y(\014)3075 1739 y(\014)3075 1789 y(\014)3103 1660 y FC(2)1140 1985 y FA(\024)1251 1872 y Fu(Z)1348 1985 y FD(d)p Fv(p)1444 1951 y Fz(0)1490 1985 y FA(j)13 b FG(^)-55 b FD(')q FG(\()p Fv(p)1653 1951 y Fz(0)1676 1985 y FG(\))p FA(j)1731 1951 y FC(2)1787 1985 y FA(\001)1829 1872 y Fu(Z)1926 1985 y FD(d)p Fv(p)23 b FA(j)p FD(K)6 b FG(\()p Fv(p)2230 1951 y Fz(0)2253 1985 y FD(;)14 b Fv(p)p FG(\))p FA(j)2431 1929 y FD(f)9 b FG(\()p FD(p)2555 1899 y Fz(0)2578 1929 y FG(\))p 2431 1966 180 4 v 2443 2042 a FD(f)g FG(\()p FD(p)p FG(\))386 2156 y(with)28 b FD(K)6 b FG(\()p Fv(p)737 2126 y Fz(0)760 2156 y FD(;)14 b Fv(p)p FG(\))23 b(:=)1016 2089 y Fu(R)1085 2156 y FD(d)p Fv(q)14 b FD(k)s FG(\(\()p Fv(q)p FD(;)g Fv(p)1442 2126 y Fz(0)1467 2156 y FG(\))g FD(k)s FG(\()p Fv(q)p FD(;)g Fv(p)p FG(\))51 b(and)739 2335 y FD(k)s FG(\()p Fv(q)p FD(;)14 b Fv(p)p FG(\))24 b(:=)46 b(\()p FD(E)1240 2347 y FB(q)1295 2335 y FG(+)18 b FD(\026)p FG(\))1460 2301 y FC(1)p FB(=)p FC(2)1721 2279 y FG(1)p 1598 2316 289 4 v 1598 2392 a FA(j)p Fv(q)h FA(\000)f Fv(p)p FA(j)1849 2368 y FC(2)2058 2279 y FG(1)p 1929 2316 300 4 v 1929 2392 a FD(E)1990 2404 y FB(q)2046 2392 y FG(+)g FD(E)2190 2404 y FB(p)2261 2335 y FG(\()p FD(E)2354 2347 y FB(p)2412 2335 y FG(+)g FD(\026)p FG(\))2577 2301 y Fz(\000)p FC(1)p FB(=)p FC(2)2733 2335 y FD(:)354 b FG(\(I.5.13\))386 2533 y(Cho)r(osing)28 b FD(f)9 b FG(\()p FD(p)p FG(\))25 b(=)g FD(p)1063 2503 y FC(3)p FB(=)p FC(2)1196 2533 y FG(and)k(estimating)g(\()p FD(E)1860 2545 y FB(q)1916 2533 y FG(+)19 b FD(E)2061 2545 y FB(p)2100 2533 y FG(\))2132 2503 y Fz(\000)p FC(1)2250 2533 y FG(b)n(y)29 b(\()p FD(q)23 b FG(+)c FD(m)p FG(\))2648 2503 y Fz(\000)p FC(1)2766 2533 y FG(for)29 b FD(m)c FA(6)p FG(=)g(0)j(\(while)386 2632 y(substituting)34 b Fv(p)g FG(=:)e FD(q)s Fv(p)1154 2602 y Fz(00)1231 2632 y FG(for)h FD(m)g FG(=)g(0\))h(in)f(the)i(in)n (tegral)d(o)n(v)n(er)g Fv(p)p 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(erator-v)-5 b(alued)26 b(function)i(in)g FD(D)2007 3360 y FB(\030)2037 3368 y Ft(0)2098 3348 y FG(\()g(EPS)f(1996,)f(JA)i (2002\).)37 b(The)28 b(dilated)386 3448 y(functions)h FD(d)788 3460 y FB(\022)826 3448 y FD(u)f FG(are)g(also)g(analytic)g (in)h FD(D)1694 3460 y FB(\030)1724 3468 y Ft(0)1761 3448 y FG(,)g(pro)n(vided)f FD(u)d FA(2)g(S)6 b FG(\()p Fx(R)2452 3418 y FC(3)2495 3448 y FG(\))20 b FA(\002)f Fx(C)2685 3418 y FC(2)2782 3448 y FG(whic)n(h)29 b(is)f(a)h(dense)386 3548 y(subspace)e(of)g FD(H)896 3563 y FC(1)p FB(=)p FC(2)1001 3548 y FG(\()p Fx(R)1087 3517 y FC(3)1130 3548 y FG(\))19 b FA(\002)f Fx(C)1318 3517 y FC(2)1412 3548 y FG(\(F)-7 b(olland)28 b(1995,)d(p.192\).)486 3648 y(F)-7 b(or)40 b FD(\022)47 b FA(2)e Fx(R)33 b FA(\\)28 b FD(D)1073 3660 y FB(\030)1103 3668 y Ft(0)1180 3648 y FG(and)41 b FD(z)48 b FA(2)d Fx(C)15 b FA(n)p Fx(R)6 b FD(;)47 b FG(one)40 b(has)g(in)n(v)-5 b(ariance)40 b(of)g(the)i(\(\014nite\))f (resolv)n(en)n(t)386 3747 y(exp)r(ectation)27 b(v)-5 b(alue)28 b(under)f(dilations,)1183 3917 y(\()p FD(u;)1424 3860 y FG(1)p 1323 3898 243 4 v 1323 3974 a FD(b)1359 3986 y FB(m)1440 3974 y FA(\000)18 b FD(z)1589 3917 y(u)p FG(\))46 b(=)g(\()p FD(d)1901 3929 y FB(\022)1953 3917 y FD(u;)2188 3860 y FG(1)p 2061 3898 296 4 v 2061 3974 a FD(b)2097 3986 y FB(m;\022)2231 3974 y FA(\000)18 b FD(z)2380 3917 y(d)2423 3929 y FB(\022)2475 3917 y FD(u)p FG(\))p FD(:)532 b FG(\(I.5.14\))386 4099 y(Ho)n(w)n(ev)n(er,)23 b(b)r(ecause)h(of)g(the)g(iden)n(tit)n(y)h(theorem)f(from)f(complex)h (analysis,)g(analyticit)n(y)f(of)h FD(d)3289 4111 y FB(\022)3327 4099 y FD(u)386 4198 y FG(and)h(of)h FD(b)674 4210 y FB(m;\022)816 4198 y FG(guaran)n(tees)d(\(I.5.14\))i(for)g(all)h FD(\022)f FA(2)f FD(D)1969 4210 y FB(\030)1999 4218 y Ft(0)2035 4198 y FD(:)49 b FG(Moreo)n(v)n(er,)23 b(since)j FA(S)32 b FG(is)26 b(dense)f(in)h FD(H)3247 4213 y FC(1)p FB(=)p FC(2)3352 4198 y FD(;)386 4304 y FG(\(I.5.14\))h(holds)g(for)g (all)g FD(u)c FA(2)g FD(H)1356 4319 y FC(1)p 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4702 y FG(\()p FD(b)2150 4714 y FC(0)p FB(m;\022)2299 4702 y FG(\))37 b(=)23 b FA(f)p FD(b)2534 4714 y FC(0)p FB(m;\022)2682 4702 y FG(\()p FD(p)p FG(\))h(:)37 b FD(p)23 b FA(2)g FG([0)p FD(;)14 b FA(1)p FG(\))p FA(g)36 b FG(=)386 4812 y FA(f)428 4741 y Fu(p)p 510 4741 411 4 v 510 4812 a FD(p)552 4788 y FC(2)589 4812 y FD(=\022)672 4788 y FC(2)728 4812 y FG(+)18 b FD(m)884 4788 y FC(2)973 4812 y FG(:)51 b FD(p)38 b FA(2)g FG([0)p FD(;)14 b FA(1)p FG(\))p FA(g)p FD(:)73 b FG(Hence)37 b(for)f FD(m)h FG(=)g(0)f(it)h(is)f(the)h (nonnegativ)n(e)e(real)h(axis)386 4911 y(rotated)h(b)n(y)g(the)h(angle) e FA(\000)p FD(y)k FG(around)c(the)i(origin,)h(and)e(for)g FD(m)j FA(6)p FG(=)f(0)e(it)h(is)f(a)g(curv)n(e)g(in)g(the)386 5011 y(complex)27 b(plane)g(in)n(tersecting)g Fx(R)34 b FG(only)27 b(at)h(the)g(p)r(oin)n(t)f FD(m)p FG(.)486 5111 y(Therefore,)c(apart)f(from)h(isolated)f(p)r(oin)n(ts,)i(the)g (resolv)n(en)n(t)e(sets)h(of)g FD(b)2628 5123 y FC(0)p FB(m;\022)2800 5111 y FG(and)g FD(b)2993 5123 y FB(m;\022)3133 5111 y FG(\(whic)n(h)386 5210 y(are)34 b(the)i(complemen)n(tary)d(sets) i(of)g(the)h(sp)r(ectrum)f FD(\033)k FG(in)c Fx(C)15 b FG(\))77 b(agree)34 b(and)h(coincide)g(due)g(to)386 5310 y(\(I.5.14\))30 b(also)f(with)i(the)g(resolv)n(en)n(t)d(set)j(of)f FD(b)1806 5322 y FB(m)1869 5310 y FD(:)58 b FG(Since)30 b(there)h(exists)f(only)g(one)g(in)n(tersection)386 5410 y(p)r(oin)n(t)e(of)f FD(\033)744 5422 y FB(ess)843 5410 y FG(\()p FD(b)911 5422 y FB(m;\022)1027 5410 y FG(\))h(with)g Fx(R)1330 5422 y FC(+)1391 5410 y FG(,)g(w)n(e)f(ha)n(v)n(e)1382 5586 y(lim)1334 5640 y FC(Im)o FB(z)r Fz(!)p FC(0)1587 5586 y FG(Im)h(\()p FD(u;)1941 5530 y FG(1)p 1841 5567 243 4 v 1841 5643 a FD(b)1877 5655 y FB(m)1958 5643 y FA(\000)18 b FD(z)2107 5586 y(u)p FG(\))46 b FD(<)g FA(1)683 b FG(\(I.5.15\))p eop %%Page: 44 50 44 49 bop 386 259 a FC(44)386 459 y FG(except)29 b(at)h(isolated)f(p)r (oin)n(ts)g(of)h Fx(R)1461 471 y FC(+)1522 459 y FD(:)56 b FG(F)-7 b(rom)29 b(this)g(it)h(follo)n(ws)f(that)h(the)g(singular)e (con)n(tin)n(uous)386 558 y(sp)r(ectrum)g(is)f(absen)n(t)g (\(Reed-Simon)h(1978,)e FA(x)p FG(XI)r(I)r(I.6,)i(XI)r(I)r(I.10;)f(EPS) g(1996\).)445 b Fm(\004)386 776 y Fn(Pr)l(o)l(of)56 b(of)31 b(\(iii\).)386 876 y FG(W)-7 b(e)28 b(ha)n(v)n(e)e(to)i(sho)n(w)e(that) i(for)f FD(m)c FG(=)g(0)p FD(;)60 b(b)27 b FG(has)g(no)g(eigen)n(v)-5 b(alues.)486 975 y(First)32 b(assume)g FD(E)37 b FA(6)p FG(=)31 b(0)i(is)f(an)h(eigen)n(v)-5 b(alue)32 b(of)g FD(b)p FG(,)i(i.e.)53 b(there)32 b(exists)h FD(u)e FA(2)h FD(H)2903 990 y FC(1)p FB(=)p FC(2)3007 975 y FG(\()p Fx(R)3094 945 y FC(3)3137 975 y FG(\))22 b FA(\002)g Fx(C)3331 945 y FC(2)386 1075 y FG(suc)n(h)27 b(that)g FD(b)14 b(u)23 b FG(=)f FD(E)5 b(u:)50 b FG(Due)28 b(to)f(the)h (scaling)e(prop)r(ert)n(y)g(\(I.5.6\),)i(w)n(e)e(ha)n(v)n(e)h(for)f FD(\022)g FA(2)d FD(D)3102 1087 y FB(\030)3132 1095 y Ft(0)3186 1075 y FA(\\)c Fx(R)3314 1087 y FC(+)1307 1253 y FG(\()p FD(d)1382 1265 y FB(\022)1434 1253 y FD(b)14 b(d)1527 1218 y Fz(\000)p FC(1)1527 1278 y FB(\022)1616 1253 y FG(\))23 b FD(u)1719 1265 y FB(\022)1802 1253 y FG(=)1923 1197 y(1)p 1923 1234 42 4 v 1923 1310 a FD(\022)1997 1253 y(b)14 b(u)2095 1265 y FB(\022)2178 1253 y FG(=)46 b FD(E)19 b(u)2417 1265 y FB(\022)3110 1253 y FG(\(I.5.16\))386 1412 y(suc)n(h)24 b(that)h FD(\022)r(E)30 b FG(is)24 b(eigen)n(v)-5 b(alue)24 b(of)g FD(b:)48 b FG(Since,)25 b(ho)n(w)n(ev)n(er,)e FD(D)2196 1424 y FB(\030)2226 1432 y Ft(0)2275 1412 y FA(\\)12 b Fx(R)2397 1424 y FC(+)2482 1412 y FG(is)25 b(o)n(v)n(ercoun)n(table,)e(there)h(is)386 1512 y(an)k(o)n(v)n(ercoun)n(table)d(basis)j(of)g(eigen)n(v)n(ectors)e FD(u)1841 1524 y FB(\022)1906 1512 y FG(whic)n(h)i(con)n(tradicts)f (the)i(separabilit)n(y)d(of)i(the)386 1611 y(Hilb)r(ert)g(space)f FD(H)963 1626 y FC(1)p FB(=)p FC(2)1067 1611 y FG(\()p Fx(R)1154 1581 y FC(3)1197 1611 y FG(\))19 b FA(\002)f Fx(C)1385 1581 y FC(2)1428 1611 y FD(:)486 1711 y FG(Assume)38 b(no)n(w)g FD(E)46 b FG(=)40 b(0)e(is)g(an)g(eigen)n(v)-5 b(alue)38 b(of)g FD(b)p FG(,)j(i.e.)69 b(there)38 b(exists)g FD(u)i FA(6)p FG(=)h(0)d(suc)n(h)g(that)386 1811 y FD(b)14 b(u)22 b FG(=)h(0)p FD(:)48 b FG(Using)26 b(the)h(partial)e(w)n(a)n(v)n (e)f(decomp)r(osition)i(in)n(tro)r(duced)g(in)g(App)r(endix)h(B,)f(w)n (e)g(ha)n(v)n(e)386 1910 y(from)h(\(B.7\))h(in)g(Mellin)g(space)782 2093 y(0)45 b(=)h(\()p FD(u;)14 b(b)g(u)p FG(\))45 b(=)1383 2014 y Fu(X)1424 2188 y FB(\027)1517 1980 y Fu(Z)1600 2000 y Fz(1)1563 2169 y(\0001)1699 2093 y FD(dt)23 b FA(j)p FD(a)1862 2059 y FC(#)1862 2114 y FB(\027)1921 2093 y FG(\()p FD(t)18 b FG(+)g FD(i=)p FG(2\))p FA(j)2252 2059 y FC(2)2312 2093 y FD(b)2348 2053 y FC(#)2348 2118 y FB(ls)2406 2093 y FG(\()p FD(t)h FA(\000)f FD(i=)p FG(2\))395 b(\(I.5.17\))386 2328 y(where)31 b FD(b)666 2288 y FC(#)666 2353 y FB(ls)724 2328 y FG(\()p FD(t)22 b FA(\000)909 2295 y FB(i)p 904 2309 34 4 v 904 2356 a FC(2)947 2328 y FG(\))30 b(=)60 b(1)20 b(+)1282 2259 y FA(p)p 1351 2259 92 4 v 69 x 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FA(\000)f Fv(p)2286 5608 y Fz(0)2309 5632 y FA(j)2332 5608 y FC(2)2416 5458 y Fu(\022)2517 5519 y FG(1)p 2488 5556 100 4 v 2488 5632 a FD(E)2549 5644 y FB(p)2629 5575 y FG(+)2777 5519 y(1)p 2736 5556 123 4 v 2736 5632 a FD(E)2797 5644 y FB(p)2831 5628 y Fl(0)2868 5458 y Fu(\023)2988 5519 y FD(f)9 b FG(\()p FD(p)p FG(\))p 2976 5556 180 4 v 2976 5632 a FD(f)g FG(\()p FD(p)3100 5608 y Fz(0)3123 5632 y FG(\))p eop %%Page: 46 52 46 51 bop 386 259 a FC(46)1293 496 y FG(+)1455 440 y FD(\015)p 1414 477 130 4 v 1414 553 a FG(4)p FD(\031)1506 529 y FC(2)1581 383 y Fu(Z)1678 496 y FD(d)p Fv(p)1774 462 y Fz(0)1820 496 y FA(j)p FD(T)1892 508 y FC(2)1929 496 y FG(\()p Fv(p)p FD(;)14 b Fv(p)2104 462 y Fz(0)2128 496 y FG(\))p FA(j)2228 440 y FD(f)9 b FG(\()p FD(p)p FG(\))p 2216 477 180 4 v 2216 553 a FD(f)g FG(\()p FD(p)2340 529 y Fz(0)2363 553 y FG(\))2405 379 y Fu(\033)3110 496 y FG(\(I.5.27\))386 691 y(The)25 b(last)f(term)h(in)g(\(I.5.27\))f(can) g(b)r(e)h(further)g(estimated)g(b)n(y)f(breaking)g FD(T)2698 703 y FC(2)2734 691 y FG(\()p Fv(p)p FD(;)14 b Fv(p)2909 661 y Fz(0)2933 691 y FG(\))25 b(from)f(\(H.1\))386 791 y(in)n(to)36 b(its)g(constituen)n(ts)h(and)f(estimating)g(eac)n(h)f (con)n(tribution)h(separately)e(as)i(indicated)h(in)386 890 y(App)r(endix)g(H.)f(Recalling)g(that)g(the)h(con)n(v)n(ergence)c (generating)i(functions)h(can)g(b)r(e)h(c)n(hosen)386 990 y(di\013eren)n(tly)c(for)f(eac)n(h)g(in)n(tegral,)h(functions)g(of) f(the)i(t)n(yp)r(e)f FD(f)9 b FG(\()p FD(p)p FG(\))45 b(=)31 b FD(p)2584 960 y FC(3)p FB(=)p FC(2)2721 990 y FG(as)h(w)n(ell)h(as)f FD(f)9 b FG(\()p FD(p)p FG(\))45 b(=)386 1111 y FD(p)428 1081 y FC(3)p FB(=)p FC(2)607 1055 y FD(p)p 556 1092 146 4 v 556 1168 a(e)p FG(\()p FD(p)p FG(\))744 1111 y(with)34 b FD(e)p FG(\()p FD(p)p FG(\))e FA(2)g(f)p FD(E)1306 1123 y FB(p)1345 1111 y FD(;)14 b(E)1443 1123 y FB(p)1503 1111 y FG(+)22 b FD(m;)14 b(p)22 b FG(+)f FD(m)p FA(g)65 b FG(are)32 b(selected)h(in)h(order)d (to)i(optimise)g(the)386 1255 y(estimates.)j(F)-7 b(urther,)26 b(the)h(follo)n(wing)e(estimate)h(is)g(used)g(in)h(the)f(ev)-5 b(aluation)26 b(of)g(the)g(in)n(tegrals)386 1354 y(o)n(v)n(er)g FD(p)606 1324 y Fz(0)629 1354 y FG(,)912 1634 y(1)p 640 1671 585 4 v 640 1688 a Fu(p)p 723 1688 350 4 v 71 x FG(\()p FD(q)s(p)837 1735 y Fz(0)861 1759 y FG(\))893 1735 y FC(2)949 1759 y FG(+)18 b(1)32 b(+)18 b FD(c)1281 1690 y FA(\024)1391 1445 y Fu(8)1391 1520 y(>)1391 1545 y(>)1391 1570 y(>)1391 1595 y(<)1391 1744 y(>)1391 1769 y(>)1391 1794 y(>)1391 1819 y(:)1585 1495 y FG(1)p 1517 1532 179 4 v 1517 1608 a(1)g(+)g FD(c)1705 1551 y(;)84 b(p)1854 1521 y Fz(0)1900 1551 y FA(\024)22 b FG(1)p FD(=q)1585 1768 y FG(1)p 1553 1805 106 4 v 1553 1881 a FD(q)s(p)1635 1857 y Fz(0)1669 1824 y FD(;)120 b(p)1854 1794 y Fz(0)1900 1824 y FD(>)22 b FG(1)p FD(=q)2176 1690 y(;)180 b(c)23 b FA(\025)f FG(0)p FD(;)37 b(q)26 b FA(\025)c FG(0)h FD(:)245 b FG(\(I.5.28\))386 2035 y(De\014ning)28 b Fv(q)23 b FG(:=)g Fv(p)p FD(=m;)k FG(denoting)g(the)h(estimate)g(of) 2031 1969 y Fu(R)2100 2035 y FD(d)p Fv(p)2196 2005 y Fz(0)2233 2035 y FA(j)p FD(T)2305 2047 y FC(2)2342 2035 y FG(\()p Fv(p)p FD(;)14 b Fv(p)2517 2005 y Fz(0)2541 2035 y FG(\))p FA(j)g FD(f)9 b FG(\()p FD(p)p FG(\))p FD(=f)g FG(\()p FD(p)2932 2005 y Fz(0)2954 2035 y FG(\))51 b(b)n(y)386 2143 y(\(4)p FD(\031)510 2113 y FC(2)547 2143 y FG(\))579 2113 y FC(2)617 2143 y FD(q)657 2113 y FC(2)694 2143 y FD(M)775 2155 y FC(2)812 2143 y FG(\()p FD(q)s FG(\))p FD(;)68 b FG(and)34 b(taking)g FD(f)9 b FG(\()p FD(p)p FG(\))33 b(:=)h FD(p)1791 2113 y FC(3)p FB(=)p FC(2)1929 2143 y FG(in)g(the)g(term)g(linear)g(in)g FD(\015)5 b FG(,)35 b(suc)n(h)f(that)g(with)386 2242 y(form)n(ula)27 b(\(A.1\),)h(the)g(substitution)g FD(q)1578 2212 y Fz(0)1624 2242 y FG(:=)23 b FD(p)1777 2212 y Fz(0)1800 2242 y FD(=mq)30 b FG(for)d FD(p)2151 2212 y Fz(0)2202 2242 y FG(and)g(\(I.5.28\),)1133 2365 y Fu(Z)1230 2478 y FD(d)p Fv(p)1326 2443 y Fz(0)1509 2422 y FG(1)p 1373 2459 315 4 v 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3480 y(\))42 b FA(\000)f FD(d\015)2818 3445 y FC(2)2855 3480 y FD(:)199 b FG(\(I)r(I.5.69\))386 3657 y(P)n(ositivit)n(y)26 b(is)i(obtained)f(for)g FD(c)1349 3669 y FC(0)1409 3657 y FD(>)c FG(0,)k(i.e.)37 b FD(\015)28 b(<)22 b FG(0)p FD(:)p FG(825)p FD(:)1178 b Fm(\004)486 3804 y FG(W)-7 b(e)36 b(note)g(that)h(due)f(to)g(the)h(smallness)e(of)i FD(e)1963 3774 y FC(2)2036 3804 y FG(\(with)g(resp)r(ect)f(to)g FD(\015)5 b FG(\))36 b(for)g(high)g(n)n(uclear)386 3904 y(c)n(harges,)25 b(the)i(critical)f(p)r(oten)n(tial)h(strength)g(for)f (p)r(ositivit)n(y)h(of)f(the)i(t)n(w)n(o-particle)d(op)r(erator)g(is) 386 4003 y(close)i(to)g(those)g(for)g(p)r(ositivit)n(y)h(of)f(the)h (Jansen-Hess)f(op)r(erator.)486 4103 y(Collecting)e(results,)i(the)g FA(j)p FD(T)12 b FA(j)p FG(-form)24 b(b)r(oundedness)j(of)f(the)h (total)f(p)r(oten)n(tial)h FD(W)38 b FG(with)27 b(form)386 4203 y(b)r(ound)g FD(<)c FG(1)j(ensures)f(that)i FD(T)h FG(+)16 b FD(W)38 b FG(is)27 b(w)n(ell)f(de\014ned)h(as)f(a)g(form)g 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y(the)j(essen)n(tial)f(sp)r(ectrum)h(of)f(the)i (transformed)d(Coulom)n(b-Dirac)g(op)r(erator)g FD(H)2938 5091 y FC(\(2\))3058 5121 y FG(and)i(that)386 5221 y(the)25 b(second-order)e(t)n(w)n(o-particle)g(in)n(teraction)h(do)r(es)h(not)g (c)n(hange)e(the)j(essen)n(tial)e(sp)r(ectrum)h(of)386 5321 y FD(H)462 5291 y FC(\(2\))551 5321 y FG(.)53 b(Moreo)n(v)n(er)30 b(w)n(e)i(will)h(pro)n(v)n(e)e(that)j(in)f(the)g(massless)f(case,)h (there)g(are)e(no)i(eigen)n(v)-5 b(alues)386 5420 y(em)n(b)r(edded)28 b(in)g(the)g(essen)n(tial)f(sp)r(ectrum.)37 b(Let)27 b(us)h(start)f(with)h(some)f(kno)n(wn)g(results.)486 5520 y(W)-7 b(e)34 b(recall)f(that)i(w)n(e)e(are)g(only)h(in)n (terested)g(in)g(exp)r(ectation)g(v)-5 b(alues)34 b(\(and)g(the)g (resulting)386 5623 y(sp)r(ectrum\))h(of)g FD(H)966 5593 y FC(\(2\))1089 5623 y FG(tak)n(en)f(with)i(states)e FD( )k FG(in)d(the)g(p)r(ositiv)n(e)f(sp)r(ectral)g(subspace)g FA(H)3165 5635 y FC(+)p FB(;)p FC(2)3308 5623 y FG(of)p eop %%Page: 68 74 68 73 bop 386 259 a FC(68)386 462 y FG(the)38 b(free)g(t)n(w)n (o-particle)e(op)r(erator)g FD(T)51 b FG(=)39 b FD(D)1796 419 y FC(\(1\))1794 485 y(0)1911 462 y FG(+)25 b FD(D)2072 419 y FC(\(2\))2070 485 y(0)2161 462 y FD(:)78 b FG(With)38 b(this)h(restriction,)g(one)e(has)386 562 y FD(\033)s FG(\()p FD(T)12 b FG(\))23 b(=)g FD(\033)719 574 y FB(ess)817 562 y FG(\()p FD(T)12 b FG(\))23 b(=)g([2)p FD(m;)14 b FA(1)p FG(\))p FD(:)50 b FG(This)28 b(is)f(easily)g(seen)g(from)407 739 y(\()456 717 y Fu(c)439 739 y FD(T)12 b( )r FG(\)\()p Fv(p)673 751 y FC(1)711 739 y FD(;)i Fv(p)801 751 y FC(2)838 739 y FG(\))47 b(=)f(\()p FD(E)1121 751 y FB(p)1155 759 y Ft(1)1224 739 y FG(+)32 b FD(E)1382 751 y FB(p)1416 759 y Ft(2)1453 739 y FG(\))1525 717 y(^)1508 739 y FD( )s FG(\()p Fv(p)1650 751 y FC(1)1688 739 y FD(;)14 b Fv(p)1778 751 y FC(2)1815 739 y FG(\))47 b(=)e(\()2036 638 y Fu(q)p 2120 638 291 4 v 2120 739 a FD(p)2162 711 y FC(2)2162 761 y(1)2217 739 y FG(+)18 b FD(m)2373 715 y FC(2)2443 739 y FG(+)2539 638 y Fu(q)p 2622 638 V 101 x FD(p)2664 711 y FC(2)2664 761 y(2)2720 739 y FG(+)g FD(m)2876 715 y FC(2)2936 739 y FG(\))3008 717 y(^)2991 739 y FD( )t FG(\()p Fv(p)3134 751 y FC(1)3171 739 y FD(;)c Fv(p)3261 751 y FC(2)3298 739 y FG(\))p FD(:)3119 855 y FG(\(I)r(I.6.1\))486 958 y(With)28 b FD(H)776 927 y FC(\(2\))893 958 y FG(from)f(\(I)r (I.5.57\))g(w)n(e)g(de\014ne)h(the)g(abbreviations)622 1114 y FD(H)698 1079 y FC(\(2\))824 1114 y FG(=)36 b FD(T)30 b FG(+)18 b FD(W)n(;)263 b(W)58 b FG(=)46 b FD(W)1786 1079 y FC(\(1\))1907 1114 y FG(+)32 b FD(W)2094 1079 y FC(\(2\))2215 1114 y FG(+)g FD(V)2379 1079 y FC(\(12\))2534 1114 y FG(+)f(2)p FD(C)2737 1070 y FC(\(12\))2731 1136 y(1)2859 1114 y FD(;)237 b FG(\(I)r(I.6.2\))386 1271 y(with)45 b FD(W)682 1241 y FC(\()p FB(k)q FC(\))797 1271 y FG(:=)23 b FD(V)975 1241 y FC(\()p FB(k)q FC(\))1073 1271 y FG(+)1158 1238 y FB(i)p 1153 1252 34 4 v 1153 1300 a FC(2)1210 1271 y FG([)p FD(W)1323 1228 y FC(\()p FB(k)q FC(\))1311 1293 y(1)1416 1271 y FD(;)14 b(B)1520 1228 y FC(\()p FB(k)q FC(\))1516 1293 y(1)1612 1271 y FG(])p FD(;)61 b(k)25 b FG(=)e(1)p FD(;)14 b FG(2)43 b(the)22 b(p)r(oten)n(tial)f(term)g(of)g(the)g(Jansen-Hess)386 1371 y(op)r(erator)26 b(for)h(particle)g FD(k)s FG(.)486 1482 y(Let)35 b FD(\033)s FG(\()p FD(D)795 1439 y FC(\()p FB(k)q FC(\))793 1504 y(0)911 1482 y FG(+)23 b FD(W)1089 1452 y FC(\()p FB(k)q FC(\))1182 1482 y FG(\))70 b(b)r(e)35 b(the)g(resp)r(ectiv)n(e)f(one-particle)g(sp)r(ectrum.)59 b(Dropping)34 b(for)g(a)386 1582 y(momen)n(t)24 b(all)g(t)n(w)n (o-particle)e(in)n(teractions,)i(the)g(sp)r(ectrum)h(of)f(the)g(sum)h (of)f(the)g(single-particle)386 1681 y(op)r(erators)i(is)h(giv)n(en)g (b)n(y)g(\(Reed-Simon)h(1980,)e(Corollary)f(to)i(Theorem)g(VI)r(I)r (I.33\))525 1854 y FD(\033)s FG(\()p FD(T)j FG(+)18 b FD(W)859 1820 y FC(\(1\))967 1854 y FG(+)g FD(W)1140 1820 y FC(\(2\))1229 1854 y FG(\))46 b(=)p 1418 1757 1539 4 v 46 w FA(f)p FD(\025)1508 1866 y FC(1)1564 1854 y FG(+)18 b FD(\025)1695 1866 y FC(2)1756 1854 y FG(:)46 b FD(\025)1873 1866 y FB(k)1937 1854 y FA(2)24 b FD(\033)s FG(\()p FD(D)2169 1811 y FC(\()p FB(k)q FC(\))2167 1876 y(0)2280 1854 y FG(+)18 b FD(W)2453 1830 y FC(\()p FB(k)q FC(\))2546 1854 y FG(\))p FD(;)37 b(k)26 b FG(=)d(1)p FD(;)14 b FG(2)p FA(g)n FD(:)140 b FG(\(I)r(I.6.3\))386 2011 y(So)24 b(if)h FD(D)642 1968 y FC(\(1\))640 2033 y(0)744 2011 y FG(+)12 b FD(W)911 1981 y FC(\(1\))1025 2011 y FG(has)24 b(a)g(b)r(ound)h(ground)f(state)g(with)h(eigen)n(v)-5 b(alue)24 b(0)e FD(<)h(\025)2748 2023 y FC(01)2842 2011 y FD(<)g(m;)47 b FG(then)25 b(the)386 2115 y(essen)n(tial)i(sp)r (ectrum)g(of)h FD(T)h FG(+)18 b FD(W)1425 2085 y FC(\(1\))1533 2115 y FG(+)g FD(W)1706 2085 y FC(\(2\))1846 2115 y FG(starts)27 b(at)g FD(\025)2227 2127 y FC(01)2316 2115 y FG(+)18 b FD(m;)51 b FG(i.e.)763 2258 y FD(\033)810 2270 y FB(ess)908 2258 y FG(\()p FD(T)30 b FG(+)18 b FD(W)1192 2224 y FC(\(1\))1300 2258 y FG(+)g FD(W)1473 2224 y FC(\(2\))1562 2258 y FG(\))46 b(=)g([)p FD(m)18 b FG(+)239 b(inf)1948 2333 y FB(\025)1987 2341 y Ft(1)2020 2333 y Fz(2)p FB(\033)r FC(\()p FB(D)2187 2303 y Ft(\(1\))2185 2351 y(0)2265 2333 y FC(+)p FB(W)2387 2317 y Ft(\(1\))2464 2333 y FC(\))2504 2258 y FD(\025)2552 2270 y FC(1)2590 2258 y FD(;)14 b FA(1)p FG(\))377 b(\(I)r(I.6.4\))386 2471 y(\(Reed-Simon)43 b(1978,)j(p.121\).)83 b(This)43 b(is)g(true)g(b)r(ecause)g(for)g(the)h(Jansen-Hess)e(op)r(erator)386 2586 y FD(D)457 2543 y FC(\(1\))455 2609 y(0)571 2586 y FG(+)24 b FD(W)750 2556 y FC(\(1\))839 2586 y FD(;)75 b FG(the)37 b(essen)n(tial)f(sp)r(ectrum)h(is)g(the)g(same)f(as)g(for)g FD(D)2581 2543 y FC(\(1\))2579 2609 y(0)2707 2586 y FG(for)h(sub)r (critical)f(p)r(o-)386 2686 y(ten)n(tial)28 b(strength,)f(namely)g([)p FD(m;)14 b FA(1)p FG(\))46 b(\(see)28 b(Theorem)f(I.3\).)486 2786 y(Let)41 b(us)f(no)n(w)h(switc)n(h)f(on)h(the)g(t)n(w)n (o-particle)e(in)n(teractions.)76 b(W)-7 b(e)41 b(aim)f(at)h(pro)n (ving)e(the)386 2885 y(follo)n(wing)26 b(prop)r(erties)h(of)h(the)g(sp) r(ectrum.)386 3005 y Fv(Theorem)i(I)s(I.1)h FG(\(Lo)r(cation)d(of)f (essen)n(tial)g(sp)r(ectrum\))p Fv(.)386 3159 y Fn(L)l(et)g FD(H)603 3129 y FC(\(2\))715 3159 y FG(=)c FD(T)h FG(+)14 b FD(W)39 b Fn(with)28 b FD(T)34 b FG(=)23 b FD(D)1492 3116 y FC(\(1\))1490 3181 y(0)1595 3159 y FG(+)14 b FD(D)1745 3116 y FC(\(2\))1743 3181 y(0)1861 3159 y Fn(and)28 b FD(W)35 b FG(=)2264 3080 y FC(2)2237 3097 y Fu(P)2220 3234 y FB(k)q FC(=1)2355 3159 y FD(W)2445 3129 y FC(\()p FB(k)q FC(\))2551 3159 y FG(+)14 b FD(V)2696 3129 y FC(\(12\))2832 3159 y FG(+)g(2)p FD(C)3018 3116 y FC(\(12\))3012 3181 y(1)3167 3159 y Fn(b)l(e)28 b(the)386 3308 y(tr)l(ansforme)l(d)37 b(Coulomb-Dir)l(ac)g(op)l(er)l(ator)h(up)e(to)g(se)l(c)l(ond)h(or)l (der)g(in)f(the)g(c)l(oupling)h(c)l(onstant)386 3419 y FD(e)425 3389 y FC(2)462 3419 y Fn(,)27 b(acting)f(on)g FA(H)944 3389 y FC(1)943 3440 y(+)p FB(;)p FC(2)1074 3419 y FG(:=)c(\(\003)1274 3376 y FC(\(1\))1274 3440 y(+)1373 3419 y FA(\012)10 b FG(\003)1506 3376 y FC(\(2\))1506 3440 y(+)1594 3419 y FG(\)\()p FA(A)p FG(\()p FD(H)1825 3431 y FC(1)1863 3419 y FG(\()p Fx(R)1949 3389 y FC(3)1993 3419 y FG(\))g FA(\002)g Fx(C)2162 3389 y FC(4)2206 3419 y FG(\))2238 3389 y FC(2)2275 3419 y FG(\))p FD(:)49 b Fn(Then)27 b(for)f(p)l(otential)g(str)l(engths)386 3519 y FD(\015)i(<)22 b FG(0)p FD(:)p FG(89)p FD(;)29 b Fn(the)g(sp)l(e)l(ctrum)g(has)i(the)f(fol)t(lowing)i(pr)l(op)l (erties)546 3639 y FG(\(i\))1052 3751 y([2)p FD(m;)14 b FA(1)p FG(\))45 b(=)h FD(\033)s FG(\()p FD(T)12 b FG(\))46 b FA(\032)g FD(\033)s FG(\()p FD(T)30 b FG(+)18 b FD(V)2141 3717 y FC(\(12\))2263 3751 y FG(\))47 b FA(\032)22 b FD(\033)s FG(\()p FD(H)2587 3717 y FC(\(2\))2677 3751 y FG(\))410 b(\(I)r(I.6.5\))522 3873 y(\(ii\))909 3986 y([2)p FD(m;)14 b FA(1)p FG(\))45 b(=)h FD(\033)1402 3998 y FB(ess)1501 3986 y FG(\()p FD(T)12 b FG(\))45 b FA(\032)h FD(\033)1829 3998 y FB(ess)1928 3986 y FG(\()p FD(T)30 b FG(+)18 b FD(V)2189 3952 y FC(\(12\))2311 3986 y FG(\))46 b FA(\032)23 b FD(\033)2524 3998 y FB(ess)2623 3986 y FG(\()p FD(H)2731 3952 y FC(\(2\))2820 3986 y FG(\))267 b(\(I)r(I.6.6\))386 4108 y Fn(and)30 b(for)h FD(\015)c(<)c FG(0)p FD(:)p FG(654)p FD(;)499 4228 y FG(\(iii\))1085 4412 y FD(\033)1132 4424 y FB(ess)1231 4412 y FG(\()p FD(H)1339 4378 y FC(\(2\))1428 4412 y FG(\))47 b(=)e FD(\033)1664 4424 y FB(ess)1763 4412 y FG(\()p FD(T)f FG(+)2028 4308 y FC(2)1985 4333 y Fu(X)1985 4512 y FB(k)q FC(=1)2119 4412 y FD(W)2209 4378 y FC(\()p FB(k)q FC(\))2334 4412 y FG(+)32 b FD(V)2498 4378 y FC(\(12\))2620 4412 y FG(\))p FD(:)444 b FG(\(I)r(I.6.7\))486 4625 y(The)29 b(b)r(ound)g(0.89)f(on)h FD(\015)34 b FG(arises)27 b(from)i(the)g (requiremen)n(t)g(that)g FD(H)2577 4595 y FC(\(2\))2695 4625 y FG(is)g(w)n(ell-de\014ned)g(\(see)386 4725 y(Prop)r(osition)d(I) r(I.3\).)386 4825 y(F)-7 b(or)37 b(the)i(pro)r(of)e(of)h(inclusions)g (\(i\))h(and)f(\(ii\))g(w)n(e)g(use)g(the)g(b)r(eha)n(viour)f(under)h (translations)386 4924 y(\(Hunziger)h(1966,)i(Reed-Simon)e(1978,)i (p.370,)h(problem)d(45\),)j(together)c(with)j(the)f(W)-7 b(eyl)386 5024 y(criterion)26 b(\(W)-7 b(eidmann)29 b(1980,)d(Theorem)g (7.22\).)386 5144 y Fv(Lemma)j(I)s(I.9)j FG(\(W)-7 b(eyl)28 b(criterion\))p Fv(.)386 5243 y Fn(L)l(et)d FD(A)h Fn(b)l(e)g(a)g (self-adjoint)h(op)l(er)l(ator)g(in)e(a)h(Hilb)l(ert)g(sp)l(ac)l(e)g FA(H)q Fn(.)37 b(Then)27 b FD(\025)c FA(2)h FD(\033)s FG(\()p FD(A)p FG(\))i Fn(i\013)g(ther)l(e)g(exists)386 5343 y(a)k(se)l(quenc)l(e)f FG(\()p FD( )881 5355 y FB(l)907 5343 y FG(\))939 5355 y FB(l)p Fz(2)p Fy(N)1081 5343 y Fn(in)h(the)g(domain)h(of)g FD(A)f Fn(with)g FA(k)p FD( )2084 5355 y FB(l)2109 5343 y FA(k)23 b FD(>)f FG(0)53 b Fn(such)30 b(that)1216 5483 y FA(k)p FG(\()p FD(A)i FA(\000)g FD(\025)p FG(\))14 b FD( )1629 5495 y FB(l)1655 5483 y FA(k)46 b(\000)-15 b(!)47 b FG(0)254 b(\()p FD(l)24 b FA(!)g(1)p FG(\))p FD(:)574 b FG(\(I)r(I.6.8\))386 5623 y Fn(If)30 b FG(\()p FD( )559 5635 y FB(l)585 5623 y FG(\))617 5635 y FB(l)p Fz(2)p Fy(N)759 5623 y Fn(c)l(an)g(b)l(e)g (chosen)g(ortho)l(gonal)i(then)d FD(\025)24 b FA(2)f FD(\033)2073 5635 y FB(ess)2172 5623 y FG(\()p FD(A)p FG(\))p FD(:)p eop %%Page: 69 75 69 74 bop 3309 259 a FC(69)486 459 y FG(In)24 b(the)h(follo)n(wing,)f (all)g(symmetric)g(op)r(erators)f(are)g(considered)g(as)h(self-adjoin)n (t)g(b)n(y)g(means)386 558 y(of)j(their)h(F)-7 b(riedric)n(hs)27 b(extension)g(\(whic)n(h)h(exists)f(for)g(sub)r(critical)g FD(\015)5 b FG(\).)386 729 y Fn(Pr)l(o)l(of)56 b(of)31 b(\(i\))f(and)g(\(ii\).)386 828 y FG(Let)25 b FD(T)581 840 y FB(a)645 828 y FG(b)r(e)g(the)g(translation)f(de\014ned)h(in)g (Lemma)g(I)r(I.5)f(and)h(let)g FD(a)e FG(:=)f FD(n)h FA(2)h Fx(N)t FD(:)54 b FG(Then)25 b(\()p FD(T)3166 840 y FB(n)3211 828 y FG(\))3243 840 y FB(n)p Fz(2)p Fy(N)386 928 y FG(induces)j(a)f(sequence)g(\()p FD( )1181 940 y FB(n)1226 928 y FG(\))1258 940 y FB(n)p Fz(2)p Fy(N)1418 928 y FG(where)g FD(a)c FA(!)g(1)28 b FG(corresp)r(onds)e(to)h FD(n)c FA(!)g(1)p FD(:)486 1044 y FG(F)-7 b(rom)32 b(the)i(pro)r(of)f (of)g(Lemma)g(I)r(I.5)g(w)n(e)g(kno)n(w)f(that)i FD(D)2250 1001 y FC(\()p FB(k)q FC(\))2248 1066 y(0)2376 1044 y FG(and)f FD(V)2610 1014 y FC(\(12\))2765 1044 y FG(are)f FD(T)2958 1056 y FB(n)3003 1044 y FG(-in)n(v)-5 b(arian)n(t,)386 1161 y(while)28 b(\()p FD( )689 1173 y FB(n)734 1161 y FD(;)14 b(W)861 1131 y FC(\()p FB(k)q FC(\))954 1161 y FD( )1008 1173 y FB(n)1053 1161 y FG(\))28 b(and)f(\()p FD( )1360 1173 y FB(n)1406 1161 y FD(;)14 b(C)1508 1118 y FC(\(12\))1502 1183 y(1)1630 1161 y FD( )1684 1173 y FB(n)1730 1161 y FG(\))28 b(tend)g(to)f(zero)g(when)h FD(n)22 b FA(!)i(1)p FD(:)j FG(Hence)457 1317 y(lim)428 1367 y FB(n)p Fz(!1)615 1317 y FA(k)p FG(\()p FD(H)765 1283 y FC(\(2\))872 1317 y FA(\000)18 b FD(\025)p FG(\))24 b FD( )1113 1329 y FB(n)1158 1317 y FA(k)1200 1283 y FC(2)1283 1317 y FG(=)75 b(lim)1394 1367 y FB(n)p Fz(!1)1567 1317 y FG(\()p FD( )1653 1329 y FB(n)1699 1317 y FD(;)28 b FG(\()p FD(H)1858 1283 y FC(\(2\))1965 1317 y FA(\000)18 b FD(\025)p 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FC(1)2438 5280 y Fu(o)2503 5372 y FA(\001)2536 5280 y Fu(h)2575 5372 y FG(\()p FD(T)30 b FG(+)18 b FD(\026)p FG(\))c(\()p FD(H)2973 5338 y FC(\(2\))3081 5372 y FG(+)k FD(\026)p FG(\))3246 5338 y Fz(\000)p FC(1)3336 5280 y Fu(i)386 5523 y FG(and)39 b(sho)n(w)f(that)i(the)f(op)r(erator)e(in)j(curly)e (brac)n(k)n(ets)g(is)h(compact)f(and)h(the)h(adjacen)n(t)f(t)n(w)n(o) 386 5623 y(factors)26 b(are)h(b)r(ounded.)p eop %%Page: 70 76 70 75 bop 386 259 a FC(70)386 462 y Fn(a\))30 b(Comp)l(actness)h(of)f FD(W)1166 474 y FC(2)1227 462 y FG(:=)23 b(\()p FD(T)29 b FG(+)18 b FD(\026)p FG(\))1613 432 y Fz(\000)p FC(1)1717 462 y FD(C)1782 419 y FC(\(12\))1776 485 y(1)1918 462 y FG(\()p FD(T)30 b FG(+)18 b FD(\026)p FG(\))2194 432 y Fz(\000)p FC(1)486 609 y FG(W)-7 b(e)31 b(apply)f(the)h(strategy)f (to)g(write)h FD(W)1732 621 y FC(2)1800 609 y FG(as)f(the)h(norm-con)n (v)n(ergen)n(t)d(limit)j(of)g(a)f(sequence)386 709 y(\()p FD(W)496 721 y FC(2)p FB(n)575 709 y FG(\))607 721 y FB(n)p Fz(2)p Fy(N)767 709 y FG(of)d(\(compact\))h(Hilb)r(ert-Sc)n (hmidt)g(op)r(erators.)35 b(If)28 b(so,)f FD(W)2555 721 y FC(2)2621 709 y FG(is)g(compact)g(to)r(o.)386 809 y(In)33 b(order)e(to)h(construct)g(the)h(Hilb)r(ert)g(Sc)n(hmidt)g(op)r (erators)e(w)n(e)h(use)g(the)h(prop)r(ert)n(y)e(that)i(an)386 908 y(op)r(erator)h FD(A)i FG(is)f(Hilb)r(ert-Sc)n(hmidt)h(i\013)g(its) g(k)n(ernel)f FD(k)2064 920 y FB(A)2154 908 y FG(is)g(square)f(in)n (tegrable)h(\(Reed-Simon)386 1008 y(1980,)26 b(Theorem)g(VI.23\).)486 1107 y(In)k(order)e(to)i(de\014ne)h(the)f(sequence)g(\()p FD(W)1760 1119 y FC(2)p FB(n)1839 1107 y FG(\))1871 1119 y FB(n)p Fz(2)p Fy(N)2003 1107 y FG(,)g(w)n(e)g(m)n(ust)g(in)n(tro)r (duce)g(con)n(v)n(ergence)e(gen-)386 1207 y(erating)j(functions)i(in)f (momen)n(tum)h(space)e(and)h(apply)g(a)g(regularisation)e(of)i(the)h (Coulom)n(b)386 1307 y(p)r(oten)n(tial.)486 1418 y(W)-7 b(e)28 b(\014rst)f(decomp)r(ose)g FD(C)1283 1375 y FC(\(12\))1277 1440 y(1)1433 1418 y FG(in)n(to)g(t)n(w)n(o)g(self-adjoin)n(t)g(op)r (erators,)1436 1593 y FD(C)1501 1550 y FC(\(12\))1495 1615 y(1)1670 1593 y FG(=)45 b FD(C)1845 1550 y FC(\(12\))1839 1615 y(1)p FB(\017)2009 1593 y FG(+)d FD(R)2180 1550 y FC(\(12\))2179 1615 y(1)p FB(\017)2301 1593 y FD(;)753 b FG(\(I)r(I.6.14\))786 1792 y FD(C)851 1749 y FC(\(12\))845 1814 y(1)p FB(\017)996 1792 y FG(:=)46 b FD(e)1169 1758 y Fz(\000)p FB(\017)p FC(\()p FB(p)1309 1766 y Ft(1)1341 1758 y FC(+)p FB(p)1426 1766 y Ft(2)1459 1758 y FC(\))1489 1792 y FD(V)1556 1758 y FC(\(12\))1678 1792 y FG(\003)1736 1749 y FC(\(1\))1736 1813 y Fz(\000)1825 1792 y FD(F)1890 1749 y FC(\(1\))1878 1814 y(0)2020 1792 y FG(+)41 b FD(F)2191 1749 y FC(\(1\))2179 1814 y(0)2280 1792 y FG(\003)2338 1749 y FC(\(1\))2338 1813 y Fz(\000)2427 1792 y FD(V)2494 1758 y FC(\(12\))2616 1792 y FD(e)2655 1758 y Fz(\000)p FB(\017)p FC(\()p FB(p)2795 1766 y Ft(1)2827 1758 y FC(+)p FB(p)2912 1766 y Ft(2)2945 1758 y FC(\))594 1979 y FD(R)658 1936 y FC(\(12\))657 2001 y(1)p FB(\017)803 1979 y FG(:=)k FD(g)976 1991 y FB(\017)1008 1979 y FD(V)1075 1945 y FC(\(12\))1197 1979 y FG(\003)1255 1936 y FC(\(1\))1255 2000 y Fz(\000)1344 1979 y FD(F)1409 1936 y FC(\(1\))1397 2001 y(0)1539 1979 y FG(+)c FD(F)1710 1936 y FC(\(1\))1698 2001 y(0)1799 1979 y FG(\003)1857 1936 y FC(\(1\))1857 2000 y Fz(\000)1946 1979 y FD(V)2013 1945 y FC(\(12\))2135 1979 y FD(g)2175 1991 y FB(\017)2207 1979 y FD(;)180 b(g)2450 1991 y FB(\017)2504 1979 y FG(:=)46 b(1)32 b FA(\000)g FD(e)2848 1945 y Fz(\000)p FB(\017)p FC(\()p FB(p)2988 1953 y Ft(1)3019 1945 y FC(+)p FB(p)3104 1953 y Ft(2)3137 1945 y FC(\))386 2126 y FG(and)21 b(secondly)-7 b(,)22 b(w)n(e)f(in)n(tro)r(duce)g(the)h(screened)e(Coulom)n(b)h (\014eld)2362 2093 y FC(1)p 2359 2107 38 4 v 2359 2155 a FB(x)2407 2126 y FD(e)2446 2096 y Fz(\000)p FB(\017x)2588 2126 y FG(with)h(its)g(F)-7 b(ourier)20 b(trans-)386 2296 y(form)582 2170 y Fu(r)p 665 2170 71 4 v 680 2240 a FG(2)p 675 2277 51 4 v 675 2353 a FD(\031)864 2240 y FG(1)p 759 2277 252 4 v 759 2353 a FD(p)801 2329 y FC(2)857 2353 y FG(+)e FD(\017)974 2329 y FC(2)1048 2296 y FG(and)28 b(decomp)r(ose)f(eac)n(h)g(of)g(the)h(t)n(w)n(o)f(momen)n (tum)h(denominators)e(in)i(the)386 2464 y(prefactor)k(of)h(the)h(k)n (ernel)f(\(I)r(I.4.10\))g(of)g FD(C)1741 2434 y FC(\(12\))1897 2464 y FG(\(whic)n(h)h(is)f(t)n(wice)h(the)g(k)n(ernel)e(of)h FD(C)3047 2421 y FC(\(12\))3041 2486 y(1)3170 2464 y FG(\))67 b(ac-)386 2564 y(cording)26 b(to)763 2714 y(1)p 745 2751 80 4 v 745 2827 a FD(p)787 2803 y FC(2)880 2770 y FG(=)46 b FD(e)1030 2782 y FB(\017)1093 2770 y FG(+)32 b FD(f)1231 2782 y FB(\017)1263 2770 y FD(;)180 b(e)1505 2782 y FB(\017)1559 2770 y FG(:=)1808 2714 y(1)p 1703 2751 252 4 v 1703 2827 a FD(p)1745 2803 y FC(2)1800 2827 y FG(+)18 b FD(\017)1917 2803 y FC(2)1964 2770 y FD(;)97 b(f)2125 2782 y FB(\017)2180 2770 y FG(:=)2485 2714 y FD(\017)2519 2684 y FC(2)p 2323 2751 396 4 v 2323 2827 a FD(p)2365 2803 y FC(2)2402 2827 y FG(\()p FD(p)2476 2803 y FC(2)2532 2827 y 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FD(R)746 3333 y FC(\(12\))745 3398 y(1)p FB(\017)868 3376 y FG(\()p FD(e)939 3388 y FB(\017)970 3376 y FD(;)14 b(e)1046 3388 y FB(\017)1078 3376 y FG(\))42 b(+)f FD(R)1322 3333 y FC(\(12\))1321 3398 y(1)p FB(\017)1444 3376 y FG(\()p FD(e)1515 3388 y FB(\017)1546 3376 y FD(;)14 b(f)1624 3388 y FB(\017)1656 3376 y FG(\))42 b(+)f FD(R)1900 3333 y FC(\(12\))1899 3398 y(1)p FB(\017)2022 3376 y FG(\()p FD(f)2095 3388 y FB(\017)2126 3376 y FD(;)14 b(e)2202 3388 y FB(\017)2234 3376 y FG(\))41 b(+)h FD(R)2478 3333 y FC(\(12\))2477 3398 y(1)p FB(\017)2599 3376 y FG(\()p FD(f)2672 3388 y FB(\017)2704 3376 y FD(;)14 b(f)2782 3388 y FB(\017)2814 3376 y FG(\))p FD(:)208 b FG(\(I)r(I.6.16\))486 3534 y(W)-7 b(e)23 b(no)n(w)f(pro)n(v)n(e)f (the)i(compactness)f(of)g FD(W)1792 3546 y FC(2)p FB(n)1894 3534 y FG(:=)36 b(\()p FD(T)20 b FG(+)9 b FD(\026)p FG(\))2275 3504 y Fz(\000)p FC(1)2378 3534 y FD(C)2443 3491 y FC(\(12\))2437 3556 y(1)p FB(\017)2565 3534 y FG(\()p FD(e)2636 3546 y FB(\017)2668 3534 y FD(;)14 b(e)2744 3546 y FB(\017)2776 3534 y FG(\))g(\()p FD(T)20 b FG(+)9 b FD(\026)p FG(\))3079 3504 y Fz(\000)p FC(1)3213 3534 y FG(with)386 3633 y FD(\017)23 b FG(:=)f(1)p FD(=n)50 b FG(via)27 b(the)h(square)e(in)n (tegrabilit)n(y)g(of)i(its)g(k)n(ernel.)35 b(Subsequen)n(tly)28 b(w)n(e)f(will)h(sho)n(w)f(that)386 3749 y(the)d(remaining)e(op)r (erators)g(from)h(the)g(decomp)r(osition)g(of)g FD(C)2300 3706 y FC(\(12\))2294 3771 y(1)2446 3749 y FG(v)-5 b(anish)24 b(in)f(the)h(limit)g FD(\017)f FA(!)g FG(0)p FD(:)386 3849 y FG(W)-7 b(e)28 b(ha)n(v)n(e)411 4039 y FD(k)454 4072 y FB(C)506 4042 y Ft(\(12\))502 4090 y(1)p Fp(\017)612 4072 y FC(\()p FB(e)669 4080 y Fp(\017)699 4072 y FB(;e)750 4080 y Fp(\017)780 4072 y FC(\))810 4039 y FG(\()p Fv(p)895 4051 y FC(1)932 4039 y FD(;)14 b Fv(p)1022 4051 y FC(2)1060 4039 y FG(;)g Fv(p)1150 4005 y Fz(0)1150 4060 y FC(1)1187 4039 y FD(;)g Fv(p)1277 4005 y Fz(0)1277 4060 y FC(2)1314 4039 y FG(\))46 b(=)g FA(\000)1636 3983 y FD(\015)5 b(e)1723 3953 y 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4467 y Ft(1)1455 4459 y Fz(j)1497 4444 y FG(+)g FD(E)1641 4458 y FB(p)1675 4438 y Fl(0)1675 4476 y Ft(1)1745 4387 y FG(\(1)h FA(\000)1940 4366 y FG(~)1921 4387 y FD(D)1992 4344 y FC(\(1\))1990 4409 y(0)2081 4387 y FG(\()p Fv(p)2166 4399 y FC(2)2222 4387 y FA(\000)f Fv(p)2358 4352 y Fz(0)2358 4407 y FC(2)2414 4387 y FG(+)g Fv(p)2550 4352 y Fz(0)2550 4407 y FC(1)2587 4387 y FG(\)\))24 b(\(1)18 b(+)2869 4366 y(~)2850 4387 y FD(D)2921 4344 y FC(\(1\))2919 4409 y(0)3010 4387 y FG(\()p Fv(p)3095 4352 y Fz(0)3095 4407 y FC(1)3133 4387 y FG(\)\))520 4708 y(+)928 4652 y(1)p 617 4689 V 617 4765 a FD(E)678 4777 y FB(p)712 4785 y Ft(1)768 4765 y FG(+)g FD(E)912 4780 y Fz(j)p Fr(p)974 4760 y Fl(0)974 4798 y Ft(2)1006 4780 y Fz(\000)p Fr(p)1100 4788 y Ft(2)1132 4780 y FC(+)p Fr(p)1225 4760 y Fl(0)1225 4798 y Ft(1)1257 4780 y Fz(j)1314 4708 y FG(\(1)g(+)1509 4687 y(~)1489 4708 y FD(D)1560 4665 y FC(\(1\))1558 4731 y(0)1649 4708 y FG(\()p Fv(p)1734 4720 y FC(1)1772 4708 y FG(\)\))24 b(\(1)18 b FA(\000)2054 4687 y FG(~)2035 4708 y FD(D)2106 4665 y FC(\(1\))2104 4731 y(0)2195 4708 y FG(\()p Fv(p)2280 4674 y Fz(0)2280 4729 y FC(2)2336 4708 y FA(\000)g Fv(p)2472 4720 y FC(2)2528 4708 y FG(+)g Fv(p)2664 4674 y Fz(0)2664 4729 y FC(1)2701 4708 y FG(\)\))24 b FD(e)2828 4674 y Fz(\000)p FB(\017)p FC(\()p FB(p)2968 4649 y Fl(0)2968 4691 y Ft(1)3000 4674 y FC(+)p FB(p)3085 4649 y Fl(0)3085 4691 y Ft(2)3117 4674 y FC(\))3147 4566 y Fu(\))3228 4708 y FD(:)386 4948 y FG(In)33 b(the)g(mo)r(dulus)g(of)g(the)g(k)n (ernel,)g FA(j)p FG(1)21 b FA(\006)1697 4927 y FG(~)1678 4948 y FD(D)1749 4905 y FC(\(1\))1747 4970 y(0)1838 4948 y FG(\()p Fv(p)p FG(\))p FA(j)34 b FG(is)e(estimated)h(b)n(y)f(2)h(and) f(the)h(energy)f(de-)386 5048 y(nominators)26 b(b)n(y)h(their)h (massless)e(expression,)h(resulting)g(in)414 5254 y FA(j)p FD(k)480 5266 y FB(W)542 5274 y Ft(2)p Fp(n)616 5254 y FG(\()p Fv(p)701 5266 y FC(1)738 5254 y FD(;)14 b Fv(p)828 5266 y FC(2)866 5254 y FG(;)g Fv(p)956 5220 y Fz(0)956 5275 y FC(1)993 5254 y FD(;)g Fv(p)1083 5220 y Fz(0)1083 5275 y FC(2)1120 5254 y FG(\))p FA(j)23 b(\024)1310 5198 y FG(4)p FD(\015)5 b(e)1439 5168 y FC(2)p 1296 5235 194 4 v 1296 5311 a FG(\(2)p FD(\031)s FG(\))1452 5287 y FC(4)1687 5198 y FG(1)p 1523 5235 370 4 v 1523 5311 a FD(p)1565 5323 y FC(1)1607 5311 y FG(+)18 b FD(p)1732 5323 y FC(2)1774 5311 y FG(+)5 b FD(\026)2161 5198 y FG(1)p 1927 5235 511 4 v 1927 5311 a FA(j)p Fv(p)2003 5323 y FC(2)2045 5311 y FA(\000)18 b Fv(p)2181 5283 y Fz(0)2181 5333 y FC(2)2218 5311 y FA(j)2241 5287 y FC(2)2283 5311 y FG(+)g FD(\017)2400 5287 y FC(2)2883 5198 y FG(1)p 2471 5235 866 4 v 2471 5311 a FA(j)p Fv(p)2547 5323 y FC(2)2589 5311 y FA(\000)g Fv(p)2725 5283 y Fz(0)2725 5333 y FC(2)2766 5311 y FG(+)h Fv(p)2903 5323 y FC(1)2944 5311 y FA(\000)f Fv(p)3080 5283 y Fz(0)3080 5333 y FC(1)3118 5311 y FA(j)3141 5287 y FC(2)3183 5311 y FG(+)g FD(\017)3300 5287 y FC(2)3077 5404 y FG(\(I)r(I.6.18\))402 5577 y FA(\001)439 5460 y Fu(\032)501 5577 y FD(e)540 5543 y Fz(\000)p FB(\017)p FC(\()p FB(p)680 5551 y Ft(1)712 5543 y FC(+)p FB(p)797 5551 y Ft(2)829 5543 y FC(\))1212 5521 y FG(1)p 883 5558 701 4 v 883 5634 a FA(j)p Fv(p)959 5646 y FC(2)1015 5634 y FA(\000)g Fv(p)1151 5605 y Fz(0)1151 5656 y FC(2)1207 5634 y FG(+)g Fv(p)1343 5646 y FC(1)1380 5634 y FA(j)g FG(+)g FD(p)1546 5605 y Fz(0)1546 5656 y FC(1)1626 5577 y FG(+)2062 5521 y(1)p 1732 5558 V 1732 5634 a FD(p)1774 5646 y FC(1)1830 5634 y FG(+)g FA(j)p Fv(p)1989 5605 y Fz(0)1989 5656 y FC(2)2045 5634 y FA(\000)g Fv(p)2181 5646 y FC(2)2237 5634 y FG(+)g Fv(p)2373 5605 y Fz(0)2373 5656 y FC(1)2410 5634 y FA(j)2457 5577 y FD(e)2496 5543 y Fz(\000)p FB(\017)p FC(\()p FB(p)2636 5517 y Fl(0)2636 5559 y Ft(1)2667 5543 y FC(+)p FB(p)2752 5517 y Fl(0)2752 5559 y Ft(2)2785 5543 y FC(\))2815 5460 y Fu(\033)3100 5521 y FG(1)p 2915 5558 411 4 v 2915 5634 a FD(p)2957 5605 y Fz(0)2957 5656 y FC(1)3013 5634 y FG(+)g FD(p)3138 5605 y Fz(0)3138 5656 y FC(2)3193 5634 y FG(+)g FD(\026)3336 5577 y(:)p eop %%Page: 71 77 71 76 bop 3309 259 a FC(71)386 459 y FG(Using)21 b(the)g(substitutions) g Fv(q)1287 471 y FC(2)1348 459 y FG(:=)h Fv(p)1511 471 y FC(2)1554 459 y FA(\000)5 b Fv(p)1677 429 y Fz(0)1677 479 y FC(2)1734 459 y FG(and)21 b Fv(q)1939 471 y FC(1)2000 459 y FG(:=)i Fv(p)2164 471 y FC(1)2206 459 y FA(\000)5 b Fv(p)2329 429 y Fz(0)2329 479 y FC(1)2371 459 y FG(+)g Fv(q)2491 471 y FC(2)2549 459 y FG(as)20 b(w)n(ell)h(as)f(the)h (estimates)386 558 y(\()p FD(p)460 570 y FC(1)518 558 y FG(+)f FD(p)645 570 y FC(2)702 558 y FG(+)g FD(\026)p FG(\))869 528 y Fz(\000)p FC(1)986 558 y FA(\024)28 b FG(1)p FD(;)69 b FG(\()p FD(p)1287 528 y Fz(0)1287 579 y FC(1)1345 558 y FG(+)20 b FD(p)1472 528 y Fz(0)1472 579 y FC(2)1529 558 y FG(+)g FD(\026)p FG(\))1696 528 y Fz(\000)p FC(1)1828 558 y FA(\024)27 b FG(1)58 b(for)30 b FD(\026)e FA(\025)g FG(1)p FD(;)58 b FG(w)n(e)30 b(ha)n(v)n(e)g(to)g (sho)n(w)g(that)g(the)386 658 y(in)n(tegral)c FD(S)33 b FG(is)27 b(\014nite,)h(de\014ned)g(b)n(y)475 847 y FD(S)f FG(:=)687 734 y Fu(Z)784 847 y FD(d)p Fv(p)880 859 y FC(1)941 847 y FD(d)p Fv(p)1037 859 y FC(2)1097 847 y FD(d)p Fv(p)1193 812 y Fz(0)1193 867 y FC(1)1254 847 y FD(d)p Fv(p)1350 812 y Fz(0)1350 867 y FC(2)1410 847 y FA(j)p FD(k)1476 859 y FB(W)1538 867 y Ft(2)p Fp(n)1612 847 y FG(\()p Fv(p)1697 859 y FC(1)1735 847 y FD(;)14 b Fv(p)1825 859 y FC(2)1862 847 y FG(;)g Fv(p)1952 812 y Fz(0)1952 867 y FC(1)1989 847 y FD(;)g Fv(p)2079 812 y Fz(0)2079 867 y FC(2)2116 847 y FG(\))p FA(j)2171 812 y FC(2)2246 847 y FA(\024)2381 791 y FG(4)p FD(\015)5 b(e)2510 760 y FC(2)p 2366 828 194 4 v 2366 904 a FG(\(2)p FD(\031)s FG(\))2522 880 y FC(4)2607 734 y Fu(Z)2704 847 y FD(d)p Fv(q)2797 859 y FC(1)2858 847 y FD(d)p Fv(q)2951 859 y FC(2)3077 847 y FG(\(I)r(I.6.19\))496 1099 y FA(\001)685 1043 y FG(1)p 529 1080 352 4 v 529 1156 a(\()p FD(q)601 1128 y FC(2)598 1178 y(2)658 1156 y FG(+)18 b FD(\017)775 1132 y FC(2)812 1156 y FG(\))844 1132 y FC(2)1079 1043 y FG(1)p 924 1080 V 924 1156 a(\()p 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b(w)n(e)g(ha)n(v)n(e)g(to)h(use)g(its)g(symmetrised)f(form)386 1567 y(\(with)i(resp)r(ect)e(to)h(particle)g(exc)n(hange\),)f(i.e.)36 b(replace)26 b(2)p FD(C)2231 1524 y FC(\(12\))2225 1589 y(1)2380 1567 y FG(b)n(y)h FD(C)2560 1524 y FC(\(12\))2554 1589 y(1)2700 1567 y FG(+)16 b FD(C)2846 1524 y FC(\(12\))2840 1589 y(2)2969 1567 y FD(:)50 b FG(F)-7 b(rom)26 b(the)386 1667 y(an)n(tisymmetry)g(prop)r(ert)n(y)g(of)g FD( )k FG(together)c(with)h(the)g(fact)g(that)g(for)g(an)n(y)f(pair)g(of)g (self-adjoin)n(t)386 1767 y(op)r(erators)j FD(A;)14 b(B)35 b FG(one)30 b(has)g FA(j)p FG(\()p FD( )s(;)14 b(AB)19 b( )s FG(\))p FA(j)42 b FG(=)28 b FA(j)p FG(\()p FD(B)t(A)14 b( )s(;)g( )s FG(\))p FA(j)43 b FG(=)27 b FA(j)p FG(\()p FD( )s(;)14 b(B)t(A)g( )s FG(\))p FA(j)p FD(;)60 b FG(the)31 b(estimate)g(of)386 1866 y FA(k)p FD(W)12 b( )s FA(k)617 1836 y FC(2)690 1866 y FG(=)23 b(\()p FD( )s(;)14 b(W)994 1836 y FC(2)1031 1866 y FD( )s FG(\))28 b(can)f(b)r(e)h(reduced)g(to) 984 2019 y(\()p FD( )s(;)g 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76 4 v 57 x FD(c)1999 3832 y FB(v)2062 3820 y FA(k)p FD(p)2146 3832 y FC(2)2196 3820 y FD(')p FA(k)g FG(=)2449 3763 y FA(p)p 2518 3763 V 57 x FD(c)2554 3832 y FB(v)2617 3820 y FA(k)p FD(p)2701 3832 y FC(2)2737 3820 y FG(\003)2795 3777 y FC(\(1\))2795 3841 y Fz(\000)2884 3820 y FD(F)2949 3777 y FC(\(1\))2937 3842 y(0)3052 3820 y FD( )s FA(k)455 3982 y FG(=)566 3924 y FA(p)p 635 3924 V 58 x FD(c)671 3994 y FB(v)733 3982 y FA(k)p FG(\003)833 3939 y FC(\(1\))833 4002 y Fz(\000)921 3982 y FD(F)986 3939 y FC(\(1\))974 4004 y(0)1076 3982 y FD(p)1118 3994 y FC(2)1168 3982 y FD( )s FA(k)g(\024)1424 3924 y(p)p 1493 3924 V 58 x FD(c)1529 3994 y FB(v)1591 3982 y FA(k)p FG(\003)1691 3939 y FC(\(1\))1691 4002 y Fz(\000)1780 3982 y FA(k)22 b(k)p FD(F)1951 3939 y FC(\(1\))1939 4004 y(0)2040 3982 y FA(k)g(k)p FD(p)2188 3994 y FC(2)2239 3982 y FD( )s FA(k)45 b(\024)2494 3924 y(p)p 2563 3924 V 58 x FD(c)2599 3994 y FB(v)2662 3982 y FA(k)p FD(F)2769 3939 y FC(\(1\))2757 4004 y(0)2857 3982 y FA(k)23 b 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FD(W)1251 5393 y FC(\(2\))1390 5423 y FG(\(when)h(the)f(cen)n(tral)g(p)r(oten)n(tial)g(strength)g FD(\015)31 b FG(is)26 b(large\),)g(w)n(e)g(are)386 5523 y(not)i(aiming)g(at)g(optimised)g(estimates.)39 b(Rather,)28 b(w)n(e)f(use)h(estimates)g(of)g(the)h(t)n(yp)r(e)f(\(for)g FD(A;)14 b(B)386 5623 y FG(self-adjoin)n(t\),)50 b FA(j)p FG(\()p FD( )s(;)14 b(AB)t( )s FG(\))p FA(j)38 b FG(=)23 b FA(j)p FG(\()p FD(A )s(;)14 b(B)t( )s FG(\))p FA(j)38 b(\024)23 b(k)p FD(A )s FA(k)18 b(\001)g(k)p FD(B)t( )s FA(k)p FD(:)p eop %%Page: 73 79 73 78 bop 3309 259 a FC(73)486 459 y FG(As)27 b(an)h(exemplary)e(case,) h(consider)614 601 y FA(j)p FG(\()p FD( )s(;)14 b(W)853 567 y FC(\(1\))943 601 y FD(V)1010 567 y FC(\(12\))1146 601 y FD( )s FG(\))p FA(j)46 b(\024)g(k)p FD(W)1547 567 y FC(\(1\))1649 601 y FD( )s FA(k)23 b(k)p FD(V)1879 567 y FC(\(12\))2015 601 y FD( )s FA(k)46 b(\024)2271 544 y(p)p 2340 544 90 4 v 57 x FD(c)2376 613 y FB(w)2453 601 y FA(k)p FD(p)2537 613 y FC(1)2587 601 y FD( )s FA(k)18 b(\001)2746 544 y(p)p 2815 544 76 4 v 57 x FD(c)2851 613 y FB(v)2913 601 y FA(k)p FD(p)2997 613 y FC(2)3048 601 y FD( )s FA(k)455 787 y FG(=)566 729 y FA(p)p 635 729 166 4 v 58 x FD(c)671 799 y FB(w)725 787 y FD(c)761 799 y FB(v)823 787 y FA(k)p FD(p)907 799 y FC(1)958 787 y FD( )s FA(k)1057 753 y FC(2)1140 787 y FG(=)1260 731 y(1)p 1260 768 42 4 v 1260 844 a(2)1335 729 y FA(p)p 1404 729 166 4 v 58 x FD(c)1440 799 y FB(w)1494 787 y FD(c)1530 799 y FB(v)1592 787 y FG(\()p FD( )s(;)28 b FG(\()p FD(p)1806 753 y FC(2)1806 808 y(1)1862 787 y FG(+)18 b FD(p)1987 753 y FC(2)1987 808 y(2)2024 787 y FG(\))c FD( )s FG(\))47 b FA(\024)2326 731 y FG(1)p 2326 768 42 4 v 2326 844 a(2)2401 729 y FA(p)p 2470 729 166 4 v 58 x FD(c)2506 799 y FB(w)2560 787 y FD(c)2596 799 y FB(v)2658 787 y FG(\()p FD( )s(;)14 b(T)2845 753 y FC(2)2895 787 y FD( )s FG(\))p FD(:)70 b FG(\(I)r(I.6.34\))486 933 y(Th)n(us,)27 b(collecting)g(results,)524 1073 y(\()p FD( )s(;)14 b(W)740 1038 y FC(2)791 1073 y FD( )s FG(\))46 b FA(\024)g FG(\()p FD(c)1105 1085 y FB(w)1191 1073 y FG(+)32 b(2)p FD(e)1369 1038 y FC(4)1438 1073 y FG(+)g(2)p FD(c)1613 1085 y FB(s)1680 1073 y FG(+)g(4)p FD(e)1858 1038 y FC(2)1894 1015 y FA(p)p 1963 1015 90 4 v 58 x FD(c)1999 1085 y FB(w)2085 1073 y FG(+)g(4)2224 1015 y FA(p)p 2293 1015 162 4 v 58 x FD(c)2329 1085 y FB(w)2383 1073 y FD(c)2419 1085 y FB(s)2486 1073 y FG(+)g(4)p FD(e)2664 1038 y FC(2)2701 1015 y FA(p)p 2770 1015 72 4 v 58 x FD(c)2806 1085 y FB(s)2855 1073 y FG(\))23 b(\()p FD( )s(;)14 b(T)3097 1038 y FC(2)3148 1073 y FD( )s FG(\))1651 1220 y(=:)23 b FD(c)1798 1186 y FC(2)1798 1241 y(0)1858 1220 y FA(k)p FD(T)i( )s FA(k)2073 1186 y FC(2)3077 1220 y FG(\(I)r(I.6.35\))386 1346 y(with)c FD(c)604 1358 y FC(0)664 1346 y FD(<)i FG(1)43 b(for)21 b FD(\015)27 b(<)c FG(0)p FD(:)p FG(654)44 b(\()p FD(Z)29 b FA(\024)23 b FG(89\))p FD(:)d FG(Hence)h(the)g(b)r(oundedness)g(of)f(\()p FD(T)c FG(+)5 b FD(\026)p FG(\)\()p FD(H)3017 1316 y FC(\(2\))3111 1346 y FG(+)g FD(\026)p 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FC(\(2\))2986 5434 y FG(\()p FD(m)j FA(\001)f FD(\022)r FG(\)\))3256 5400 y Fz(\000)p FC(1)1498 5623 y FG(=)46 b FD(T)24 b FG(\()p FD(H)1790 5588 y FC(\(2\))1880 5623 y FG(\))1912 5588 y Fz(\000)p FC(1)2001 5623 y FG(\()p FD(m)19 b FA(\001)f FD(\022)r FG(\))p FD(:)815 b FG(\(I)r(I.6.39\))p eop %%Page: 74 80 74 79 bop 386 259 a FC(74)386 459 y FG(Hence)34 b(our)f(op)r(erator)f (is)i(dilational)f(in)n(v)-5 b(arian)n(t)33 b(with)h FD(m)g FG(b)r(eing)g(absorb)r(ed)f(in)h(the)g(dilation)386 558 y(parameter)27 b FD(\022)825 528 y Fz(0)873 558 y FG(:=)d FD(m)19 b FA(\001)g FD(\022)r(:)53 b FG(Since)28 b(dilation)g(as)g(a)g(unitary)g(transformation)f(do)r(es)h(not)g(c)n (hange)386 658 y(the)f(norm,)f(w)n(e)g(get)g FA(k)p FD(T)e FG(\()p FD(H)1248 628 y FC(\(2\))1338 658 y FG(\))1370 628 y Fz(\000)p FC(1)1459 658 y FG(\()p FD(m)p FG(\))p FA(k)37 b FG(=)23 b FA(k)p FD(T)h FG(\()p FD(H)1986 628 y FC(\(2\))2075 658 y FG(\))2107 628 y Fz(\000)p FC(1)2197 658 y FG(\()p FD(m)f FG(=)f(0\))p FA(k)37 b(\024)22 b FD(c)50 b FG(with)27 b(the)g(constan)n(t)386 758 y FD(c)h FG(from)f(the)h FD(m)23 b FG(=)f(0)28 b(estimate)f(giv)n(en)g(ab)r(o)n (v)n(e.)386 975 y Fn(c\))j(Bounde)l(dness)g(of)h FG(\()p FD(H)1180 987 y FB(v)1238 975 y FG(+)18 b FD(\026)p FG(\))1403 945 y Fz(\000)p FC(1)1492 975 y FG(\()p FD(T)30 b FG(+)18 b FD(\026)p FG(\))486 1122 y(F)-7 b(rom)21 b(Lemma)h(I.4)g(w)n(e)f(ha)n (v)n(e)g FA(k)p FG(\()p FD(H)1551 1134 y FB(v)1598 1122 y FG(+)7 b FD(\026)p FG(\))1752 1092 y Fz(\000)p FC(1)1855 1122 y FG(\()p FD(T)18 b FG(+)7 b FD(\026)p FG(\))p FA(k)37 b(\024)23 b FD(c)45 b FG(if)22 b FA(k)p FG(\()p FD(T)c FG(+)7 b FD(\026)p FG(\))14 b(\()p FD(H)2836 1134 y FB(v)2883 1122 y FG(+)7 b FD(\026)p FG(\))3037 1092 y Fz(\000)p FC(1)3127 1122 y FA(k)36 b(\024)23 b FD(c:)386 1222 y FG(Ho)n(w)n(ev)n(er,)j(for)h FD(\015)h(<)23 b FG(0)p FD(:)p FG(654,)j(the)i(latter)f(follo)n(ws)g(immediately)g(from)h(the)g (pro)r(of)f(of)h(b)r(ounded-)386 1333 y(ness)19 b(of)h(\()p FD(T)15 b FG(+)s 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FD(x)650 1996 y FC(1)717 1984 y FG(and)g FD(x)926 1996 y FC(2)992 1984 y FG(go)f(to)h(in\014nit)n(y)-7 b(,)29 b(k)n(eeping)f(sim)n (ultaneously)f Fv(x)2423 1996 y FC(1)2485 1984 y FG(=)d Fv(x)2624 1996 y FC(2)2690 1984 y FG(\(Reed-Simon)29 b(1978,)386 2084 y(p.120\).)486 2302 y(Concerning)d(the)h(absence)g(of) g(eigen)n(v)-5 b(alues)26 b(in)h(the)h(massless)d(case,)i(the)g(same)g (holds)g(true)386 2401 y(as)g(in)h(the)g(one-particle)e(case.)386 2518 y Fv(Theorem)k(I)s(I.2)h FG(\(Absence)d(of)g(eigen)n(v)-5 b(alues)26 b(in)i(massless)f(case\))p Fv(.)386 2618 y Fn(L)l(et)37 b FD(m)g FG(=)g(0)g Fn(and)h(the)f(critic)l(al)i(p)l (otential)f(str)l(ength)f FD(\015)2142 2630 y FC(0)2216 2618 y FG(=)g(0)p FD(:)p FG(986)f Fn(as)i(in)f(Pr)l(op)l(osition)i (II.3.)386 2718 y(Then)30 b(the)g(sp)l(e)l(ctrum)f(of)i FD(H)1263 2687 y FC(\(2\))1381 2718 y Fn(has)g(no)f(eigenvalues)h(for)f FD(\015)e(<)23 b(\015)2418 2730 y FC(0)2455 2718 y FD(:)386 2906 y Fn(Pr)l(o)l(of)p FG(.)107 b(W)-7 b(e)28 b(sho)n(w)g(that)g (\(i\),)i FD(H)1454 2875 y FC(\(2\))1571 2906 y FG(has)e(no)g(eigen)n (v)-5 b(alues)27 b FA(6)p FG(=)d(0)p FD(;)52 b FG(and)28 b(\(ii\))h(that)g FD(E)g FG(=)24 b(0)k(is)g(no)386 3005 y(eigen)n(v)-5 b(alue.)486 3105 y(F)e(or)33 b(\(i\))h(w)n(e)g(argue)e (as)h(in)i(the)f(one-particle)e(case)h(\(Theorem)h(I.3\(iii\)\).)56 b(F)-7 b(rom)33 b(\(I)r(I.6.38\))386 3208 y(w)n(e)42 b(kno)n(w)h(that)g FD(H)1027 3177 y FC(\(2\))1159 3208 y FG(scales)f(under)g(dilations)h(according)e(to)i FD(d)2558 3220 y FB(\022)2596 3208 y FD(H)2672 3177 y FC(\(2\))2761 3208 y FG(\()p FD(m)48 b FG(=)h(0\))14 b FD(d)3159 3172 y Fz(\000)p FC(1)3159 3233 y FB(\022)3310 3208 y FG(=)396 3282 y FC(1)p 396 3296 34 4 v 396 3343 a FB(\022)453 3315 y FD(H)529 3284 y FC(\(2\))618 3315 y FG(\()p FD(m)31 b FG(=)f(0\))p FD(:)62 b FG(Therefore,)32 b(assume)g FD( )j FG(is)d(an)g(eigenfunction)g(of)g FD(H)2691 3284 y FC(\(2\))2812 3315 y FG(with)h(eigen)n(v)-5 b(alue)386 3425 y FD(E)49 b FA(6)p FG(=)43 b(0)p FD(;)c FG(then)i FD(d)953 3437 y FB(\022)991 3425 y FD( )i FG(is)c(also)g(an)h (eigenfunction)g(of)g FD(H)2195 3395 y FC(\(2\))2324 3425 y FG(\(with)g(eigen)n(v)-5 b(alue)39 b FD(\022)r(E)5 b FG(\))p FD(:)85 b FG(This)386 3525 y(con)n(tradicts)26 b(the)i(separabilit)n(y)e(of)i(the)g(Hilb)r(ert)g(space)f(\()p FD(H)2251 3540 y FC(1)p FB(=)p FC(2)2356 3525 y FG(\()p Fx(R)p FG(\))2474 3495 y FC(3)2536 3525 y FA(\002)18 b Fx(C)2673 3495 y FC(4)2716 3525 y FG(\))2748 3495 y FC(2)2786 3525 y FD(:)486 3625 y FG(F)-7 b(or)36 b(\(ii\))i(w)n(e)f (also)f(follo)n(w)g(the)i(pro)r(of)e(of)h(Theorem)f(I.3\(iii\).)66 b(Assume)37 b(there)g(exists)g(an)386 3728 y(eigenfunction)f FD( )41 b FA(6)p FG(=)c(0)e(to)i FD(H)1365 3698 y FC(\(2\))1490 3728 y FG(with)f(eigen)n(v)-5 b(alue)36 b FD(E)42 b FG(=)37 b(0)p FD(:)f FG(Then)g(from)g(\(I)r(I.5.61\))g(and)g(a)386 3827 y(partial)29 b(w)n(a)n(v)n(e)g(analysis,)h(together)f(with)i(the)g (Mellin)f(transform)f(prop)r(erties)h(giv)n(en)f(in)i(BSS)386 3927 y(\(2002\),)26 b(see)h(App)r(endix)i(C)f(for)f(a)g(generalisation) e(to)j(the)g(t)n(w)n(o-particle)e(case,)g(w)n(e)i(ha)n(v)n(e)707 4115 y(0)46 b(=)g(\()p FD( )s(;)14 b(H)1108 4080 y FC(\(2\))1211 4115 y FD( )s FG(\))46 b FA(\025)1457 3998 y Fu(\022)1518 4115 y FG(1)32 b FA(\000)g FD(\015)18 b FG(\()1847 4058 y(1)p 1792 4096 151 4 v 1792 4172 a FD(\015)1835 4184 y FB(B)s(R)1985 4115 y FG(+)32 b FD(e)2121 4080 y FC(2)2181 4058 y FD(\031)2231 4028 y FC(2)p 2181 4096 88 4 v 2204 4172 a FG(4)2293 4115 y(\))42 b(+)31 b FD(d)14 b(\015)2568 4080 y FC(2)2605 3998 y Fu(\023)2703 4115 y FG(\()p FD( )s(;)g FG(2)p FD(p)2913 4127 y FC(1)2964 4115 y FD( )s FG(\))513 4360 y(=)46 b(2)693 4247 y Fu(Z)790 4360 y FD(d)p Fv(p)886 4372 y FC(2)960 4282 y Fu(X)1002 4456 y FB(\027)1094 4247 y Fu(Z)1177 4268 y Fz(1)1140 4436 y(\0001)1276 4360 y FD(dt)1386 4290 y Fu(\014)1386 4340 y(\014)1414 4360 y FD(a)1458 4326 y FC(#)1458 4381 y FB(\027;p)1545 4389 y Ft(2)1582 4360 y FG(\()p FD(t)19 b FG(+)f FD(i=)p FG(2\))1891 4290 y Fu(\014)1891 4340 y(\014)1917 4306 y FC(2)1992 4243 y Fu(\022)2053 4360 y FG(1)31 b FA(\000)h FD(\015)19 b FG(\()2381 4304 y(1)p 2327 4341 151 4 v 2327 4417 a FD(\015)2370 4429 y FB(B)s(R)2520 4360 y FG(+)32 b FD(e)2656 4326 y FC(2)2716 4304 y FD(\031)2766 4274 y FC(2)p 2716 4341 88 4 v 2739 4417 a FG(4)2828 4360 y(\))41 b(+)g FD(d)14 b(\015)3112 4326 y FC(2)3149 4243 y Fu(\023)3224 4360 y FD(:)3077 4535 y FG(\(I)r(I.6.40\))386 4635 y(If)24 b FD(\015)k(<)23 b FG(0)p FD(:)p FG(986)p FD(;)f FG(the)j(factor)e(in)h (brac)n(k)n(ets)f(is)h(strictly)f(p)r(ositiv)n(e)h(and)g(hence)g FA(j)p FD(a)2766 4605 y FC(#)2766 4656 y FB(\027;p)2853 4664 y Ft(2)2890 4635 y FG(\()p FD(t)11 b FG(+)3055 4602 y FB(i)p 3050 4616 34 4 v 3050 4664 a FC(2)3093 4635 y FG(\))p FA(j)3148 4605 y FC(2)3223 4635 y FG(=)22 b(0)386 4735 y(almost)k(ev)n(erywhere.)34 b(The)26 b(remaining)g(part)f(of)i (the)f(pro)r(of)g(can)g(b)r(e)g(copied)g(from)g(the)h(one)f(of)386 4834 y(Theorem)h(I.3\(iii\),)h(to)f(sho)n(w)g(that)h FD( )e FG(=)d(0)p FD(;)k FG(a)g(con)n(tradiction.)948 b Fm(\004)p eop %%Page: 75 81 75 80 bop 3309 259 a FC(75)386 459 y Fq(I)s(I)s(I.)37 b(Outlo)s(ok:)49 b(The)38 b(general)f Fg(N)11 b Fq(-particle)36 b(case)486 676 y FG(F)-7 b(or)33 b FD(N)43 b FG(fermionic)33 b(particles,)i(w)n(e)f(de\014ne)g(the)g(p)r(ositiv)n(e)g(sp)r(ectral)f (subspace)h(of)g(the)g(free)386 791 y FD(N)9 b FG(-particle)24 b(Dirac)g(op)r(erator)f(b)n(y)h(means)h(of)f FA(H)1866 803 y FC(+)p FB(;N)2023 791 y FG(:=)37 b(\003)2206 803 y FC(+)p FB(;N)2353 791 y FA(A)2433 723 y Fu(\000)2471 791 y FD(H)2540 806 y FC(1)p FB(=)p FC(2)2644 791 y FG(\()p Fx(R)2731 761 y FC(3)2774 791 y FG(\))19 b FA(\002)f Fx(C)2962 761 y FC(4)3005 723 y Fu(\001)3043 741 y FB(N)3120 791 y FD(;)48 b FG(to)24 b(b)r(e)386 898 y(considered)30 b(as)h(subspace)f(of)h(the)h(Hilb)r(ert)g(space)e(\003)2070 910 y FC(+)p FB(;N)2203 898 y FA(A)p FG(\()p FD(L)2358 910 y FC(2)2396 898 y FG(\()p Fx(R)2482 868 y FC(3)2525 898 y FG(\))21 b FA(\002)g Fx(C)2718 868 y FC(4)2761 898 y FG(\))2793 868 y FB(N)2856 898 y FD(;)61 b FG(where)30 b(\003)3241 910 y FC(+)p FB(;N)386 998 y FG(is)d(the)h(pro)r(duct)g(of) g(the)f(free)h(pro)5 b(jectors)26 b(for)h(eac)n(h)f(particle,)1473 1152 y(\003)1531 1164 y FC(+)p FB(;N)1687 1152 y FG(:=)46 b(\003)1879 1109 y FC(\(1\))1879 1173 y(+)1982 1152 y FA(\001)14 b(\001)g(\001)f FG(\003)2150 1109 y FC(\()p FB(N)6 b FC(\))2150 1173 y(+)2265 1152 y FD(:)863 b FG(\(I)r(I)r(I.1\)) 386 1301 y(and)40 b(the)g(spin)g(of)g(eac)n(h)g(particle)f(is)h (assumed)f(to)h(b)r(e)2213 1268 y FC(1)p 2213 1282 34 4 v 2213 1330 a(2)2256 1301 y FD(:)84 b FG(A)40 b(state)g FD( )47 b FA(2)d(H)2951 1313 y FC(+)p FB(;N)3125 1301 y FG(can)39 b(b)r(e)386 1401 y(considered)33 b(as)g(a)g(sup)r(erp)r (osition)g(of)h(Slater)f(determinan)n(ts)h(\(see)f(section)h(I)r (I.1\),)h(suc)n(h)f(that)386 1557 y(eac)n(h)j(summand)h(of)f FD( )k FG(con)n(tains)c(a)g(pro)r(duct)g(of)h FD(N)47 b FG(one-particle)36 b(states,)2898 1478 y FB(N)2888 1494 y Fu(Q)2873 1630 y FB(i)p FC(=1)2995 1557 y FD(')3049 1514 y FC(\()p FB(\013)p FC(\))3049 1585 y FB(\033)r FC(\()p FB(i)p FC(\))3169 1557 y FG(\()p Fv(x)3251 1569 y FB(i)3280 1557 y FG(\))p FD(;)386 1704 y FG(where)31 b FD(\033)i FA(2)d FD(S)846 1716 y FB(N)940 1704 y FG(is)i(a)f(p)r(erm) n(utation)g(and)h FD(\013)g FG(en)n(umerates)e(the)i(basis)f(of)g(the)h (single-particle)386 1804 y(Hamiltonian)26 b FD(h)910 1774 y FC(\()p FB(k)q FC(\))1002 1804 y FD(:)49 b FG(Since)27 b(op)r(erators)d(acting)h(on)h(distinct)g(particles)g(comm)n(ute,)g (the)g(ab)r(o)n(v)n(e)386 1903 y(de\014nition)39 b(of)g(our)g(Hilb)r (ert)g(space)g(agrees)e(with)i(the)h(con)n(v)n(en)n(tional)d (de\014nition)i FA(H)3120 1915 y FC(+)p FB(;N)3310 1903 y FG(=)386 2005 y(\003)444 1975 y FB(N)444 2026 y(n)p FC(=1)573 2005 y FA(H)643 2017 y FC(+)757 2005 y FG(where)30 b(\003)1058 1975 y FB(N)1058 2026 y(n)p FC(=1)1218 2005 y FG(sym)n(b)r(olises)f(the)i FD(N)9 b FG(-fold)30 b(an)n(tisymmetric)g (tensor)g(pro)r(duct)h(of)f(the)386 2105 y(single-particle)35 b(Hilb)r(ert)i(spaces)f FA(H)1555 2117 y FC(+)1648 2105 y FG(:=)i(\003)1832 2117 y FC(+)1901 2105 y FG(\()p FD(H)2002 2120 y FC(1)p FB(=)p FC(2)2106 2105 y FG(\()p Fx(R)2193 2075 y FC(3)2236 2105 y FG(\))25 b FA(\002)f Fx(C)2436 2075 y FC(4)2479 2105 y FG(\))75 b(\(see)37 b(e.g.)63 b(Ho)r(ev)n(er)36 b(and)386 2229 y(Sieden)n(top)27 b(1999\).)36 b(In)28 b(particular,)e(one)h(has)g(\003)1893 2186 y FC(\()p FB(k)q FC(\))1893 2250 y(+)1986 2229 y FD( )f FG(=)c FD( )54 b FG(for)27 b(all)g FD(k)f FG(=)d(1)p FD(;)14 b(:::;)g(N)t(:)486 2329 y FG(The)34 b(transformation)f(sc)n (heme)h(for)f(the)i(Coulom)n(b-Dirac)e(op)r(erator)f(\(I)r(I.1\),)37 b(in)n(tro)r(duced)386 2429 y(in)26 b(section)g(I)r(I.2)g(for)g(the)g (t)n(w)n(o-particle)e(case,)i(can)g(readily)f(b)r(e)h(generalised)f(to) h(an)g(atom)f(with)386 2540 y FD(N)38 b FG(electrons,)28 b FD(N)34 b(>)25 b FG(2)p FD(:)53 b FG(Iterating)28 b(the)i(represen)n (tation)d(\(I)r(I.2.5\))i(of)f(the)i(pro)5 b(jector)27 b FD(P)3154 2497 y FC(\()p FB(k)q FC(\))3142 2560 y(+)3275 2540 y FG(for)386 2662 y(particle)g FD(k)s(;)h FG(to)f(get)g FD(P)1087 2619 y FC(\()p FB(k)q FC(\))1075 2683 y(+)1217 2662 y FG(=)22 b(\003)1362 2619 y FC(\()p FB(k)q FC(\))1362 2683 y(+)1473 2662 y FG(+)c FD(F)1621 2619 y FC(\()p FB(k)q FC(\))1609 2684 y(0)1746 2662 y FG(+)g FD(R)q FG(\(\()p FD(e)1996 2632 y FC(2)2033 2662 y FG(\))2065 2632 y FC(2)2103 2662 y FG(\))p FD(;)51 b FG(and)27 b(de\014ning)h(the) g(pro)r(ducts)661 2828 y FA(F)721 2840 y FB(k)784 2828 y FG(:=)46 b(\003)976 2785 y FC(\(1\))976 2849 y(+)1078 2828 y FA(\001)14 b(\001)g(\001)g FG(\003)1247 2785 y FC(\()p FB(k)q Fz(\000)p FC(1\))1247 2849 y(+)1448 2828 y FD(F)1513 2785 y FC(\()p FB(k)q FC(\))1501 2850 y(0)1628 2828 y FG(\003)1686 2785 y FC(\()p FB(k)q FC(+1\))1686 2849 y(+)1877 2828 y FA(\001)g(\001)g(\001)f FG(\003)2045 2785 y FC(\()p FB(N)6 b FC(\))2045 2849 y(+)2160 2828 y FD(;)180 b(k)26 b FA(2)d(f)p FG(1)p FD(;)14 b(:::;)g(N)9 b FA(g)p FD(;)273 b FG(\(I)r(I)r(I.2\))386 2971 y(the)28 b(t)n(w)n(o-particle)e(in)n(teraction)g(term)i(of)g(\(I)r(I.1\))g(is)f (expanded)g(according)f(to)954 3194 y FD(P)1007 3206 y FC(+)p FB(;N)1213 3091 y(N)1182 3116 y Fu(X)1177 3294 y FB(nn)2015 676 y FG(\()p FD( )s(;)28 b FG(\()p FD(V)2254 642 y FC(\()p FB(nk)q FC(\))2420 676 y FG(+)k FD(C)2582 642 y FC(\()p FB(nk)q FC(\))2716 676 y FG(\))23 b FD( )s FG(\))291 b(\(I)r(I)r(I.9\))386 941 y(where)25 b(the)g(op)r(erators)f(on)h(the)g (r.h.s.)36 b(of)25 b(\(I)r(I)r(I.6\))h(and)f(\(I)r(I)r(I.8\))h(are)e (abbreviated)h(b)n(y)g FD(B)3118 898 y FC(\()p FB(k)q FC(\))3114 951 y FB(m)3211 941 y FG(,)g(the)386 1044 y(Jansen-Hess)20 b(op)r(erator)h(of)h(particle)f FD(k)s FG(,)i(and)f FD(C)1872 1014 y FC(\()p FB(nk)q FC(\))2006 1044 y FG(,)i(the)e(second-order)e(in)n(teraction)h(b)r(et)n(w)n(een) 386 1144 y(particles)27 b FD(n)g FG(and)h FD(k)s FG(.)486 1243 y(F)-7 b(rom)27 b(Prop)r(osition)f(I.5)h(w)n(e)g(ha)n(v)n(e)1316 1412 y(\()p FD( )s(;)14 b(B)1509 1378 y FC(\()p FB(k)q FC(\))1505 1432 y FB(m)1625 1412 y FD( )s FG(\))46 b FA(\025)g FD(c)p FG(\()p FD(\015)5 b FG(\))23 b(\()p FD( )s(;)14 b(D)2239 1369 y FC(\()p FB(k)q FC(\))2237 1434 y(0)2355 1412 y FD( )s FG(\))666 b(\(I)r(I)r(I.10\))386 1589 y(with)21 b FD(c)p FG(\()p FD(\015)5 b FG(\))37 b(=)23 b(1)5 b FA(\000)1011 1551 y FB(\015)p 966 1569 127 4 v 966 1617 a(\015)1001 1625 y Fp(B)r(R)1122 1589 y FA(\000)1201 1551 y FB(\015)1240 1526 y Ft(2)p 1201 1569 71 4 v 1220 1617 a FC(8)1282 1589 y FG(\()1324 1556 y FB(\031)p 1324 1570 41 4 v 1328 1617 a FC(2)1380 1589 y FA(\000)1463 1556 y FC(2)p 1459 1570 V 1459 1617 a FB(\031)1510 1589 y FG(\))1542 1558 y FC(2)1624 1589 y FG(from)20 b(\(I.4.53\),)h(where)g(w)n(e)f(ha)n(v)n(e)g(used)g(that)h FD( )27 b FA(2)c(H)3241 1601 y FC(+)p FB(;N)386 1723 y FG(suc)n(h)k(that)h FD(E)819 1693 y FC(\()p FB(k)q FC(\))912 1723 y FD( )e FG(=)d FD(D)1151 1680 y FC(\()p FB(k)q FC(\))1149 1746 y(0)1243 1723 y FD( )s(:)486 1823 y FG(Moreo)n(v)n(er,)e(the)i(second-order)e(t)n(w)n(o-particle)h(p)r (oten)n(tial)h(is)g(estimated)g(from)g(Lemma)g(I)r(I.4,)493 2029 y FA(j)p FG(\()p FD( )s(;)14 b(C)707 1994 y FC(\()p FB(nk)q FC(\))855 2029 y FD( )s FG(\))p FA(j)46 b(\024)g FD(\015)5 b(e)1211 1994 y FC(2)1281 1973 y FD(\031)1331 1942 y FC(2)p 1281 2010 88 4 v 1304 2086 a FG(4)1401 2029 y(\()p FD( )s(;)28 b FG(\()p FD(D)1644 1986 y FC(\()p FB(n)p FC(\))1642 2051 y(0)1774 2029 y FG(+)k FD(D)1942 1986 y FC(\()p FB(k)q FC(\))1940 2051 y(0)2035 2029 y FG(\))23 b FD( )s FG(\))46 b(=)g FD(\015)5 b(e)2423 1994 y FC(2)2493 1973 y FD(\031)2543 1942 y FC(2)p 2493 2010 V 2516 2086 a FG(2)2613 2029 y(\()p FD( )s(;)14 b(D)2810 1986 y FC(\(1\))2808 2051 y(0)2913 2029 y FD( )s FG(\))108 b(\(I)r(I)r(I.11\))386 2226 y(where)32 b(the)g(symmetry)g(of)p 1270 2158 58 4 v 32 w FD( )s( )j FG(with)e(resp)r(ect)f(to)g(particle)f (exc)n(hange)g(has)g(b)r(een)i(exploited.)386 2377 y(Hence)669 2298 y FB(N)655 2315 y Fu(P)634 2452 y FB(k)q(>n)802 2377 y FG(=)901 2337 y FB(N)6 b FC(\()p FB(N)g Fz(\000)p FC(1\))p 901 2358 255 4 v 1012 2406 a(2)1219 2377 y FG(can)29 b(b)r(e)g(used,)g(together)e(with)i(\()p FD( )s(;)14 b(N)9 b(D)2492 2334 y FC(\(1\))2490 2399 y(0)2596 2377 y FD( )s FG(\))38 b(=)2844 2298 y FB(N)2829 2315 y Fu(P)2813 2452 y FB(k)q FC(=1)2934 2377 y FG(\()p FD( )s(;)14 b(D)3131 2334 y FC(\()p FB(k)q FC(\))3129 2399 y(0)3237 2377 y FD( )s FG(\))p FD(:)386 2547 y FG(Finally)27 b(one)h(can)f(tak)n(e)g (adv)-5 b(an)n(tage)26 b(of)h FD(V)1706 2517 y FC(\()p FB(nk)q FC(\))1863 2547 y FA(\025)c FG(0)k(to)g(drop)g(the)h(resp)r (ectiv)n(e)f(term.)486 2646 y(Therefore,)f(the)i(total)f(op)r(erator)f FD(H)1637 2616 y FC(\(2\))1754 2646 y FG(can)h(b)r(e)h(estimated)g(b)n (y)711 2880 y(\()p FD( )s(;)14 b(H)913 2846 y FC(\(2\))1025 2880 y FD( )s FG(\))47 b FA(\025)1302 2776 y FB(N)1272 2801 y Fu(X)1271 2980 y FB(k)q FC(=1)1392 2880 y FG(\()p FD( )s(;)1532 2763 y Fu(\024)1576 2880 y FD(c)p FG(\()p FD(\015)5 b FG(\))32 b FA(\000)g FD(\015)5 b(e)1940 2846 y FC(2)2010 2824 y FD(\031)2060 2794 y FC(2)2097 2824 y FG(\()p FD(N)28 b FA(\000)18 b FG(1\))p 2010 2861 371 4 v 2174 2937 a(4)2404 2763 y Fu(\025)2485 2880 y FD(D)2556 2837 y FC(\()p FB(k)q FC(\))2554 2902 y(0)2672 2880 y FD( )s FG(\))p FD(:)326 b FG(\(I)r(I)r(I.12\))386 3111 y(W)-7 b(e)42 b(ha)n(v)n(e)e(stabilit)n(y)-7 b(,)44 b(ev)n(en)d(p)r (ositivit)n(y)-7 b(,)45 b(for)40 b(all)h FD(\015)46 b FG(for)41 b(whic)n(h)g(the)h(expression)e(in)h(square)386 3211 y(brac)n(k)n(ets)26 b(is)h(nonnegativ)n(e.)486 3311 y(Th)n(us)g(w)n(e)g(ha)n(v)n(e)g(pro)n(v)n(ed)386 3439 y Fv(Prop)s(osition)j(I)s(I)s(I.1)h FG(\(Stabilit)n(y)d(of)g FD(N)9 b FG(-electron)26 b(ions)h(and)h(atoms\))p Fv(.)386 3542 y Fn(L)l(et)k FD(\015)i FG(=)28 b FD(Z)6 b(e)804 3512 y FC(2)841 3542 y FD(;)56 b(N)38 b FA(\024)28 b FD(Z)39 b Fn(b)l(e)33 b(the)g(numb)l(er)f(of)i(ele)l(ctr)l(ons,)g(and)f (let)g FD(H)2592 3512 y FC(\(2\))2700 3542 y Fn(b)l(e)g(the)g(tr)l (ansforme)l(d)386 3641 y FD(N)9 b Fn(-p)l(article)30 b(Coulomb-Dir)l(ac)h(op)l(er)l(ator)g(up)f(to)f(se)l(c)l(ond)h(or)l (der)h(in)f(the)g(p)l(otential)g(str)l(ength)f FD(\015)5 b(:)386 3741 y Fn(Then)1281 3863 y FD(H)1357 3829 y FC(\(2\))1492 3863 y FA(\025)46 b FG(0)368 b Fn(for)54 b FD(\015)d FA(\024)46 b FD(\015)2417 3875 y FB(N)3110 3863 y FG(\(I)r(I)r(I.13\)) 386 4003 y Fn(with)30 b FD(\015)609 4015 y FB(N)702 4003 y Fn(the)g(smal)t(lest)g(solution)g(to)767 4222 y FG(1)41 b FA(\000)1018 4166 y FD(\015)p 966 4203 151 4 v 966 4279 a(\015)1009 4291 y FB(B)s(R)1168 4222 y FA(\000)1284 4166 y FD(\015)1332 4136 y FC(2)p 1284 4203 85 4 v 1306 4279 a FG(8)1416 4105 y Fu(\022)1487 4166 y FD(\031)p 1487 4203 51 4 v 1491 4279 a FG(2)1580 4222 y FA(\000)1691 4166 y FG(2)p 1686 4203 V 1686 4279 a FD(\031)1747 4105 y Fu(\023)1808 4122 y FC(2)1886 4222 y FA(\000)h FD(\015)5 b(e)2080 4188 y FC(2)2149 4166 y FD(\031)2199 4136 y FC(2)2237 4166 y FG(\()p FD(N)27 b FA(\000)18 b FG(1\))p 2149 4203 371 4 v 2314 4279 a(4)2576 4222 y(=)46 b(0)381 b(\(I)r(I)r(I.14\))386 4437 y Fn(wher)l(e)30 b FD(\015)663 4449 y FB(B)s(R)794 4437 y FG(=)1008 4404 y FC(2)p 891 4418 268 4 v 891 4465 a FB(\031)r(=)p FC(2+2)p FB(=\031)1168 4437 y FD(:)386 4550 y Fn(F)-6 b(or)30 b(neutr)l(al)f(atoms)h FG(\()p FD(N)i FG(=)23 b FD(Z)6 b FG(\))29 b Fn(we)h(have)h(stability)g (for)g FD(\015)c FA(\024)c FG(0)p FD(:)p FG(446)44 b(\()p FD(Z)29 b FA(\024)23 b FG(61\))p FD(:)486 4679 y FG(Since)31 b(the)h(l.h.s.)48 b(of)32 b(\(I)r(I)r(I.14\))f(is)h(decreasing)d(b)r (oth)j(with)g FD(N)40 b FG(and)31 b FD(\015)5 b FG(,)32 b(it)g(follo)n(ws)f(that)g(for)386 4778 y FD(\015)d FA(\024)22 b FG(0)p FD(:)p FG(446)p FD(;)k FG(w)n(e)h(ha)n(v)n(e)g(stabilit)n(y)g (for)g(all)g(ions)h(with)g FD(N)j FA(\024)23 b FD(Z)q(:)486 4878 y FG(Alternativ)n(ely)-7 b(,)23 b(one)h(migh)n(t)f(think)h(of)g (estimating)f FD(C)2156 4848 y FC(\()p FB(nk)q FC(\))2314 4878 y FG(b)n(y)g FD(V)2492 4848 y FC(\()p FB(nk)q FC(\))2650 4878 y FG(via)g(Prop)r(osition)f(I)r(I.2.)386 4978 y(Ho)n(w)n(ev)n(er,) 30 b(ev)n(en)h(with)h(the)f(conjecture)g FD(C)36 b FG(=)28 b(1)p FD(;)j FG(one)g(w)n(ould)g(need)g(the)h(restriction)e FD(\015)k FA(\024)3299 4945 y FC(1)p 3295 4959 41 4 v 3295 5006 a FB(\031)386 5077 y FG(\(i.e.)i FD(Z)28 b FA(\024)23 b FG(43\))46 b(for)22 b(a)h(p)r(ositiv)n(e)g(t)n(w)n (o-particle)e(in)n(teraction)i(term.)35 b(This)23 b(b)r(ound)h(is)f(lo) n(w)n(er)f(than)386 5177 y(the)28 b(ab)r(o)n(v)n(e)e(v)-5 b(alue,)28 b FD(Z)g FA(\024)23 b FG(61)p FD(:)486 5324 y FG(W)-7 b(e)23 b(remark)f(that)i(stabilit)n(y)f(of)g(matter)g(for)g (the)g(Bro)n(wn-Ra)n(v)n(enhall)e(op)r(erator)g(w)n(as)i(sho)n(wn)386 5423 y(b)n(y)35 b(Balinsky)f(and)h(Ev)-5 b(ans)35 b(\(1999\))f(in)i (the)f(case)g(of)g FD(K)41 b FG(n)n(uclei)35 b(and)g(one)g(electron,)i (and)e(b)n(y)386 5523 y(Ho)r(ev)n(er)j(and)h(Sieden)n(top)g(\(1999\))f (in)i(the)f(case)f(of)i FD(K)k FG(n)n(uclei)39 b(and)g FD(N)48 b FG(electrons)39 b(\(at)g(zero)386 5623 y(mass\))27 b(for)g FD(\015)h FA(\024)22 b FD(\015)950 5635 y FB(B)s(R)1104 5623 y FG(\()p FD(Z)29 b FA(\024)23 b FG(124\))49 b(and)28 b FD(\015)f FA(\024)c FG(0)p FD(:)p FG(64)45 b(\()p FD(Z)29 b FA(\024)22 b FG(88\))p FD(;)50 b FG(resp)r(ectiv)n(ely)-7 b(.)p eop %%Page: 78 84 78 83 bop 386 259 a FC(78)486 459 y FG(The)36 b(con)n(tribution)f(of)h (t)n(w)n(o-particle)f(second-order)e(terms)j(\(neglected)h(in)f(the)g (Bro)n(wn-)386 558 y(Ra)n(v)n(enhall)26 b(op)r(erator\))g(is)i(usually) f(considered)g(to)g(b)r(e)h(unimp)r(ortan)n(t)g(b)r(ecause)f(of)g(the)h (small-)386 658 y(ness)e(of)h(their)f(coupling)g(constan)n(t)g FD(e)1556 628 y FC(2)1643 658 y FG(\(with)h(resp)r(ect)g(to)f FD(\015)5 b FG(\))p FD(:)50 b FG(Ho)n(w)n(ev)n(er,)25 b(since)h(these)h(terms)386 758 y(tend)d(to)f(b)r(e)g(negativ)n(e)g (\(whic)n(h)g(w)n(e)g(conjecture)g(from)g(the)g(negativit)n(y)g(prop)r (ert)n(y)f(of)h(the)h(k)n(ernel)386 860 y(of)e FD(C)540 830 y FC(\()p FB(nk)q FC(\))674 860 y FG(,)i(see)e(\(I)r(I.4.10\)\),)i (and)e(since)g(they)h(o)r(ccur)f(with)h(a)f(w)n(eigh)n(t)g(prop)r (ortional)f(to)h FD(N)3133 830 y FC(2)3170 860 y FD(;)h FG(they)386 960 y(will)j(coun)n(teract)e(stabilit)n(y)h(for)g(large)f FD(N)34 b FG(in)26 b(a)f(non-negligible)f(w)n(a)n(y)-7 b(.)35 b(That)26 b(b)r(ecomes)f(eviden)n(t)386 1060 y(from)f(the)h (large)e(reduction)h(of)g(the)h(critical)e(p)r(oten)n(tial)i(strength)f (for)f(atoms)h(as)g(compared)f(to)386 1159 y(one-)k(or)g(t)n(w)n (o-electron)e(ions)i(found)h(in)g(this)g(w)n(ork.)486 1731 y FE(Ac)m(kno)m(wledgmen)m(t)486 1949 y FG(It)22 b(is)h(a)f(pleasure)f(to)h(thank)h(Heinz)f(Sieden)n(top)h(for)f (directing)f(m)n(y)i(in)n(terest)f(to)g(this)h(in)n(trigu-)386 2049 y(ing)g(topic,)i(for)e(incorp)r(orating)f(me)i(in)f(his)h(researc) n(h)e(group,)h(and)g(for)h(his)f(con)n(tin)n(ual)g(help)h(and)386 2148 y(supp)r(ort)j(during)g(the)h(course)f(of)g(this)h(w)n(ork.)486 2295 y(I)19 b(should)g(also)g(lik)n(e)g(to)g(thank)g(Hub)r(ert)h(Kalf)f (for)g(his)h(b)r(eautiful)g(lecture)f(on)h(sp)r(ectral)e(theory)-7 b(,)386 2395 y(for)25 b(his)h(in)n(terest)f(in)h(m)n(y)f(w)n(ork)f(and) i(for)f(his)g(willingness)g(and)h(great)e(patience)h(to)h(discuss)f(m)n (y)386 2494 y(problems.)486 2641 y(In)g(particular)f(I)i(wish)f(to)h (thank)f(Alex)h(Sob)r(olev)f(for)g(in)n(tro)r(ducing)f(me)i(in)n(to)f (the)h(theory)f(of)386 2741 y(pseudo)r(di\013eren)n(tial)c(op)r (erators)e(and)i(for)g(guiding)g(me)h(patien)n(tly)f(in)h(\014nding)f (suitable)g(unitary)386 2841 y(transformation)26 b(sc)n(hemes)h(during) g(m)n(y)g(sta)n(ys)g(at)g(the)h(Univ)n(ersit)n(y)f(of)g(Sussex)h(in)f (2002.)486 2987 y(I)e(also)f(wish)i(to)f(thank)g(Alexei)h(Ian)n(tc)n (henk)n(o)e(for)h(a)g(fruitful)h(collab)r(oration,)e(and)h(for)g(man)n (y)386 3087 y(discussions)h(during)i(his)f(visit)h(in)g(Munic)n(h)g(in) f(2003.)486 3234 y(I)e(am)g(also)f(grateful)g(to)h(Edgardo)e(Sto)r(c)n (kmey)n(er,)i(Marcel)f(Griesemer)g(and)h(Rup)r(ert)g(Lasser)386 3334 y(for)i(enligh)n(tening)g(discussions.)486 3480 y(I)40 b(greatly)g(appreciate)g(the)h(friendship)g(and)f(op)r(enness)h (of)f(Klaus)g(W)-7 b(ol\013hardt,)44 b(Heinz)386 3580 y(Sieden)n(top,)32 b(Hub)r(ert)g(Kalf,)f(Heiner)g(Steinlein,)i(Helm)n (ut)f(Z)2281 3584 y(\177)2281 3580 y(osc)n(hinger)d(and)i(Otto)g(F)-7 b(orster)30 b(at)386 3680 y(the)h(Munic)n(h)f(Institute)h(of)f (Mathematics)g(and)g(Rup)r(ert)h(Lasser)e(at)h(the)h(Munic)n(h)f(Cen)n (ter)g(of)386 3779 y(Mathematics)25 b(to)g(a)g(ph)n(ysicist)h(who)f (has)g(got)f(an)i(a\013ection)f(for)g(math.)36 b(This)26 b(has)e(created)h(an)386 3879 y(exquisite)i(atmosphere)g(for)g(adv)-5 b(ancing)27 b(in)g(scien)n(ti\014c)h(researc)n(h.)p eop %%Page: 79 85 79 84 bop 3309 259 a FC(79)386 459 y Fq(App)s(endix)38 b(A)386 676 y FE(Compilation)c(of)h(in)m(tegrals)486 832 y FG(Let)681 765 y Fu(R)663 914 y Fy(R)710 898 y Ft(3)760 832 y FD(d)p Fv(p)46 b FG(=)1020 748 y Fz(1)1017 765 y Fu(R)1020 908 y FC(0)1092 832 y FD(p)1134 802 y FC(2)1171 832 y 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5159 y FC(1)2028 5327 y(0)2126 5195 y FD(dy)p 2126 5232 87 4 v 2148 5308 a(y)2246 5251 y(q)2283 5263 y FC(1)2321 5251 y FG(\()p FD(y)s FG(\))37 b(=)45 b(1)p FD(:)501 b FG(\(A.10\))486 5423 y(The)26 b(in)n(tegrals)g(can)g(b)r(e)h(found)g (with)h(the)f(help)g(of)f(Gradsh)n(teyn)g(and)h(Ryzhik)g(\(1965\);)e (the)386 5523 y(in)n(tegrals)i(\(A.4\))h(and)g(\(A.9\))h(are)e(pro)n (vided)g(b)n(y)h(EPS)f(\(1996\),)g(\(A.7\))i(b)n(y)e(BSS)i(\(2002\),)e (while)386 5623 y(\(A.8\))h(is)f(based)h(on)f(an)g(in)n(tegral)g(giv)n (en)f(b)n(y)i(Tix)f(\(1998;)f(see)h(also)g(\(C.6\))h(and)f(\(C.7\)\).)p eop %%Page: 80 86 80 85 bop 386 259 a FC(80)486 459 y FG(W)-7 b(e)25 b(also)g(pro)n(vide) f(some)h(\014nite)h(in)n(tegrals)e(whic)n(h)h(are)f(needed)i(for)f(the) g(pro)r(of)g(of)h(Theorem)386 558 y(I.4.)36 b(F)-7 b(or)27 b FD(a)c(>)g FG(0)p FD(;)417 642 y Fu(Z)500 662 y FB(a)464 831 y FC(0)554 755 y FD(dy)17 b(y)699 721 y FC(5)p FB(=)p FC(2)817 755 y FD(q)854 767 y FC(0)891 755 y FG(\()p FD(y)s FG(\))38 b(=)1148 699 y(2)p 1148 736 42 4 v 1148 812 a(7)1213 638 y Fu(\024)1257 755 y FD(a)1311 698 y Ft(7)p 1311 707 29 3 v 1311 741 a(2)1367 755 y FG(ln)1450 635 y Fu(\014)1450 684 y(\014)1450 734 y(\014)1450 784 y(\014)1488 699 y FG(1)18 b(+)g FD(a)p 1488 736 187 4 v 1488 812 a FG(1)g FA(\000)g FD(a)1685 635 y Fu(\014)1685 684 y(\014)1685 734 y(\014)1685 784 y(\014)1745 755 y FA(\000)g FG(2)c(arctan)2128 691 y FA(p)p 2197 691 44 4 v 64 x FD(a)k FG(+)g(4)2384 691 y FA(p)p 2453 691 V 64 x FD(a)g FG(+)2608 699 y(4)p 2608 736 42 4 v 2608 812 a(5)2674 755 y FD(a)2728 698 y Ft(5)p 2727 707 29 3 v 2727 741 a(2)2788 755 y FA(\000)g FG(ln)2954 635 y Fu(\014)2954 684 y(\014)2954 734 y(\014)2954 784 y(\014)2992 699 y FG(1)g(+)3135 639 y FA(p)p 3204 639 44 4 v 60 x FD(a)p 2992 736 257 4 v 2992 812 a FG(1)g FA(\000)3135 752 y(p)p 3204 752 44 4 v 60 x FD(a)3258 635 y Fu(\014)3258 684 y(\014)3258 734 y(\014)3258 784 y(\014)3299 638 y(\025)3142 904 y FG(\(A.11\))404 1071 y Fu(Z)487 1092 y FB(a)450 1260 y FC(0)541 1184 y FD(dy)25 b(y)694 1150 y FC(3)p FB(=)p FC(2)822 1184 y FD(q)859 1196 y FC(0)896 1184 y FG(\()p FD(y)s FG(\))46 b(=)1171 1128 y(2)p 1171 1165 42 4 v 1171 1241 a(5)1236 1067 y Fu(\024)1280 1184 y FD(a)1324 1150 y FC(5)p FB(=)p FC(2)1442 1184 y FG(ln)1525 1064 y Fu(\014)1525 1114 y(\014)1525 1164 y(\014)1525 1213 y(\014)1563 1128 y FG(1)18 b(+)g FD(a)p 1563 1165 187 4 v 1563 1241 a FG(1)g FA(\000)g FD(a)1760 1064 y Fu(\014)1760 1114 y(\014)1760 1164 y(\014)1760 1213 y(\014)1829 1184 y FG(+)1945 1128 y(4)p 1945 1165 42 4 v 1945 1241 a(3)2019 1184 y FD(a)2063 1150 y FC(3)p FB(=)p FC(2)2209 1184 y FG(+)41 b(2)14 b(arctan)2615 1121 y FA(p)p 2684 1121 44 4 v 63 x FD(a)42 b FA(\000)f FG(ln)2959 1064 y Fu(\014)2959 1114 y(\014)2959 1164 y(\014)2959 1213 y(\014)2997 1068 y FA(p)p 3066 1068 V 60 x FD(a)18 b FG(+)g(1)p 2997 1165 257 4 v 2997 1182 a FA(p)p 3066 1182 44 4 v 60 x FD(a)g FA(\000)g FG(1)3263 1064 y Fu(\014)3263 1114 y(\014)3263 1164 y(\014)3263 1213 y(\014)3313 1067 y(\025)3142 1334 y FG(\(A.12\))443 1501 y Fu(Z)526 1521 y FB(a)489 1690 y FC(0)580 1614 y FD(dy)26 b(y)734 1580 y FC(1)p FB(=)p FC(2)861 1614 y FD(q)898 1626 y FC(0)935 1614 y FG(\()p FD(y)s FG(\))46 b(=)1210 1558 y(2)p 1210 1595 42 4 v 1210 1671 a(3)1275 1497 y Fu(\024)1319 1614 y FD(a)1363 1580 y FC(3)p FB(=)p FC(2)1481 1614 y FG(ln)1564 1494 y Fu(\014)1564 1543 y(\014)1564 1593 y(\014)1564 1643 y(\014)1602 1558 y FG(1)18 b(+)g FD(a)p 1602 1595 187 4 v 1602 1671 a FG(1)g FA(\000)g FD(a)1799 1494 y Fu(\014)1799 1543 y(\014)1799 1593 y(\014)1799 1643 y(\014)1868 1614 y FG(+)41 b(4)2016 1550 y FA(p)p 2085 1550 44 4 v 64 x FD(a)g FA(\000)g FG(2)14 b(arctan)2576 1550 y FA(p)p 2645 1550 V 64 x FD(a)42 b FA(\000)f FG(ln)2920 1494 y Fu(\014)2920 1543 y(\014)2920 1593 y(\014)2920 1643 y(\014)2957 1558 y FG(1)18 b(+)3100 1498 y FA(p)p 3170 1498 V 3170 1558 a FD(a)p 2957 1595 257 4 v 2957 1671 a FG(1)g FA(\000)3100 1611 y(p)p 3170 1611 44 4 v 3170 1671 a FD(a)3223 1494 y Fu(\014)3223 1543 y(\014)3223 1593 y(\014)3223 1643 y(\014)3274 1497 y(\025)3142 1763 y FG(\(A.13\))535 1930 y Fu(Z)618 1951 y FB(a)582 2119 y FC(0)672 2043 y FD(dy)828 1987 y FG(1)p 792 2024 113 4 v 792 2049 a FA(p)p 862 2049 44 4 v 862 2100 a FD(y)938 2043 y(q)975 2055 y FC(0)1013 2043 y FG(\()p FD(y)s FG(\))46 b(=)g(2)1320 1980 y FA(p)p 1388 1980 V 1388 2043 a FD(a)37 b FG(ln)1552 1923 y Fu(\014)1552 1973 y(\014)1552 2023 y(\014)1552 2073 y(\014)1590 1987 y FG(1)18 b(+)g FD(a)p 1590 2024 187 4 v 1590 2100 a FG(1)g FA(\000)g FD(a)1787 1923 y Fu(\014)1787 1973 y(\014)1787 2023 y(\014)1787 2073 y(\014)1856 2043 y FG(+)41 b(4)14 b(arctan)2262 1980 y FA(p)p 2331 1980 44 4 v 63 x FD(a)41 b FA(\000)h FG(2)14 b(ln)2661 1923 y Fu(\014)2661 1973 y(\014)2661 2023 y(\014)2661 2073 y(\014)2699 1927 y FA(p)p 2768 1927 V 60 x FD(a)k FG(+)g(1)p 2699 2024 257 4 v 2699 2041 a FA(p)p 2768 2041 44 4 v 60 x FD(a)g FA(\000)g FG(1)2965 1923 y Fu(\014)2965 1973 y(\014)2965 2023 y(\014)2965 2073 y(\014)3142 2043 y FG(\(A.14\))515 2260 y Fu(Z)598 2281 y Fz(1)561 2449 y FB(a)682 2373 y FD(dy)838 2317 y FG(1)p 802 2354 113 4 v 802 2378 a FA(p)p 871 2378 44 4 v 52 x FD(y)948 2373 y(q)985 2385 y FC(0)1022 2373 y FG(\()p FD(y)s FG(\))46 b(=)g(2)1343 2256 y Fu(\024)1386 2373 y FD(\031)f FA(\000)1584 2309 y(p)p 1653 2309 V 64 x FD(a)14 b FG(ln)1794 2253 y Fu(\014)1794 2302 y(\014)1794 2352 y(\014)1794 2402 y(\014)1832 2317 y FG(1)k(+)g FD(a)p 1832 2354 187 4 v 1832 2430 a FG(1)g FA(\000)g FD(a)2029 2253 y Fu(\014)2029 2302 y(\014)2029 2352 y(\014)2029 2402 y(\014)2098 2373 y FA(\000)41 b FG(2)14 b(arctan)2504 2309 y FA(p)p 2573 2309 44 4 v 64 x FD(a)42 b FG(+)f(ln)2848 2253 y Fu(\014)2848 2302 y(\014)2848 2352 y(\014)2848 2402 y(\014)2885 2257 y FA(p)p 2954 2257 V 60 x FD(a)19 b FG(+)f(1)p 2885 2354 257 4 v 2885 2371 a FA(p)p 2954 2371 44 4 v 59 x FD(a)h FA(\000)f FG(1)3151 2253 y Fu(\014)3151 2302 y(\014)3151 2352 y(\014)3151 2402 y(\014)3202 2256 y(\025)3142 2529 y FG(\(A.15\))529 2696 y Fu(Z)612 2716 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b(fact,)534 5250 y Fu(Z)617 5271 y Fz(1)580 5439 y FC(0)701 5363 y FD(dq)781 5375 y FC(1)941 5307 y FD(\017)975 5277 y FC(2)p 852 5344 250 4 v 852 5420 a FD(q)892 5391 y FC(2)889 5442 y(1)947 5420 y FG(+)18 b FD(\017)1064 5396 y FC(2)1248 5307 y FG(1)p 1144 5344 249 4 v 1144 5420 a FD(q)1181 5432 y FC(1)1237 5420 y FG(+)g FD(m)1449 5363 y FG(=)79 b(lim)1560 5417 y FB(R)p Fz(!1)1872 5307 y FD(\017)1906 5277 y FC(2)p 1766 5344 283 4 v 1766 5420 a FD(m)1839 5396 y FC(2)1895 5420 y FG(+)18 b FD(\017)2012 5396 y FC(2)2073 5250 y Fu(Z)2156 5271 y FB(R)2119 5439 y FC(0)2224 5363 y FD(dq)2304 5375 y FC(1)2355 5246 y Fu(\022)2426 5307 y FA(\000)p FD(q)2528 5319 y FC(1)2584 5307 y FG(+)g FD(m)p 2426 5344 314 4 v 2458 5420 a(q)2498 5391 y FC(2)2495 5442 y(1)2554 5420 y FG(+)g FD(\017)2671 5396 y FC(2)2791 5363 y FG(+)3011 5307 y(1)p 2907 5344 249 4 v 2907 5420 a FD(q)2944 5432 y FC(1)3000 5420 y FG(+)g FD(m)3166 5246 y Fu(\023)489 5592 y FG(=)697 5536 y FD(m\017)p 609 5573 283 4 v 609 5649 a(m)682 5625 y FC(2)738 5649 y FG(+)g FD(\017)855 5625 y FC(2)916 5500 y Fu(\020)995 5536 y FD(\017)p 975 5573 73 4 v 975 5649 a(m)1095 5592 y FG(ln)1208 5536 y FD(\017)p 1188 5573 V 1188 5649 a(m)1312 5592 y FG(+)1428 5536 y FD(\031)p 1428 5573 51 4 v 1432 5649 a FG(2)1489 5500 y Fu(\021)1584 5592 y FA(\024)37 b FD(c)1722 5604 y FC(1)1759 5592 y FD(\017)k FG(+)g FD(c)1976 5604 y FC(2)2013 5592 y FD(\017)2047 5558 y FC(2)2098 5592 y FG(ln)14 b FD(\017)41 b FG(+)h FD(O)r FG(\()p FD(\017)2494 5558 y FC(2)2531 5592 y FG(\))p FD(;)181 b FG(\()p FD(m)23 b FA(6)p FG(=)g(0\))p FD(:)102 b FG(\(D.6\))p eop %%Page: 86 92 86 91 bop 386 259 a FC(86)386 459 y FG(The)40 b(in)n(tegral)g(\(D.6\))h (div)n(erges)e(logarithmically)g(with)i FD(m)f FG(for)g(\014xed)h FD(\017)f FG(as)g FD(m)k FA(!)h FG(0)p FD(;)40 b FG(and)386 558 y(tends)34 b(to)731 526 y FB(\031)p 731 540 41 4 v 735 587 a FC(4)815 558 y FG(if)g FD(m)f FG(=)f FD(\017)g FG(=)1278 526 y FC(1)p 1274 540 42 4 v 1274 587 a FB(n)1357 558 y FA(!)h FG(0)p FD(:)66 b FG(Th)n(us)33 b FD(I)1856 570 y FC(2)1927 558 y FG(is)h(no)f(longer)f(b)r(ounded)i(b)n(y)f FD(\017)g FG(for)g(all)g FD(p)3222 570 y FC(1)3259 558 y FD(;)14 b(p)3338 570 y FC(2)386 658 y FG(when)33 b FD(m)e FA(!)h FG(0)p FD(:)64 b FG(In)33 b(order)e(to)i(include)g(the)g (massless)f(case)f(in)i(the)h(pro)r(of,)f(a)f(more)g(careful)386 758 y(decomp)r(osition)27 b(should)g(replace)g(\(I)r(I.6.15\).)486 857 y(Collecting)g(results)1099 1015 y FA(j)p FG(\()p FD( )s(;)14 b(A)1310 1027 y FB(\017)1356 1015 y FD( )s FG(\))p FA(j)46 b(\024)g FD(\017)23 b FA(k)p FD( )s FA(k)1823 981 y FC(2)1896 1015 y FG(\()p FD(c)33 b FG(+)e FD(C)20 b(\017)14 b FG(ln)g FD(\017)32 b FG(+)g FD(O)r FG(\()p FD(\017)p FG(\)\))1651 1196 y FA(\000)-14 b(!)46 b FG(0)331 b(\()p FD(\017)23 b FA(!)h FG(0)p FD(;)96 b(m)23 b FA(6)p FG(=)g(0\))363 b(\(D.7\))386 1336 y(with)28 b(suitable)g(constan)n(ts)e FD(c;)14 b(C)q(:)486 1447 y FG(In)35 b(a)f(similar)h(w)n(a)n(y)-7 b(,)36 b(the)f(op)r(erator)e(in)n(v)n(olving)h FD(C)2068 1404 y FC(\(12\))2062 1469 y(1)p FB(\017)2191 1447 y FG(\()p FD(f)2264 1459 y FB(\017)2295 1447 y FD(;)14 b(e)2371 1459 y FB(\017)2403 1447 y FG(\))35 b(can)g(b)r(e)g(estimated) h(b)n(y)e(the)386 1564 y(r.h.s.)76 b(of)41 b(\(D.7\),)k(while)c(the)g (third)g(term,)j FD(C)1921 1521 y FC(\(12\))1915 1586 y(1)p FB(\017)2044 1564 y FG(\()p FD(f)2117 1576 y FB(\017)2149 1564 y FD(;)14 b(f)2227 1576 y FB(\017)2258 1564 y FG(\))p FD(;)41 b FG(leads)g(to)f(an)h(estimate)g(whic)n(h)386 1664 y(v)-5 b(anishes)27 b(to)g(higher)g(order)g(in)g FD(\017:)386 1882 y Fo(b\))32 b(Estimate)i(of)f Fk(B)1056 1896 y Fi(\017)486 2028 y FG(W)-7 b(e)28 b(start)f(from)961 2208 y FA(j)p FG(\()p FD( )s(;)14 b(B)1173 2220 y FB(\017)1219 2208 y FD( )s FG(\))p FA(j)46 b(\024)37 b FD(c)1515 2173 y Fz(0)1515 2228 y FC(0)1566 2095 y Fu(Z)1663 2208 y FD(d)p Fv(p)1759 2220 y FC(1)1819 2208 y FD(d)p Fv(p)1915 2220 y FC(2)1976 2208 y FA(j)2016 2186 y FG(^)1999 2208 y FD( )s 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Ft(2)2008 2669 y FC(\))2038 2611 y Fu(\021)2101 2590 y(Z)2198 2703 y FD(d)p Fv(p)2294 2669 y Fz(0)2294 2723 y FC(1)2355 2703 y FD(d)p Fv(p)2451 2669 y Fz(0)2451 2723 y FC(2)2769 2647 y FG(1)p 2521 2684 538 4 v 2521 2760 a FA(j)p Fv(p)2597 2772 y FC(2)2653 2760 y FA(\000)18 b Fv(p)2789 2731 y Fz(0)2789 2782 y FC(2)2826 2760 y FA(j)2849 2736 y FC(2)2905 2760 y FG(+)g FD(\017)3022 2736 y FC(2)649 2981 y FA(\001)1122 2925 y FG(1)p 682 2962 922 4 v 682 3038 a FA(j)p Fv(p)758 3050 y FC(2)814 3038 y FA(\000)g Fv(p)950 3010 y Fz(0)950 3060 y FC(2)1006 3038 y FG(+)g Fv(p)1142 3050 y FC(1)1197 3038 y FA(\000)g Fv(p)1333 3010 y Fz(0)1333 3060 y FC(1)1371 3038 y FA(j)1394 3014 y FC(2)1449 3038 y FG(+)g FD(\017)1566 3014 y FC(2)1957 2925 y FG(1)p 1646 2962 664 4 v 1646 3038 a FD(E)1707 3053 y Fz(j)p Fr(p)1769 3061 y Ft(2)1802 3053 y Fz(\000)p Fr(p)1896 3033 y Fl(0)1896 3071 y Ft(2)1928 3053 y FC(+)p Fr(p)2021 3061 y Ft(1)2053 3053 y Fz(j)2095 3038 y FG(+)g FD(E)2239 3052 y FB(p)2273 3033 y Fl(0)2273 3071 y Ft(1)2538 2925 y FG(1)p 2353 2962 411 4 v 2353 3038 a FD(p)2395 3010 y Fz(0)2395 3060 y FC(1)2451 3038 y FG(+)g FD(p)2576 3010 y Fz(0)2576 3060 y FC(2)2631 3038 y FG(+)g FD(\026)2807 2925 y(p)2849 2937 y FC(1)p 2807 2962 80 4 v 2807 3038 a FD(p)2849 3010 y Fz(0)2849 3060 y FC(1)2896 2981 y FD(:)263 b FG(\(D.9\))386 3182 y(When)41 b(\(D.9\))g(is)g(estimated)f(b)n(y)g(setting)h FD(\017)j FG(=)g(0)d(in)f(the)h(t)n(w)n(o)f(denominators)f(and)i(when) 386 3282 y(\()p FD(p)460 3252 y Fz(0)460 3303 y FC(1)507 3282 y FG(+)10 b FD(p)624 3252 y Fz(0)624 3303 y FC(2)670 3282 y FG(+)g FD(\026)p FG(\))827 3252 y Fz(\000)p FC(1)952 3282 y FA(\024)23 b FG(1)p FD(=p)1166 3252 y Fz(0)1166 3303 y FC(1)1248 3282 y FG(is)g(used,)h(the)g(in)n(tegral)e(coincides)g (with)i(one)f(giv)n(en)f(in)i(the)f(pro)r(of)g(of)386 3382 y(Lemma)j(I)r(I.4.)36 b(F)-7 b(rom)25 b(\(I)r(I.5.5\))h(to)g(\(I)r (I.5.8\),)g(together)f(with)h(the)h(estimate)e FD(\017)h FG(of)g(the)g(prefactor)386 3481 y(according)g(to)h(\(I)r(I.6.22\))g (one)h(\014nds)1580 3640 y FD(J)1626 3652 y FC(1)1709 3640 y FA(\024)46 b FD(c)1856 3606 y Fz(0)1893 3640 y FD(\017)18 b FA(\001)h FD(\031)2037 3606 y FC(6)2074 3640 y FD(=)p FG(2)p FD(:)960 b FG(\(D.10\))486 3798 y(F)-7 b(or)27 b(the)i(second)f(in)n(tegral)f FD(J)1397 3810 y FC(2)1463 3798 y FG(w)n(e)h(estimate)h(b)n(y)f(setting)g FD(\017)c FG(=)g(0)29 b(as)e(b)r(efore)h(and)h(mak)n(e)e(the)386 3897 y(substitutions)h Fv(q)931 3909 y FC(2)992 3897 y FG(:=)22 b Fv(p)1155 3867 y Fz(0)1155 3918 y FC(2)1211 3897 y FA(\000)c Fv(p)1347 3909 y FC(2)1412 3897 y FG(for)27 b Fv(p)1592 3867 y Fz(0)1592 3918 y FC(2)1680 3897 y FG(and)g Fv(q)1891 3909 y FC(1)1952 3897 y FG(:=)c Fv(q)2113 3909 y FC(2)2169 3897 y FG(+)18 b Fv(p)2305 3867 y Fz(0)2305 3918 y FC(1)2370 3897 y FG(for)27 b Fv(p)2550 3867 y Fz(0)2550 3918 y FC(1)2587 3897 y FD(:)51 b FG(Then)723 4097 y FD(J)769 4109 y FC(2)853 4097 y FA(\024)46 b FD(c)1000 4063 y Fz(0)1241 4041 y FG(1)p 1056 4078 411 4 v 1056 4154 a FD(p)1098 4166 y FC(1)1153 4154 y FG(+)18 b FD(p)1278 4166 y FC(2)1334 4154 y FG(+)g FD(\026)1491 3984 y Fu(Z)1587 4097 y FD(d)p Fv(q)1680 4109 y FC(1)1741 4097 y FD(d)p Fv(q)1834 4109 y FC(2)1923 4041 y FG(1)p 1905 4078 78 4 v 1905 4154 a FD(q)1945 4126 y FC(2)1942 4176 y(2)2186 4041 y FG(1)p 2026 4078 363 4 v 2026 4154 a FA(j)p Fv(q)2099 4166 y FC(1)2155 4154 y FA(\000)g Fv(p)2291 4166 y FC(1)2328 4154 y FA(j)2351 4130 y FC(2)2591 4041 y FG(1)p 2431 4078 362 4 v 2431 4154 a FD(E)2492 4166 y FB(p)2526 4174 y Ft(1)2582 4154 y FG(+)g FD(E)2726 4166 y FB(q)2756 4174 y Ft(1)3141 4097 y FG(\(D.11\))1181 4388 y FA(\001)1218 4271 y Fu(\024)1262 4296 y(\020)1312 4388 y FG(1)41 b FA(\000)g FD(e)1540 4353 y Fz(\000)p FB(\017)p FC(\()p FB(p)1680 4328 y Fl(0)1680 4370 y Ft(1)1712 4353 y FC(+)p FB(p)1797 4328 y Fl(0)1797 4370 y Ft(2)1829 4353 y FC(\))1859 4296 y Fu(\021)2140 4332 y FG(1)p 1956 4369 411 4 v 1956 4445 a FD(p)1998 4416 y Fz(0)1998 4467 y FC(1)2053 4445 y FG(+)18 b 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FD(f)9 b FG(\()p FD(x)p FG(\))47 b(=)e FD(f)9 b FG(\(0\))41 b(+)18 b FD(x)1836 5517 y(d)p 1813 5554 91 4 v 1813 5630 a(dx)1927 5573 y(f)9 b FG(\()p FD(x)p FG(\))2088 5452 y Fu(\014)2088 5502 y(\014)2088 5552 y(\014)2088 5602 y(\014)2116 5656 y FB(x)p FC(=0)2284 5573 y FG(+)41 b FD(O)r FG(\()p FD(x)2534 5539 y FC(2)2572 5573 y FG(\))p FD(:)514 b FG(\(D.13\))p eop %%Page: 87 93 87 92 bop 3309 259 a FC(87)386 459 y FG(Since)39 b FD(f)9 b FG(\(0\))43 b(=)f(0)p FD(;)d FG(one)g(has)g(from)g(the)g(mean)g(v)-5 b(alue)40 b(theorem)e(the)i(estimate)82 b FA(j)p FD(f)9 b FG(\()p FD(x)p FG(\))p FA(j)57 b(\024)386 567 y FD(x)25 b FA(\001)506 506 y(p)p 575 506 84 4 v 61 x FG(2)p FD(p)13 b(e)711 537 y Fz(\000)773 515 y Ft(1)p 773 524 29 3 v 773 557 a(2)815 567 y FD(;)76 b FG(and)37 b(the)h(in)n(tegral)e (corresp)r(onding)f(to)i(the)g(linear)f(term)i(in)f FD(x)h FG(pro)n(vides)d(a)386 667 y(con)n(v)n(ergen)n(t)25 b(ma)5 b(joran)n(t:)684 877 y FD(M)32 b FG(:=)931 764 y Fu(Z)1028 877 y FD(d)p Fv(q)1121 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FA(\024)23 b FG(1)p FD(=q)1596 1164 y FC(1)1683 1152 y FG(and)1912 1023 y Fu(p)p 1995 1023 260 4 v 70 x FD(p)2037 1064 y Fz(0)2037 1115 y FC(1)2092 1093 y FG(+)18 b FD(p)2217 1064 y Fz(0)2217 1115 y FC(2)p 1877 1133 411 4 v 1877 1209 a FD(p)1919 1180 y Fz(0)1919 1231 y FC(1)1975 1209 y FG(+)g FD(p)2100 1180 y Fz(0)2100 1231 y FC(2)2155 1209 y FG(+)g FD(\026)2335 1152 y FA(\024)2583 1096 y FG(1)p 2433 1133 343 4 v 2433 1149 a Fu(p)p 2516 1149 260 4 v 70 x FD(p)2558 1190 y Fz(0)2558 1241 y FC(1)2613 1219 y FG(+)g FD(p)2738 1190 y Fz(0)2738 1241 y FC(2)2822 1152 y FA(\024)2957 1096 y FG(1)p 2920 1133 117 4 v 2920 1240 a FD(p)2962 1175 y Fl(0)2994 1170 y Ft(1)p 2994 1179 29 3 v 2994 1212 a(2)2962 1263 y FC(1)3097 1152 y FG(one)27 b(has)1079 1435 y FD(M)55 b FA(\024)1325 1322 y Fu(Z)1432 1379 y FD(d)p Fv(q)1525 1391 y FC(1)p 1432 1416 131 4 v 1461 1492 a FD(q)1498 1504 y FC(1)1767 1379 y FG(1)p 1606 1416 363 4 v 1606 1492 a FA(j)p Fv(q)1679 1504 y FC(1)1735 1492 y FA(\000)18 b Fv(p)1871 1504 y FC(1)1909 1492 y FA(j)1932 1468 y FC(2)1993 1322 y Fu(Z)2100 1379 y FD(d)p Fv(q)2193 1391 y FC(2)p 2100 1416 131 4 v 2126 1492 a FD(q)2166 1464 y FC(2)2163 1514 y(2)2440 1379 y FG(1)p 2273 1416 376 4 v 2273 1504 a FA(j)p Fv(q)2346 1516 y FC(1)2402 1504 y FA(\000)h Fv(q)2536 1516 y FC(2)2573 1504 y FA(j)2606 1453 y Ft(3)p 2606 1462 29 3 v 2606 1495 a(2)2658 1435 y FD(:)460 b FG(\(D.15\))386 1641 y(With)26 b(the)g(substitution)h Fv(p)c FG(:=)f Fv(q)1436 1653 y FC(2)1488 1641 y FA(\000)15 b Fv(q)1618 1653 y FC(1)1655 1641 y FD(;)49 b FG(the)26 b(second)f(in)n(tegral)f(can)i(b)r (e)g(estimated)f(with)h(the)386 1741 y(help)i(of)f(App)r(endix)i(A,)397 1827 y Fu(Z)504 1884 y FD(d)p Fv(p)p 504 1921 97 4 v 505 2009 a FD(p)557 1957 y Ft(3)p 557 1966 29 3 v 557 2000 a(2)786 1884 y FG(1)p 644 1921 326 4 v 644 1997 a FA(j)p Fv(p)18 b FG(+)g Fv(q)871 2009 y FC(1)909 1997 y FA(j)932 1973 y FC(2)1025 1940 y FG(=)1146 1884 y(2)p FD(\031)p 1146 1921 92 4 v 1155 1997 a(q)1192 2009 y FC(1)1261 1827 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190 4 v 2791 2237 a FA(p)p 2860 2237 80 4 v 52 x FD(p)2902 2301 y FC(1)2970 2232 y FD(:)60 b FG(Insertion)386 2387 y(in)n(to)27 b(\(D.11\))h(giv)n(es)1050 2577 y FD(J)1096 2589 y FC(2)1179 2577 y FA(\024)46 b FD(c)1326 2542 y Fz(0)1567 2520 y FG(1)p 1382 2557 411 4 v 1382 2633 a FD(p)1424 2645 y FC(1)1479 2633 y FG(+)18 b FD(p)1604 2645 y FC(2)1660 2633 y FG(+)g FD(\026)1826 2577 y(M)32 b(x)1986 2504 y FA(p)p 2055 2504 42 4 v 73 x FG(2)14 b FD(e)2150 2542 y Fz(\000)2211 2520 y Ft(1)p 2211 2529 29 3 v 2211 2562 a(2)2272 2577 y FA(\001)k FD(p)2355 2589 y FC(1)2439 2577 y FA(\024)45 b FD(c)2608 2513 y FA(p)p 2677 2513 34 4 v 64 x FD(\017)430 b FG(\(D.17\))386 2766 y(with)28 b FD(c)g FG(a)f(suitable)g(constan)n(t)g(indep)r(enden)n(t)i(of)e FD(p)1959 2778 y FC(1)2024 2766 y FG(and)g FD(p)2227 2778 y FC(2)2264 2766 y FD(:)386 2984 y Fo(c\))32 b(Estimate)i(of)f(r) -5 b(emaining)33 b(terms)486 3131 y FG(In)i(the)g(remaining)f(three)g (con)n(tributions)g(to)h FD(W)2061 3143 y FC(2)2122 3131 y 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y Fz(\000)p FC(1)2349 4011 y FD( )s FG(\))c FA(j)46 b(\024)g FD(c)23 b(\017)2735 3954 y Ft(1)p 2734 3963 29 3 v 2734 3996 a(2)2800 4011 y FA(k)p FD( )s FA(k)2941 3976 y FC(2)2977 4011 y FD(:)141 b FG(\(D.19\))486 4186 y(The)27 b(op)r(erator)f(b)r(oundedness)i(follo)n(ws)e(from)h(\(D.19\)) h(b)n(y)f(means)h(of)f(Lemma)h(I.3.)386 4522 y Fq(App)s(endix)38 b(E)386 4739 y FE(Estimate)33 b(of)i(the)f(Jansen-Hess)i(term)d Fe(B)2052 4753 y Fd(2)p Fc(m)2205 4739 y FE(in)i(co)s(ordinate)h(space) 486 4896 y FG(Our)23 b(goal)g(is)h(to)g(sho)n(w)g(b)r(oundedness)g(of)g (the)h(in)n(tegral)e(in)h(\(I.4.24\),)2627 4830 y Fu(R)2696 4896 y FD(d)p Fv(x)2789 4866 y Fz(0)2826 4896 y FA(j)p FD(k)s FG(\()p Fv(x)p FD(;)14 b Fv(x)3064 4866 y Fz(0)3089 4896 y FG(\))p FA(j)3179 4856 y FB(f)7 b FC(\()p FB(x)p FC(\))p 3168 4877 151 4 v 3168 4925 a FB(f)g FC(\()p FB(x)3271 4909 y Fl(0)3293 4925 y FC(\))3329 4896 y FD(;)386 5010 y FG(relativ)n(e)26 b(to)i(the)g(Coulom)n(b)f(\014eld)h FD(\015)5 b(=x;)50 b FG(for)27 b(arbitrary)e(mass)i FD(m)c FA(\025)g FG(0)p FD(:)486 5110 y FG(W)-7 b(e)27 b(tak)n(e)g FD(f)9 b FG(\()p FD(x)p FG(\))24 b(=)e FD(x)28 b FG(and)g(obtain)f (with)h(\(I.4.23\))e(and)i(\(I.3.27\))e(for)h(the)h(k)n(ernel)f FD(k)j FG(of)e FD(B)3256 5122 y FC(2)p FB(m)3352 5110 y FD(;)657 5302 y(k)s FG(\()p Fv(x)p FD(;)14 b Fv(x)872 5268 y Fz(0)896 5302 y FG(\))47 b(=)e FD(c)1121 5314 y FC(0)1172 5189 y Fu(Z)1269 5302 y FD(d)p Fv(p)24 b FD(e)1428 5268 y FB(i)p Fr(p)n(x)1571 5189 y Fu(Z)1668 5302 y FD(d)p Fv(p)1764 5268 y Fz(0)1811 5302 y FD(e)1850 5268 y Fz(\000)p FB(i)p Fr(p)1967 5243 y Fl(0)1989 5268 y Fr(x)2029 5243 y Fl(0)2092 5189 y Fu(Z)2189 5302 y FD(d)p Fv(p)2285 5268 y Fz(00)2506 5246 y FG(1)p 2361 5283 334 4 v 2361 5359 a FA(j)p Fv(p)2437 5335 y Fz(00)2498 5359 y FA(\000)18 b Fv(p)p FA(j)2657 5335 y FC(2)2894 5246 y FG(1)p 2737 5283 357 4 v 2737 5359 a FA(j)p Fv(p)2813 5335 y Fz(00)2874 5359 y FA(\000)g Fv(p)3010 5335 y Fz(0)3033 5359 y FA(j)3056 5335 y FC(2)1097 5575 y FA(\001)p FG(\(1)32 b FA(\000)1342 5554 y FG(~)1323 5575 y FD(D)1392 5587 y FC(0)1429 5575 y FG(\()p Fv(p)1514 5541 y Fz(00)1556 5575 y FG(\)\))1649 5458 y Fu(\022)1870 5519 y FG(1)p 1720 5556 342 4 v 1720 5632 a FD(E)1781 5644 y FB(p)1815 5628 y Fl(00)1879 5632 y FG(+)18 b FD(E)2023 5644 y FB(p)2113 5575 y FG(+)2390 5519 y(1)p 2229 5556 365 4 v 2229 5632 a FD(E)2290 5644 y FB(p)2324 5628 y Fl(00)2388 5632 y FG(+)g FD(E)2532 5644 y FB(p)2566 5628 y Fl(0)2603 5458 y Fu(\023)3189 5575 y FG(\(E.1\))p eop %%Page: 88 94 88 93 bop 386 259 a FC(88)386 465 y FG(with)28 b FD(c)611 477 y FC(0)671 465 y FG(:=)855 432 y FC(1)p 792 446 159 4 v 792 493 a(\(2)p FB(\031)r FC(\))918 477 y Ft(3)1018 428 y FB(\015)1057 402 y Ft(2)p 984 445 140 4 v 984 493 a FC(16)p FB(\031)1091 477 y Ft(4)1133 465 y FD(:)51 b FG(W)-7 b(e)28 b(recall)f(the)h(Coulom)n(bic)f(in)n(tegrals,)1488 573 y Fu(Z)1585 686 y FD(d)p Fv(q)c FD(e)1740 651 y FB(i)p Fr(qx)1898 629 y FG(1)p 1880 667 78 4 v 1880 743 a FD(q)1920 719 y FC(2)2013 686 y 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y FA(k)36 b FG(=)23 b(2)p FD(;)48 b FG(the)386 2487 y(sym)n(b)r(ol)19 b(class)f(of)h FD(C)999 2457 y FC(\(12\))1141 2487 y FG(is)g(not)h(c)n(hanged)e(up)r (on)i(replacing)e(\(1)r FA(\000)2378 2466 y FG(~)2360 2487 y FD(D)2431 2444 y FC(\(1\))2429 2509 y(0)2519 2487 y FG(\()p Fv(q)2601 2499 y FC(1)2641 2487 y FG(+)r Fv(p)2761 2457 y Fz(0)2761 2508 y FC(1)2798 2487 y FG(\)\)\(1)r(+)3024 2466 y(~)3005 2487 y FD(D)3076 2444 y FC(\(1\))3074 2509 y(0)3165 2487 y FG(\()p Fv(p)3250 2457 y Fz(0)3250 2508 y FC(1)3287 2487 y FG(\)\))386 2587 y(b)n(y)27 b(4)g(in)h FD(I)703 2599 y FC(1)768 2587 y FG(and)f(similarly)g(in)g FD(I)1401 2599 y FC(2)1439 2587 y FD(:)51 b FG(Note,)27 b(ho)n(w)n(ev)n(er,)f(that)h(this)h(is)f(a)g(rather)g(crude)g (estimate.)486 2686 y(Moreo)n(v)n(er,)33 b(w)n(e)h(can)h(replace)e Fv(q)1509 2698 y FC(1)1581 2686 y FG(and)i Fv(q)1800 2698 y FC(2)1872 2686 y FG(in)g(the)g(energy)e(denominators)h(of)g FD(I)3060 2698 y FC(1)3133 2686 y FG(and)g FD(I)3337 2698 y FC(2)386 2786 y 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FG(\))2368 5578 y FD(:)792 b FG(\(H.2\))p eop %%Page: 92 98 92 97 bop 386 259 a FC(92)386 494 y FG(W)-7 b(e)30 b(tak)n(e)e FD(f)9 b FG(\()p FD(p)p FG(\))26 b(:=)g FD(p)1050 464 y FC(5)p FB(=)p FC(2)1154 494 y FD(=)p FG(\()p FD(E)1289 506 y FB(p)1347 494 y FG(+)19 b FD(m)p FG(\))56 b(and)29 b(mak)n(e)g(the)h(substitutions)f Fv(q)2665 464 y Fz(00)2734 494 y FG(:=)2867 438 y Fv(p)2920 408 y Fz(00)p 2858 475 113 4 v 2858 551 a FD(mq)3024 494 y FG(and)g Fv(q)3237 464 y Fz(0)3287 494 y FG(:=)455 638 y Fv(p)508 608 y Fz(0)p 396 675 196 4 v 396 751 a FD(mq)s(q)549 727 y Fz(00)649 694 y FG(for)24 b Fv(p)826 664 y Fz(00)893 694 y FG(and)h Fv(p)1105 664 y Fz(0)1128 694 y FG(,)g(resp)r(ectiv)n (ely)-7 b(.)35 b(W)-7 b(e)25 b(estimate)g FA(j)p Fw(\033)6 b Fv(^)-51 b(p)2259 664 y Fz(00)2315 694 y Fw(\033)6 b Fv(^)-51 b(p)2428 664 y Fz(0)2452 694 y FA(j)38 b FG(b)n(y)25 b(1)f(and)g(set)h Fv(q)e FG(:=)g Fv(p)p FD(=m:)386 838 y FG(Then)28 b(the)g(angular)e(in)n(tegrations)g(can)h(b)r(e)h(p)r 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b(and)33 b(then)g(use)f(the)386 1596 y(estimate)27 b(\(I.5.28\))g(to)h(obtain) 730 1810 y FD(I)53 b FA(\024)46 b FG(4)p FD(\031)1022 1776 y FC(3)1059 1810 y FD(q)1099 1776 y FC(4)1291 1754 y FG(1)p 1160 1791 304 4 v 1160 1807 a Fu(p)p 1243 1807 221 4 v 73 x FD(q)1283 1856 y FC(2)1339 1880 y FG(+)18 b(1)1760 1754 y(1)p 1506 1791 549 4 v 1506 1880 a(\()1538 1807 y Fu(p)p 1622 1807 221 4 v 1622 1880 a FD(q)1662 1856 y FC(2)1717 1880 y FG(+)g(1)g(+)g(1\))2017 1856 y FC(2)2101 1668 y Fu(")2150 1697 y(Z)2233 1717 y FC(1)p FB(=q)2196 1886 y FC(0)2350 1810 y FD(dq)2433 1776 y Fz(00)2476 1810 y FD(q)2516 1751 y Fl(00)2567 1753 y Ft(3)p 2567 1762 29 3 v 2567 1796 a(2)2637 1810 y FG(ln)2720 1690 y Fu(\014)2720 1739 y(\014)2720 1789 y(\014)2720 1839 y(\014)2757 1754 y FG(1)g(+)g FD(q)2940 1724 y Fz(00)p 2757 1791 226 4 v 2757 1867 a FG(1)g FA(\000)g FD(q)2940 1843 y Fz(00)2993 1690 y Fu(\014)2993 1739 y(\014)2993 1789 y(\014)2993 1839 y(\014)1326 2111 y FG(+)1442 2055 y(1)p 1424 2092 78 4 v 1424 2168 a FD(q)1464 2144 y FC(2)1525 1998 y Fu(Z)1608 2019 y Fz(1)1571 2187 y FC(1)p FB(=q)1692 2111 y FD(dq)1775 2077 y Fz(00)1887 2055 y FG(1)p 1842 2092 134 4 v 1842 2180 a FD(q)1882 2134 y Fl(00)1932 2129 y Ft(1)p 1932 2138 29 3 v 1932 2171 a(2)2012 2111 y FG(ln)2095 1991 y Fu(\014)2095 2041 y(\014)2095 2090 y(\014)2095 2140 y(\014)2133 2055 y FG(1)g(+)g FD(q)2316 2025 y Fz(00)p 2133 2092 226 4 v 2133 2168 a FG(1)g FA(\000)g FD(q)2316 2144 y Fz(00)2368 1991 y Fu(\014)2368 2041 y(\014)2368 2090 y(\014)2368 2140 y(\014)2419 1969 y(#)843 2370 y FG(=)46 b(8)p FD(\031)1046 2336 y FC(3)1083 2370 y FD(q)1123 2336 y FC(2)1324 2314 y FG(1)p 1193 2351 304 4 v 1193 2367 a Fu(p)p 1276 2367 221 4 v 73 x FD(q)1316 2416 y FC(2)1372 2440 y FG(+)18 b(1)1792 2314 y(1)p 1539 2351 549 4 v 1539 2440 a(\()1571 2367 y Fu(p)p 1654 2367 221 4 v 73 x FD(q)1694 2416 y FC(2)1750 2440 y FG(+)g(1)g(+)g(1\))2050 2416 y FC(2)2134 2253 y Fu(\024)2178 2370 y FD(\031)45 b FA(\000)2432 2314 y FG(4)p 2386 2351 134 4 v 2386 2439 a(5)p FD(q)2478 2388 y Ft(1)p 2477 2397 29 3 v 2477 2430 a(2)2543 2370 y FG(ln)2626 2250 y Fu(\014)2626 2299 y(\014)2626 2349 y(\014)2626 2399 y(\014)2664 2314 y FG(1)18 b(+)g FD(q)p 2664 2351 184 4 v 2664 2427 a FG(1)g FA(\000)g FD(q)2857 2250 y Fu(\014)2857 2299 y(\014)2857 2349 y(\014)2857 2399 y(\014)610 2622 y FG(+)711 2505 y Fu(\022)782 2566 y FG(2)p FD(q)864 2536 y FC(2)p 782 2603 119 4 v 821 2679 a FG(5)943 2622 y FA(\000)g FG(2)1068 2505 y Fu(\023)1156 2622 y FG(arctan)1445 2566 y(1)p 1411 2603 110 4 v 1411 2627 a FA(p)p 1480 2627 41 4 v 52 x FD(q)1572 2622 y FG(+)1678 2505 y Fu(\022)1739 2622 y FG(1)32 b FA(\000)1920 2566 y FD(q)1960 2536 y FC(2)p 1920 2603 78 4 v 1937 2679 a FG(5)2007 2505 y Fu(\023)2096 2622 y FG(ln)2179 2502 y Fu(\014)2179 2552 y(\014)2179 2601 y(\014)2179 2651 y(\014)2216 2562 y FG(1)18 b(+)2359 2510 y FA(p)p 2428 2510 41 4 v 52 x FD(q)p 2216 2603 253 4 v 2216 2679 a FG(1)g FA(\000)2359 2627 y(p)p 2428 2627 41 4 v 52 x FD(q)2478 2502 y Fu(\014)2478 2552 y(\014)2478 2601 y(\014)2478 2651 y(\014)2548 2622 y FG(+)2684 2566 y(4)p 2664 2603 84 4 v 2664 2679 a(15)2757 2566 y FA(p)p 2826 2566 41 4 v 56 x FD(q)2889 2505 y Fu(\025)2947 2622 y FD(:)213 b FG(\(H.4\))386 2817 y(Due)31 b(to)f(the)g(ab)r(o)n(v)n(e)f (c)n(hoice)g(of)i FD(f)9 b FG(\()p FD(p)p FG(\))p FD(;)30 b FG(the)g(r.h.s.)45 b(of)30 b(\(H.4\))p FA(\030)d FD(q)2407 2764 y Ft(5)p 2407 2773 29 3 v 2407 2806 a(2)2480 2817 y FG(for)i FD(q)i FA(!)c FG(0)p FD(;)44 b FG(assuring)29 b(that)386 2916 y(its)j(con)n(tribution)g(to)g FD(M)1171 2928 y FC(2)1208 2916 y FG(\()p FD(q)s FG(\))h(de\014ned)g(b)r(elo)n(w) f(\(I.5.28\))f(is)h(\014nite.)52 b(The)32 b(in)n(tegrals)f(o)r (ccurring)386 3016 y(here)h(and)h(in)g(the)g(remaining)e(con)n (tributions)h(to)h FD(T)2047 3028 y FC(2)2116 3016 y FG(are)f(listed)h(in)g(App)r(endix)h(A,)f(starting)386 3115 y(from)27 b(\(A.11\).)486 3215 y(It)e(should)g(b)r(e)h(noted)f (that)h(the)f(Sob)r(olev)g(represen)n(tation)e(for)i(the)h(Jansen-Hess) e(op)r(erator)386 3315 y(cannot)h(b)r(e)h(used)f(in)h(the)g(virial)e (theorem.)36 b(The)26 b(reason)d(is)j(that)g(the)f(transformation)f(op) r(era-)386 3426 y(tors)e FD(U)614 3396 y Fz(0)605 3446 y FC(0)665 3426 y FG(in)i(\(I.4.1\))f(linking)g FD(b)1311 3438 y FB(m)1397 3426 y FG(and)g FD(B)1621 3383 y FC(\(2\))1617 3436 y FB(m)1733 3426 y FG(do)g(dep)r(end)h(on)f(the)h(mass)e FD(m)h FG(and)g(hence)g(in\015uence)386 3525 y(the)28 b(deriv)-5 b(ativ)n(es.)386 3861 y Fq(App)s(endix)38 b(J)386 4079 y FE(On)c(the)h(exp)s(ectation)g(v)-6 b(alue)35 b(of)g Fe(F)1756 4093 y Fd(0)1835 4079 y FE(in)g(the)g(p)s(ositiv)m(e)g (sp)s(ectral)g(subspace)486 4226 y FG(W)-7 b(e)32 b(will)g(sho)n(w)f (that)h(for)f(the)i(\014rst-order)d(expansion)h(term)g FD(F)2505 4238 y FC(0)2575 4226 y FG(of)h(the)g(exact)f(pro)5 b(jector)386 4326 y FD(P)439 4338 y FC(+)494 4326 y FG(,)28 b(one)f(has)g(\()p FD( )s(;)14 b(F)1024 4338 y FC(0)1076 4326 y FD( )s FG(\))37 b(=)23 b(0)50 b(if)28 b FD( )e FA(2)e(H)1687 4338 y FC(+)p FB(;N)1821 4326 y FD(:)486 4425 y FG(F)-7 b(or)32 b(a)g(state)g FD( )k FG(in)d(the)g(p)r(ositiv)n (e)g(sp)r(ectral)f(subspace)g(of)g(the)h(free)g(Dirac)f(op)r(erator,)g (one)386 4536 y(has)25 b(for)g(an)n(y)g FD(l)f FA(2)g(f)p FG(1)p FD(;)14 b(:::;)g(N)9 b FA(g)21 b FG(:)69 b(\003)1456 4493 y FC(\()p FB(l)p FC(\))1456 4557 y(+)1547 4536 y FD( )40 b FG(=)23 b FD( )s(:)48 b FG(Therefore)25 b(the)h(exp)r (ectation)f(v)-5 b(alue)25 b(of)h FD(F)3188 4493 y FC(\()p FB(l)p FC(\))3176 4558 y(0)3265 4536 y FD(;)60 b(l)386 4636 y FG(sp)r(ecifying)28 b(the)f(particle)h(on)f(whic)n(h)g FD(F)1620 4648 y FC(0)1686 4636 y FG(is)g(acting,)g(can)g(b)r(e)h (written)g(in)g(the)g(form)857 4786 y(\()p FD( )s(;)14 b(F)1048 4743 y FC(\()p FB(l)p FC(\))1036 4809 y(0)1139 4786 y FD( )s FG(\))47 b(=)e(\(\003)1475 4743 y FC(\()p FB(l)p FC(\))1475 4807 y(+)1566 4786 y FD( )s(;)14 b(F)1725 4743 y FC(\()p FB(l)p FC(\))1713 4809 y(0)1816 4786 y FG(\003)1874 4743 y FC(\()p FB(l)p FC(\))1874 4807 y(+)1965 4786 y FD( )s FG(\))47 b(=)e(\()p FD( )s(;)14 b FG(\003)2395 4743 y FC(\()p FB(l)p FC(\))2395 4807 y(+)2487 4786 y FD(F)2552 4743 y FC(\()p FB(l)p FC(\))2540 4809 y(0)2643 4786 y FG(\003)2701 4743 y FC(\()p FB(l)p FC(\))2701 4807 y(+)2792 4786 y FD( )s FG(\))p FD(:)299 b FG(\(J.1\))386 4925 y(Dropping)32 b(the)h(index)g FD(l)h FG(again,)f(w)n(e)f(use)h (the)g(relation)f(\(I.3.26\))g(b)r(et)n(w)n(een)h(the)g(sym)n(b)r(ol)f (and)386 5025 y(the)c(k)n(ernel)f(of)g(an)g(op)r(erator)f(to)i(extract) f(from)g(\(I)r(I.3.19\))753 5203 y FD(k)796 5215 y FB(F)838 5223 y Ft(0)875 5203 y FG(\()p Fv(p)p FD(;)14 b Fv(p)1050 5169 y Fz(0)1074 5203 y FG(\))46 b(=)g FA(\000)1411 5147 y FD(\015)p 1338 5184 194 4 v 1338 5260 a FG(\(2)p FD(\031)s FG(\))1494 5236 y FC(2)1711 5147 y FG(1)p 1574 5184 315 4 v 1574 5260 a FA(j)p Fv(p)19 b FA(\000)f Fv(p)1805 5236 y Fz(0)1828 5260 y FA(j)1851 5236 y FC(2)2072 5147 y FG(1)p 1931 5184 324 4 v 1931 5260 a FD(E)1992 5272 y FB(p)2049 5260 y FG(+)g FD(E)2193 5272 y FB(p)2227 5256 y Fl(0)2288 5203 y FG(\(1)32 b FA(\000)2510 5182 y FG(~)2491 5203 y FD(D)2560 5215 y FC(0)2597 5203 y FG(\()p Fv(p)p FG(\))2747 5182 y(~)2728 5203 y FD(D)2797 5215 y FC(0)2834 5203 y FG(\()p Fv(p)2919 5169 y Fz(0)2943 5203 y FG(\)\))196 b(\(J.2\))386 5394 y(suc)n(h)27 b(that)h(the)g (r.h.s.)37 b(of)27 b(\(J.1\))h(is)f(written)h(as)603 5575 y(\()p FD( )s(;)14 b FG(\003)787 5587 y FC(+)856 5575 y FD(F)909 5587 y FC(0)961 5575 y FG(\003)1019 5587 y FC(+)1087 5575 y FD( )s FG(\))47 b(=)e FA(\000)1481 5519 y FD(\015)p 1408 5556 194 4 v 1408 5632 a FG(\(2)p FD(\031)s FG(\))1564 5608 y FC(2)1645 5519 y FG(1)p 1645 5556 42 4 v 1645 5632 a(4)1724 5462 y Fu(Z)1770 5651 y Fy(R)1817 5634 y Ft(3)p Fp(N)5 b Fl(\000)p Ft(3)1985 5575 y FD(dQ)2108 5462 y Fu(Z)2204 5575 y FD(d)p Fv(p)14 b FD(d)p Fv(p)2410 5541 y Fz(0)2603 5519 y FG(1)p 2467 5556 315 4 v 2467 5632 a FA(j)p Fv(p)k FA(\000)g Fv(p)2697 5608 y Fz(0)2721 5632 y FA(j)2744 5608 y FC(2)2965 5519 y FG(1)p 2824 5556 324 4 v 2824 5632 a FD(E)2885 5644 y FB(p)2942 5632 y FG(+)g FD(E)3086 5644 y FB(p)3120 5628 y Fl(0)p eop %%Page: 93 99 93 98 bop 3309 259 a FC(93)734 472 y FA(\001)p 771 382 229 4 v 806 450 a FG(^)771 472 y FD( )825 484 y FB(Q)882 472 y FG(\()p Fv(p)p FG(\))23 b(\(1)32 b(+)1245 451 y(~)1225 472 y FD(D)1294 484 y FC(0)1331 472 y FG(\()p Fv(p)p FG(\)\))24 b(\(1)32 b FA(\000)1726 451 y FG(~)1707 472 y FD(D)1776 484 y FC(0)1813 472 y FG(\()p Fv(p)p FG(\))1964 451 y(~)1944 472 y FD(D)2013 484 y FC(0)2051 472 y FG(\()p Fv(p)2136 438 y Fz(0)2159 472 y FG(\)\))24 b(\(1)32 b(+)2469 451 y(~)2450 472 y FD(D)2519 484 y FC(0)2556 472 y FG(\()p Fv(p)2641 438 y Fz(0)2664 472 y FG(\)\))2786 450 y(^)2752 472 y FD( )2806 484 y FB(Q)2862 472 y FG(\()p Fv(p)2947 438 y Fz(0)2971 472 y FG(\))p FD(;)177 b FG(\(J.3\))386 589 y(where)30 b FD(Q)h FG(comprises)e(the)i(co)r(ordinates)e(\(resp)r (ectiv)n(e)i(momen)n(ta\))f(of)h(the)g(remaining)f FD(N)f FA(\000)20 b FG(1)386 689 y(particles.)36 b(Ho)n(w)n(ev)n(er,)26 b(it)i(is)f(easily)g(v)n(eri\014ed)g(that)923 824 y(\(1)32 b(+)1146 803 y(~)1126 824 y FD(D)1195 836 y FC(0)1232 824 y FG(\()p Fv(p)p FG(\)\))24 b(\(1)32 b FA(\000)1627 803 y FG(~)1608 824 y FD(D)1677 836 y FC(0)1714 824 y FG(\()p Fv(p)p FG(\))1874 803 y(~)1855 824 y FD(D)1924 836 y FC(0)1961 824 y FG(\()p Fv(p)2046 790 y Fz(0)2069 824 y FG(\)\))24 b(\(1)32 b(+)2379 803 y(~)2360 824 y FD(D)2429 836 y FC(0)2466 824 y FG(\()p Fv(p)2551 790 y Fz(0)2574 824 y FG(\)\))47 b(=)f(0)365 b(\(J.4\))386 965 y(since)609 944 y(~)589 965 y FD(D)660 935 y FC(2)658 985 y(0)721 965 y FG(=)22 b(1)p FD(:)51 b FG(Th)n(us)27 b(\()p FD( )s(;)14 b(F)1313 977 y FC(0)1365 965 y FD( )s FG(\))37 b(=)23 b(0)p FD(:)486 1064 y FG(W)-7 b(e)24 b(w)n(an)n(t)f(to)h(add)g(that)g(also)f(\003)1477 1076 y Fz(\000)1546 1064 y FD(F)1599 1076 y FC(0)1651 1064 y FG(\003)1709 1076 y Fz(\000)1787 1064 y FG(=)g(0)p FD(;)47 b FG(suc)n(h)23 b(that)h FD(F)2399 1076 y FC(0)2474 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FG(one)k(has)g(\003)2436 1357 y FC(\()p FB(l)p FC(\))2436 1420 y(+)2513 1400 y FD( )f FG(=)d FD( )s(;)51 b FG(the)28 b(exp)r(ectation)386 1499 y(v)-5 b(alue)27 b(turns)h(in)n(to)1071 1634 y(\()p FD( )s(;)14 b(A)1259 1600 y FC(\()p FB(nk)q FC(\))1408 1634 y FD(F)1473 1591 y FC(\()p FB(l)p FC(\))1461 1656 y(0)1564 1634 y FD( )s FG(\))46 b(=)g(\()p FD( )s(;)14 b(A)1998 1600 y FC(\()p FB(nk)q FC(\))2146 1634 y FG(\003)2204 1591 y FC(\()p FB(l)p FC(\))2204 1655 y(+)2295 1634 y FD(F)2360 1591 y FC(\()p FB(l)p FC(\))2348 1656 y(0)2451 1634 y FG(\003)2509 1591 y FC(\()p FB(l)p FC(\))2509 1655 y(+)2600 1634 y FD( )s FG(\))514 b(\(J.5\))386 1791 y(b)r(ecause)21 b(\003)745 1748 y FC(\()p FB(l)p FC(\))745 1812 y(+)844 1791 y FG(comm)n(utes)h(with)g FD(A)1474 1761 y FC(\()p FB(nk)q FC(\))1630 1791 y FG(for)f(distinct)h(particles.)34 b(Ho)n(w)n(ev)n(er,)21 b(the)i(r.h.s.)34 b(of)22 b(\(J.5\))386 1914 y(v)-5 b(anishes)27 b(since)g(\003)973 1871 y FC(\()p FB(l)p FC(\))973 1935 y(+)1064 1914 y FD(F)1129 1871 y FC(\()p FB(l)p FC(\))1117 1936 y(0)1220 1914 y FG(\003)1278 1871 y FC(\()p FB(l)p FC(\))1278 1935 y(+)1392 1914 y FG(=)c(0)50 b(as)27 b(sho)n(wn)g(ab)r(o)n(v)n(e.)p eop %%Page: 94 100 94 99 bop 386 259 a FC(94)386 459 y Fq(References)386 748 y FG(Abramo)n(witz)30 b(M.)i(and)f(Stegun)g(I.A.)h(1965:)42 b Fn(Handb)l(o)l(ok)34 b(of)g(Mathematic)l(al)h(F)-6 b(unctions)32 b(with)386 848 y(F)-6 b(ormulas,)30 b(Gr)l(aphs)h(and)g (Mathematic)l(al)g(T)-6 b(ables)p FG(.)38 b(Do)n(v)n(er)27 b(Publications,)f(New)i(Y)-7 b(ork)386 1019 y(\(BBHS\))42 b(Bac)n(h)e(V.,)46 b(Barbaroux)39 b(J.M.,)44 b(Hel\013er)e(B.)f(and)h (Sieden)n(top)f(H.)h(1999:)62 b(On)41 b(the)386 1118 y(stabilit)n(y)34 b(of)h(the)g(relativistic)e(electron-p)r(ositron)g (\014eld.)58 b(Comm)n(un.)g(Math.)g(Ph)n(ys.)e Fv(201)p FG(,)386 1218 y(445-460)386 1389 y(Balinsky)36 b(A.A.)i(and)f(Ev)-5 b(ans)37 b(W.D.)h(1998:)54 b(On)37 b(the)h(virial)e(theorem)h(for)g (the)h(relativistic)386 1489 y(op)r(erator)25 b(of)h(Bro)n(wn)g(and)g (Ra)n(v)n(enhall,)g(and)g(the)h(absence)f(of)h(em)n(b)r(edded)g(eigen)n (v)-5 b(alues.)35 b(Lett.)386 1588 y(Math.)i(Ph)n(ys.)f Fv(44)p FG(,)27 b(233-248)386 1759 y(Balinsky)36 b(A.A.)i(and)e(Ev)-5 b(ans)37 b(W.D.)h(1999:)53 b(Stabilit)n(y)37 b(of)g(one-electron)f (molecules)g(in)h(the)386 1859 y(Bro)n(wn-Ra)n(v)n(enhall)24 b(mo)r(del.)38 b(Comm)n(un.)e(Math.)h(Ph)n(ys.)f Fv(202)p FG(,)27 b(481-500)386 2030 y(Bethe)d(H.A.)g(and)g(Salp)r(eter)g(E.E.)f (1957:)33 b Fn(Quantum)25 b(Me)l(chanics)j(of)f(One)f(and)g(Two)i(Ele)l (ctr)l(on)386 2130 y(A)n(toms)p FG(.)36 b(Springer-V)-7 b(erlag,)25 b(Berlin)386 2301 y(Bro)n(wn)j(G.E.)i(and)f(Ra)n(v)n (enhall)g(D.G.)h(1951:)39 b(On)30 b(the)g(in)n(teraction)e(of)i(t)n(w)n (o)f(electrons.)42 b(Pro)r(c.)386 2401 y(Ro)n(y)-7 b(.)36 b(So)r(c.)h(London)27 b Fv(A208)p FG(,)g(552-559)386 2572 y(\(BSS\))k(Brummelh)n(uis)g(R.,)h(Sieden)n(top)f(H.)g(and)g(Sto)r (c)n(kmey)n(er)e(E.)i(2002:)41 b(The)31 b(ground)f(state)386 2671 y(energy)21 b(of)i(relativistic)e(one-electron)g(atoms)h (according)f(to)h(Jansen)g(and)g(Hess.)35 b(Do)r(c.)h(Math.)386 2771 y Fv(7)p FG(,)27 b(167-182)386 2942 y(Burenk)n(o)n(v)d(V.I.)i(and) g(Ev)-5 b(ans)25 b(W.D.)i(1998:)34 b(On)26 b(the)g(ev)-5 b(aluation)25 b(of)h(the)g(norm)g(of)f(an)h(in)n(tegral)386 3042 y(op)r(erator)38 b(asso)r(ciated)g(with)i(the)f(stabilit)n(y)g(of) h(one-electron)e(atoms.)71 b(Pro)r(c.)g(Ro)n(y)-7 b(.)72 b(So)r(c.)386 3141 y(\(Edin)n(burgh\))27 b Fv(128A)p FG(,)g(993-1005)386 3312 y(Darwin)f(C.G.)h(1928:)34 b(The)26 b(w)n(a)n(v)n(e)f(equations)h(of)g(the)h(electron.)35 b(Pro)r(c.)h(Ro)n(y)-7 b(.)35 b(So)r(c.)i(\(London\))386 3412 y Fv(A118)p FG(,)27 b(654-679)386 3583 y(Datta)36 b(N.,)i(F)-7 b(ern\023)-42 b(andez)36 b(R.)g(and)g(F)-7 b(r)1556 3587 y(\177)1556 3583 y(ohlic)n(h)35 b(J.)h(1999:)52 b(E\013ectiv)n(e)35 b(hamiltonians)h(and)f(phase)386 3683 y(diagrams)26 b(for)h(tigh)n(t-binding)g(mo)r(dels.)37 b(Journal)26 b(of)i(Stat.)37 b(Ph)n(ys.)f Fv(96)p FG(,)27 b(545-611)386 3854 y(Dirac)g(P)-7 b(.A.M.)28 b(1928:)35 b(The)28 b(quan)n(tum)f(theory)g(of)g(the)h(electron.)37 b(Pro)r(c.)e(Ro)n(y)-7 b(.)37 b(So)r(c.)f(London)386 3954 y Fv(A117)p FG(,)27 b(610-624)386 4125 y(\(DES\))32 b(Dolb)r(eault)f(J.,)h(Esteban)e(M.J.)h(and)g(S)n(\023)-39 b(er)n(\023)g(e)28 b(E.)j(2000:)42 b(V)-7 b(ariational)30 b(c)n(haracterisation)386 4224 y(for)d(eigen)n(v)-5 b(alues)26 b(of)i(Dirac)f(op)r(erators.)35 b(Calc.)h(V)-7 b(ar.)37 b Fv(10)p FG(,)27 b(321-347)386 4395 y(Douglas)34 b(M.)h(and)g(Kroll)f (N.M.)i(1974:)50 b(Quan)n(tum)34 b(electro)r(dynamical)g(corrections)f (to)i(the)386 4495 y(\014ne)28 b(structure)f(of)g(helium.)38 b(Ann.)f(Ph)n(ys.)f(\(N.Y.\))29 b Fv(82)p FG(,)e(89-155)386 4666 y(Durand)f(Ph.)36 b(and)27 b(Malrieu)f(J.-P)-7 b(.)25 b(1987:)35 b(E\013ectiv)n(e)26 b(Hamiltonians)g(and)g(pseudo)g(p)r (oten)n(tials)386 4766 y(as)k(to)r(ols)h(for)g(rigorous)e(mo)r (delling.)47 b(In)32 b Fn(A)n(b)g(initio)i(Metho)l(ds)h(in)e(Quantum)e 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b(no-pair)g(formalism)f(with)i (external-\014eld)f(pro)5 b(jection)39 b(op)r(erators.)73 b(Ph)n(ys.)i(Rev.)386 1745 y Fv(A33)p FG(,)27 b(3742-3748)386 1916 y(Ho)r(ev)n(er)41 b(G.)h(and)f(Sieden)n(top)h(H.)g(1999:)63 b(Stabilit)n(y)42 b(of)g(the)g(Bro)n(wn-Ra)n(v)n(enhall)c(op)r(erator.) 386 2016 y(Math.)f(Ph)n(ys.)f(Electron.)g(J.)27 b Fv(5)p FG(\(6\),)g(1-11)386 2187 y(Hunzik)n(er)19 b(W.)h(1966:)32 b(On)19 b(the)h(sp)r(ectra)f(of)h(Sc)n(hr)1875 2191 y(\177)1875 2187 y(odinger)e(m)n(ultiparticle)i(Hamiltonians.)34 b(Helv.)386 2286 y(Ph)n(ys.)i(Acta)27 b Fv(39)p FG(,)h(451-462)386 2458 y(\(IJA\))d(Ian)n(tc)n(henk)n(o)f(A.)h(and)g(Jakubassa-Am)n (undsen)d(D.H.)k(2003:)34 b(On)25 b(the)g(p)r(ositivit)n(y)g(of)f(the) 386 2557 y(Jansen-Hess)i(op)r(erator)g(for)h(arbitrary)e(mass.)37 b(Ann.)g(Henri)28 b(P)n(oincar)n(\023)-39 b(e)24 b Fv(4)p FG(,)j(1-17)386 2728 y(\(JA\))32 b(Jakubassa-Am)n(undsen)d(D.H.)k (2002:)42 b(The)32 b(essen)n(tial)f(sp)r(ectrum)g(of)h(relativistic)f (one-)386 2828 y(electron)c(ions)g(in)h(the)g(Jansen-Hess)e(mo)r(del.) 37 b(Math.)g(Ph)n(ys.)f(Electron.)f(J.)28 b Fv(8)p FG(\(3\),)f(1-30)386 2999 y(Jansen)19 b(G.)h(and)f(Hess)h(B.A.)g(1989:)31 b(Revision)19 b(of)h(the)g(Douglas-Kroll)d(transformation.)33 b(Ph)n(ys.)386 3099 y(Rev.)k Fv(A39)p FG(,)27 b(6016-6017)386 3270 y(Kadison)34 b(R.V.)j(and)e(Ringrose)f(J.R.)i(1983:)51 b Fn(F)-6 b(undamentals)37 b(of)h(the)g(The)l(ory)g(of)g(Op)l(er)l (ator)386 3369 y(A)n(lgebr)l(as)28 b FG(V)-7 b(ol.)37 b Fv(1)p FG(.)g(Academic)27 b(Press,)f(New)i(Y)-7 b(ork)386 3541 y(Kato)24 b(T.)h(1966:)34 b Fn(Perturb)l(ation)27 b(The)l(ory)i(for)f(Line)l(ar)g(Op)l(er)l(ators)p FG(,)e(V)-7 b(ol.)36 b Fv(132)24 b FG(of)h Fn(Grund)t(lehr)l(en)386 3640 y(der)30 b(mathematischen)h(Wissenschaften)p FG(.)38 b(Springer-V)-7 b(erlag,)25 b(Berlin)386 3811 y(Landau)h(L.D.)g(and)g (Lifsc)n(hitz)h(E.M.)f(1965:)34 b Fn(Quantenme)l(chanik)p FG(,)27 b(V)-7 b(ol.)36 b Fv(I)s(I)s(I)27 b FG(of)f Fn(L)l(ehrbuch)j (der)386 3911 y(The)l(or)l(etischen)i(Physik)p FG(.)39 b(Ak)-5 b(ademie-V)e(erlag,)26 b(Berlin)386 4082 y(Lieb)i(E.H.)h(and)f (Y)-7 b(au)28 b(H.-T.)h(1988:)36 b(The)29 b(stabilit)n(y)f(and)g (instabilit)n(y)g(of)g(relativistic)g(matter.)386 4182 y(Comm)n(un.)37 b(Math.)g(Ph)n(ys.)e Fv(118)p FG(,)27 b(177-213)386 4353 y(Mittleman)f(M.H.)h(1981:)34 b(Theory)25 b(of)g(relativistic)h(e\013ects)g(on)f(atoms:)35 b (Con\014guration-space)386 4452 y(Hamiltonian.)i(Ph)n(ys.)e(Rev.)i Fv(A24)p FG(,)28 b(1167-1175)386 4624 y(Morse)c(P)-7 b(.M.)26 b(and)f(F)-7 b(esh)n(bac)n(h)25 b(H.)h(1953:)34 b Fn(Metho)l(ds)29 b(of)g(The)l(or)l(etic)l(al)h(Physics)d FG(part)e Fv(I)s(I)p FG(,)h FA(x)p FG(9.)36 b(Mc)386 4723 y(Gra)n(w)26 b(Hill,)j(New)e(Y)-7 b(ork)386 4894 y(P)n(auli)30 b(W.)i(1927:)42 b(Zur)30 b(Quan)n(tenmec)n(hanik)g(des)h (magnetisc)n(hen)g(Elektrons.)46 b(Z.)31 b(Ph)n(ys.)46 b Fv(43)p FG(,)386 4994 y(601-623)386 5165 y(P)n(auli)29 b(W.)i(1958:)41 b(Die)31 b(allgemeinen)f(Prinzipien)f(der)h(W)-7 b(ellenmec)n(hanik,)31 b(in)g Fn(Handbuch)i(der)386 5265 y(Physik)p FG(,)c(V)-7 b(ol.)37 b Fv(V)28 b FG(T)-7 b(eil)28 b(1.)36 b(Springer-V)-7 b(erlag,)25 b(Berlin,)j(1-168)386 5436 y(P)n(earson)35 b(D.B.)j(1988:)55 b Fn(Quantum)37 b(Sc)l(attering)i(and)g(Sp)l(e)l(ctr)l(al)g(The)l(ory)p FG(.)69 b(Academic)37 b(Press,)386 5535 y(New)28 b(Y)-7 b(ork)p eop %%Page: 96 102 96 101 bop 386 259 a FC(96)386 459 y FG(Reed)35 b(M.)h(and)f(Simon)g (B.)g(1980:)51 b Fn(F)-6 b(unctional)37 b(A)n(nalysis)f FG(V)-7 b(ol.)59 b Fv(I)36 b FG(of)f Fn(Metho)l(ds)j(of)g(Mo)l(dern)386 558 y(Mathematic)l(al)32 b(Physics)p FG(.)38 b(Academic)28 b(Press,)e(New)i(Y)-7 b(ork)386 729 y(Reed)26 b(M.)h(and)f(Simon)g(B.)g (1978:)34 b Fn(A)n(nalysis)29 b(of)g(Op)l(er)l(ators)e FG(V)-7 b(ol.)36 b Fv(IV)27 b FG(of)f Fn(Metho)l(ds)k(of)f(Mo)l(dern) 386 829 y(Mathematic)l(al)j(Physics)p FG(.)38 b(Academic)28 b(Press,)e(New)i(Y)-7 b(ork)386 1000 y(Rose,)27 b(E.M.)g(1961:)35 b Fn(R)l(elativistic)c(Ele)l(ctr)l(on)f(The)l(ory)p FG(.)38 b(John)27 b(Wiley)h(and)g(Sons,)f(New)h(Y)-7 b(ork)386 1171 y(Sob)r(olev)24 b(A.)i(2003:)33 b(In)n(tegrated)24 b(densit)n(y)h(of)g(states)f(for)g(the)i(p)r(erio)r(dic)e(Sc)n(hr)2775 1175 y(\177)2775 1171 y(odinger)g(op)r(erator)386 1271 y(in)k(dimension)f(t)n(w)n(o.)36 b(Univ)n(ersit)n(y)27 b(of)h(Sussex)f(preprin)n(t)386 1442 y(Sob)r(olev)f(A.)g(2004:)35 b(Asymptotics)26 b(of)g(the)h(in)n(tegrated)e(densit)n(y)i(of)f(states) g(for)g(p)r(erio)r(dic)g(ellip-)386 1542 y(tic)f(pseudo-di\013eren)n (tial)f(op)r(erators)f(in)i(dimension)g(one.)35 b(Revista)24 b(Matematica)h(Ib)r(eroamer-)386 1641 y(icana,)i(in)h(prin)n(t)386 1812 y(Suc)n(her)f(J.)g(1958:)35 b(PhD)28 b(Thesis)f(\(Colum)n(bia)h (Univ)n(ersit)n(y)-7 b(,)26 b(USA\))386 1984 y(Suc)n(her)31 b(J.)h(1987:)43 b(Relativistic)31 b(man)n(y-electron)f(Hamiltonians.)49 b(Ph)n(ys.)f(Scripta)31 b Fv(36)p FG(,)i(271-)386 2083 y(281)386 2254 y(Suc)n(her)40 b(J.)h(1980:)61 b(F)-7 b(oundations)40 b(of)h(the)g(relativistic)g(theory)f(of)g(man)n (y-electron)f(atoms.)386 2354 y(Ph)n(ys.)d(Rev.)h Fv(A22)p FG(,)27 b(348-362)386 2525 y(Sto)r(c)n(kmey)n(er)h(E.)g(2002:)38 b(Electrones)28 b(relativistas)f(en)i(un)h(con)n(texto)e(de)h(la)f (estabilidad)h(de)g(la)386 2625 y(materia.)36 b(PhD)28 b(Thesis)f(\(P)n(on)n(ti\014cia)f(Catholic)h(Univ)n(ersiy)g(of)h (Chile\))386 2796 y(T)-7 b(a)n(ylor)17 b(M.E.)h(1981:)30 b Fn(Pseudo)l(di\013er)l(ential)23 b(Op)l(er)l(ators)p FG(.)34 b(Princeton)18 b(Univ)n(ersit)n(y)f(Press,)i(Prince-)386 2895 y(ton)386 3067 y(Thaller)27 b(B.)g(1992:)35 b Fn(The)c(Dir)l(ac)f (Equation)p FG(.)38 b(Springer-V)-7 b(erlag,)25 b(Berlin)386 3238 y(Tix)d(C.)g(1997:)32 b(Self-adjoin)n(tness)22 b(and)g(sp)r (ectral)f(prop)r(erties)h(of)g(a)f(pseudo-relativistic)g(Hamil-)386 3337 y(tonian)27 b(due)h(to)g(Bro)n(wn)e(and)h(Ra)n(v)n(enhall.)36 b(Preprin)n(t)26 b(mp-arc/97-441)386 3508 y(Tix)20 b(C.)h(1998:)31 b(Strict)21 b(p)r(ositivit)n(y)f(of)h(a)f(relativistic)f(Hamiltonian)i (due)f(to)h(Bro)n(wn)e(and)h(Ra)n(v)n(en-)386 3608 y(hall.)37 b(Bull.)g(London)27 b(Math.)37 b(So)r(c.)f Fv(30)p FG(,)28 b(283-290)386 3779 y(De)19 b(V)-7 b(ries)18 b(E.)g(1970:)30 b(F)-7 b(oldy-W)g(outh)n(uysen)18 b(transformations)e(and)j(related)e (problems.)33 b(F)-7 b(ortsc)n(hr.)386 3879 y(Ph)n(ysik)26 b Fv(18)p FG(,)i(149-182)386 4050 y(W)-7 b(eidmann)28 b(J.)f(1980:)35 b Fn(Line)l(ar)c(Op)l(er)l(ators)f(in)f(Hilb)l(ert)h (Sp)l(ac)l(es)p FG(.)38 b(Springer-V)-7 b(erlag,)25 b(Berlin)386 4221 y(W)-7 b(erner)27 b(D.)h(1995:)35 b Fn(F)-6 b(unktionalanalysis)p FG(.)39 b(Springer-V)-7 b(erlag,)25 b(Berlin)386 4392 y(W)-7 b(olf)32 b(A.,)i(Reiher)d(M.)i(and)e(Hess)h(B.A.)g(2002:)44 b(The)32 b(generalised)e(Douglas-Kroll)f(transfor-)386 4492 y(mation.)37 b(J.)27 b(Chem.)37 b(Ph)n(ys.)f Fv(117)p FG(,)27 b(9215-9226)386 4663 y(W)-7 b(olf)36 b(A.,)j(Reiher)d(M.)g(and) g(Hess)f(B.A.)i(2004:)52 b(T)-7 b(ransgressing)33 b(theory)i(b)r (oundaries:)53 b(The)386 4762 y(generalised)32 b(Douglas-Kroll)f (transformation.)54 b(In)34 b Fn(R)l(e)l(c)l(ent)g(A)l(dvanc)l(es)h(in) h(R)l(elativistic)g(Ef-)386 4862 y(fe)l(cts)g(in)g(Chemistry)p FG(,)i(eds.)57 b(K.)34 b(Hirao)g(and)g(Y.)h(Ishik)-5 b(a)n(w)n(a.)56 b(W)-7 b(orld)34 b(Scien)n(ti\014c)h(Publishing,)386 4962 y(Singap)r(ore.)h(In)27 b(prin)n(t.)p eop %%Page: 97 103 97 102 bop 3309 259 a FC(97)427 445 y FE(Notations)427 744 y FA(A)464 b FG(an)n(tisymmetrisation)27 b(with)h(resp)r(ect)f(to)h (in)n(terc)n(hange)e(of)h(t)n(w)n(o)g(particles)427 848 y FD(A)489 818 y FC(\()p FB(n)p FC(\))957 848 y FG(op)r(erator)f FD(A)i FG(relating)f(to)g(the)h FD(n)p FG(-th)g(particle)427 948 y FD(A)489 917 y Fz(\003)957 948 y FG(adjoin)n(t)g(of)f(op)r (erator)f FD(A)427 1047 y(a)p FG(\()p Fv(x)p FD(;)14 b Fv(p)p FG(\))282 b(sym)n(b)r(ol)27 b(of)h(op)r(erator)e FD(A)p FG(,)i(de\014ned)g(b)n(y)957 1156 y(\()p FD(A')p FG(\)\()p Fv(x)p FG(\))d(=)e(\(2)p FD(\031)s FG(\))1520 1126 y Fz(\000)1582 1104 y Ft(3)p 1582 1113 29 3 v 1582 1146 a(2)1639 1089 y Fu(R)1708 1156 y FD(d)p Fv(p)14 b FD(a)p FG(\()p Fv(x)p FD(;)g Fv(p)p FG(\))g FD(e)2119 1126 y FB(i)p Fr(p)n(x)2239 1156 y FG(^)-55 b FD(')p FG(\()p Fv(p)p FG(\))429 1256 y(^)-44 b FD(a)p FG(\()p Fv(q)p FD(;)14 b Fv(p)p FG(\))282 b(F)-7 b(ourier)27 b(transform)f(of)i(sym)n(b)r(ol)f(with)h(resp)r(ect)f(to)h Fv(x)427 1356 y FG([)p FD(A;)14 b(B)t FG(])318 b FD(AB)23 b FA(\000)18 b FD(B)t(A)51 b FG(\(comm)n(utator)27 b(of)h FD(A)f FG(and)h FD(B)t FG(\))427 1455 y Fw(\013)467 b FG(v)n(ector)27 b(of)g(Dirac)g(matrices)g(in)h Fx(C)2010 1425 y FC(4)p FB(;)p FC(4)427 1605 y FD(\014)957 1488 y Fu(\022)1060 1555 y FD(I)1186 1654 y FA(\000)p FD(I)1335 1488 y Fu(\023)1433 1605 y FA(2)23 b Fx(C)1565 1575 y FC(4)q FB(;)p FC(4)427 1754 y Fx(C)491 b FG(complex)27 b(space)427 1854 y FD(D)496 1866 y FC(0)957 1854 y FG(free)h (one-particle)e(Dirac)h(op)r(erator)447 1939 y(~)427 1960 y FD(D)496 1972 y FC(0)957 1960 y FD(D)1026 1972 y FC(0)1063 1960 y FD(=)p FA(j)p FD(D)1197 1972 y FC(0)1234 1960 y FA(j)111 b FG(\(an)27 b(op)r(erator)f(of)i(norm)f(unit)n(y\))427 2069 y FD(d)470 2081 y FB(\022)957 2069 y FG(dilation)h(op)r(erator,)e (de\014ned)i(b)n(y)f FD(d)2064 2081 y FB(\022)2107 2069 y FG(^)-47 b FD(u)o FG(\()p Fv(p)p FG(\))24 b(=)f FD(\022)2419 2039 y Fz(\000)2481 2017 y Ft(3)p 2481 2026 V 2481 2059 a(2)2528 2069 y FG(^)-47 b FD(u)p FG(\()p Fv(p)p FD(=\022)r FG(\))427 2169 y FD(d!)435 b FG(area)26 b(elemen)n(t)i(on)f(the)h(unit) h(sphere)e FD(S)2193 2139 y FC(2)427 2315 y FD(f)477 2285 y FC(#)957 2315 y FG(Mellin)h(transform,)f(de\014ned)h(b)n(y)f FD(f)2069 2285 y FC(#)2127 2315 y FG(\()p FD(t)p FG(\))d(=)f(\(2)p FD(\031)s FG(\))2489 2285 y Fz(\000)2551 2263 y Ft(1)p 2551 2272 V 2551 2305 a(2)2615 2232 y Fz(1)2612 2248 y Fu(R)2616 2392 y FC(0)2687 2315 y FD(dp)14 b(f)9 b FG(\()p FD(p)p FG(\))14 b FD(p)2998 2285 y Fz(\000)p FB(it)p Fz(\000)3160 2263 y Ft(1)p 3160 2272 V 3160 2305 a(2)427 2470 y FD(\015)487 b FG(p)r(oten)n(tial)28 b(strength)427 2570 y FD(\015)470 2582 y FB(B)s(R)957 2570 y FG(2)p FD(=)p FG(\()1083 2537 y FB(\031)p 1083 2551 41 4 v 1087 2599 a FC(2)1152 2570 y FG(+)1249 2537 y FC(2)p 1245 2551 V 1245 2599 a FB(\031)1296 2570 y FG(\))427 2670 y FD(\015)470 2682 y FB(c)957 2670 y FG(critical)f(p)r(oten)n(tial)h (strength)427 2769 y FD(\015)470 2781 y FB(J)957 2769 y FG(1.006)427 2869 y(\000)478 b(gamma)27 b(function)427 2969 y FD(H)461 b(D)1026 2981 y FC(0)1082 2969 y FG(+)18 b FD(V)69 b FG(\(Dirac)28 b(op)r(erator\))427 3068 y FA(H)460 b FG(Hilb)r(ert)29 b(space)d(\(complete)i(metric)g(space)f (with)h(scalar)e(pro)r(duct\))427 3168 y FA(H)497 3180 y FC(+)p FB(;)p FC(1)957 3168 y FG(\003)1015 3180 y FC(+)1084 3168 y FG(\()p FD(H)1185 3183 y FC(1)p FB(=)p FC(2)1290 3168 y FG(\()p Fx(R)1376 3138 y FC(3)1419 3168 y FG(\))19 b FA(\002)f Fx(C)1607 3138 y FC(4)1650 3168 y FG(\))427 3284 y FA(H)497 3296 y FC(+)p FB(;N)957 3284 y FG(\003)1015 3241 y FC(\(1\))1015 3305 y(+)1123 3284 y FA(\012)g(\001)c(\001)g(\001) k(\012)g FG(\003)1462 3241 y FC(\()p FB(N)6 b FC(\))1462 3305 y(+)1590 3284 y FA(A)p FG(\()p FD(H)1757 3299 y FC(1)p FB(=)p FC(2)1862 3284 y FG(\()p Fx(R)1948 3254 y FC(3)1992 3284 y FG(\))18 b FA(\002)g Fx(C)2179 3254 y FC(4)2223 3284 y FG(\))2255 3254 y FB(N)2318 3284 y FD(;)60 b(N)36 b FG(the)28 b(n)n(um)n(b)r(er)g(of)f(particles)427 3384 y FD(H)496 3399 y FC(1)p FB(=)p FC(2)601 3384 y FG(\()p Fx(R)687 3354 y FC(3)730 3384 y FG(\))195 b(Sob)r(olev)27 b(space)g(of)h(order)e(1)p FD(=)p FG(2)49 b(\(form)28 b(domain)f(of)h FD(D)2663 3396 y FC(0)2700 3384 y FG(\))427 3484 y FD(H)496 3496 y FC(1)534 3484 y FG(\()p Fx(R)620 3454 y FC(3)663 3484 y FG(\))262 b(Sob)r(olev)27 b(space)g(of)h(order)e (1)50 b(\(domain)28 b(of)f FD(D)2383 3496 y FC(0)2420 3484 y FG(\))427 3583 y(Im)h FD(z)364 b FG(imaginary)26 b(part)h(of)h FD(z)427 3683 y(k)470 3695 y FB(A)525 3683 y FG(\()p Fv(p)p FD(;)14 b Fv(p)700 3653 y Fz(0)723 3683 y FG(\))202 b(k)n(ernel)27 b(of)h(op)r(erator)d FD(A)p FG(,)j(de\014ned)g(b)n(y)g(\()p FD(A')p FG(\)\()p Fv(p)p FG(\))d(=)2555 3616 y Fu(R)2624 3683 y FD(d)p Fv(p)2720 3653 y Fz(0)2758 3683 y FD(k)2801 3695 y FB(A)2855 3683 y FG(\()p Fv(p)p FD(;)14 b Fv(p)3030 3653 y Fz(0)3054 3683 y FG(\))g FD(')p FG(\()p Fv(p)3239 3653 y Fz(0)3263 3683 y FG(\))427 3783 y FD(L)484 3795 y FC(2)521 3783 y FG(\()p Fx(R)607 3752 y FC(3)651 3783 y FG(\))274 b(Hilb)r(ert)29 b(space)d(of)i(\(equiv)-5 b(alence)27 b(class)g(of)6 b(\))28 b(square-in)n(tegrable)d(functions)957 3882 y(with)j(domain)g Fx(R)1496 3852 y FC(3)427 3982 y FG(\003)485 3994 y FC(+)957 3982 y FG(pro)5 b(jection)27 b(on)n(to)g(the)h(p)r(ositiv)n(e)f(sp)r (ectral)g(subspace)g(of)g FD(D)2811 3994 y FC(0)427 4081 y FG(\003)485 4093 y Fz(\000)957 4081 y FG(pro)5 b(jection)27 b(on)n(to)g(the)h(negativ)n(e)e(sp)r(ectral)h(subspace)g(of)h FD(D)2832 4093 y FC(0)427 4181 y Fx(N)481 4193 y FC(0)957 4181 y FG(space)f(of)h(natural)f(n)n(um)n(b)r(ers)g Fx(N)h FA([)19 b(f)p FG(0)p FA(g)427 4281 y FG(\012)487 4293 y FB(\027)957 4281 y FG(v)n(ector)27 b(spherical)f(harmonic)427 4380 y(\011DO)337 b(pseudo)r(di\013eren)n(tial)27 b(op)r(erator)f (\(de\014ned)j(b)n(y)e(its)h(sym)n(b)r(ol\))427 4480 y Fx(R)476 b FG(real)27 b(space)427 4580 y Fx(R)481 4592 y FC(+)957 4580 y FG(p)r(ositiv)n(e)g(real)g(space)427 4679 y(Re)h FD(z)365 b FG(real)27 b(part)g(of)h FD(z)427 4779 y FA(S)480 b FG(Sc)n(h)n(w)n(artz)26 b(space)h(of)h(in\014nitely)g (di\013eren)n(tiable,)f(rapidly)g(decreasing)957 4878 y(functions)427 4978 y FD(\033)474 4990 y FC(1)512 4978 y FD(;)14 b(\033)596 4990 y FC(2)634 4978 y FD(;)g(\033)718 4990 y FC(3)957 4978 y FG(P)n(auli)27 b(matrices)g(in)h Fx(C)1657 4948 y FC(2)p FB(;)p FC(2)427 5078 y FD(\033)s FG(\()p FD(H)7 b FG(\))340 b(sp)r(ectrum)28 b(of)g FD(H)427 5177 y(\033)474 5189 y FB(ac)545 5177 y FG(\()p FD(H)7 b FG(\))272 b(absolutely)27 b(con)n(tin)n(uous)g(sp)r(ectrum)g(of)h FD(H)427 5277 y(\033)474 5289 y FB(ess)573 5277 y FG(\()p FD(H)7 b FG(\))244 b(essen)n(tial)27 b(sp)r(ectrum)h(of)f FD(H)427 5377 y(\033)474 5389 y FB(p)513 5377 y FG(\()p FD(H)7 b FG(\))304 b(p)r(oin)n(t)28 b(sp)r(ectrum)g(of)f FD(H)58 b FG(\(set)28 b(of)f(eigen)n(v)-5 b(alues)27 b(of)g FD(H)7 b FG(\))427 5476 y FD(\033)474 5488 y FB(sc)540 5476 y FG(\()p FD(H)g FG(\))277 b(singular)27 b(con)n(tin)n(uous)f(sp)r (ectrum)i(of)g FD(H)427 5576 y(T)481 b FG(kinetic)28 b(energy)f(op)r(erator)427 5675 y FD(T)476 5687 y FB(a)957 5675 y FG(translation)g(op)r(erator,)f(de\014ned)i(b)n(y)f FD(T)2186 5687 y FB(a)2226 5675 y FD(')p FG(\()p Fv(x)p FG(\))d(=)f FD(')p FG(\()p Fv(x)c FG(+)f Fv(a)p FG(\))p eop %%Page: 98 104 98 103 bop 386 259 a FC(98)427 445 y FD(U)484 457 y FB(k)940 445 y FG(unitary)27 b(transformation)f(op)r(erator)427 545 y FD(V)465 b FA(\000)p FD(\015)5 b(=x)50 b FG(\(Coulom)n(b)27 b(p)r(oten)n(tial\),)h(except)g(in)f(section)h(I.2.d)427 645 y FD(Y)475 657 y FB(lM)940 645 y FG(spherical)f(harmonic)427 744 y Fx(Z)489 756 y Fz(\000)940 744 y FG(space)g(of)g(negativ)n(e)g (in)n(tegers)427 844 y(\()14 b FA(\001)g FD(;)28 b FA(\001)14 b FG(\))310 b(scalar)26 b(pro)r(duct)i(in)f(the)h(Hilb)r(ert)h(space)d FD(L)2292 856 y FC(2)427 953 y FG(\()p FD(';)14 b(\036)p FG(\))940 886 y Fu(R)1009 953 y FD(d)p Fv(x)p 1126 880 170 4 v 24 w FD(')p FG(\()p Fv(x)p FG(\))24 b FD(\036)p FG(\()p Fv(x)p FG(\))441 1052 y(^)-56 b FD(')459 b FG(F)-7 b(ourier)27 b(transform)f(of)i FD(')427 1162 y FG(\()13 b(^)-55 b FD(')q(;)562 1140 y FG(^)551 1162 y FD(\036)p FG(\))940 1095 y Fu(R)1009 1162 y FD(d)p Fv(p)p 1129 1090 172 4 v 37 w FG(^)g FD(')p FG(\()p Fv(p)p FG(\))1334 1140 y(^)1324 1162 y FD(\036)p FG(\()p Fv(p)p FG(\))47 b(=)e(\()p FD(';)14 b(\036)p FG(\))427 1261 y FA(k)46 b(k)383 b FG(norm)27 b(in)h FD(L)1311 1273 y FC(2)1348 1261 y FG(\()p Fx(R)1434 1231 y FC(3)1477 1261 y FG(\))427 1365 y FA(k)p FD(A')p FA(k)313 b FG(\()p FD(A';)14 b(A')p FG(\))1273 1335 y FC(1)p FB(=)p FC(2)427 1465 y FA(k)p FD(A)p FA(k)402 b FG(sup)940 1538 y Fz(k)p FB(')p Fz(k)p FC(=1)1150 1465 y FA(k)p FD(A')p FA(k)p eop %%Page: 99 105 99 104 bop 386 459 a FE(Leb)s(enslauf)427 726 y FG(17.10.47)145 b(Geb)r(oren)27 b(in)h(M)1380 730 y(\177)1378 726 y(unc)n(hen)427 826 y(21.7.67)187 b(Abitur)28 b(am)g(St.Anna)g(Gymnasium,)f(M)2169 830 y(\177)2167 826 y(unc)n(hen)427 926 y(1967-1972)79 b(Ph)n(ysikstudium)27 b(an)h(der)f(T)-7 b(ec)n(hnisc)n(hen)27 b(Univ)n(ersit)2491 930 y(\177)2491 926 y(at)g(M)2670 930 y(\177)2668 926 y(unc)n(hen)427 1025 y(12.7.72)187 b(Diplom)870 1125 y(Thema)28 b(der)f(Arb)r(eit:)h("Breathing)e(Mo)r (des")427 1224 y(1972-1978)79 b(Wissensc)n(haftlic)n(her)27 b(Mitarb)r(eiter)g(an)g(der)g(TU)h(M)2493 1228 y(\177)2491 1224 y(unc)n(hen)427 1324 y(26.6.75)187 b(Promotion)26 b(an)h(der)h(TU)g(M)1769 1328 y(\177)1767 1324 y(unc)n(hen)f(b)r(ei)h (Prof.)f(H.)h(Sc)n(hmidt)870 1424 y(Thema)g(der)f(Arb)r(eit:)h("T)-7 b(ransferprozesse,)24 b(Coulom)n(banregung)870 1523 y(und)k (Strahlungsph)1517 1527 y(\177)1517 1523 y(anomene)f(b)r(ei)h(Sc)n(h)n (w)n(erionenst)2511 1527 y(\177)2511 1523 y(o\031en")427 1623 y(1978-1980)79 b(F)-7 b(orsc)n(h)n(ungsstip)r(endiat/Lektor)25 b(an)i(den)h(Univ)n(ersit)2543 1627 y(\177)2543 1623 y(aten)f(Bergen,)f(Oslo)h(\(Norw)n(egen\))870 1723 y(und)h(Aarh)n(us)f (\(D)1418 1727 y(\177)1418 1723 y(anemark\))427 1822 y(6.6.80)229 b(Heirat)27 b(v)n(on)g(P)n(er)g(A.)h(Am)n(undsen)427 1922 y(1981-1985)79 b(Wissensc)n(haftlic)n(her)27 b(Angestellter)g(an)g (der)g(TU)h(M)2514 1926 y(\177)2512 1922 y(unc)n(hen)427 2022 y(4.8.85)229 b(Geburt)28 b(der)f(T)-7 b(o)r(c)n(h)n(ter)27 b(Ragnhild)g(Elisab)r(eth)427 2121 y(1987-1991)79 b(Wissensc)n(haftlic) n(her)27 b(Angestellter)g(an)g(der)g(Ludwig-Maximilians-Univ)n(ersit) 3386 2125 y(\177)3386 2121 y(at)f(M)3564 2125 y(\177)3562 2121 y(unc)n(hen)427 2221 y(18.1.89)187 b(Habilitation)28 b(an)f(der)g(Univ)n(ersit)1923 2225 y(\177)1923 2221 y(at)g(F)-7 b(rankfurt)870 2320 y(Thema)28 b(der)f(Arb)r(eit:)h("Mo)r (delle)f(zur)g(Besc)n(hreibung)g(des)g(Elektron)n(transfers)870 2420 y(in)h(atomaren)e(Kollisionen")427 2520 y(seit)i(1989)122 b(F)-7 b(akult)1104 2524 y(\177)1104 2520 y(atsmitglied)28 b(am)f(F)-7 b(ac)n(h)n(b)r(ereic)n(h)26 b(Ph)n(ysik)h(der)g(Univ)n (ersit)2871 2524 y(\177)2871 2520 y(at)g(F)-7 b(rankfurt)427 2619 y(1992-1996)79 b(Wissensc)n(haftlic)n(her)27 b(Angestellter)g(an)g (der)g(Univ)n(ersit)2617 2623 y(\177)2617 2619 y(at)g(F)-7 b(rankfurt)427 2719 y(1996-2000)79 b(Mathematikstudium)28 b(an)g(der)f(LMU)h(M)2191 2723 y(\177)2189 2719 y(unc)n(hen)427 2819 y(30.3.00)187 b(Diplom)870 2918 y(Thema)28 b(der)f(Arb)r(eit:)h ("F)-7 b(uc)n(hssc)n(he)26 b(Grupp)r(en)i(und)g(die)g(durc)n(h)f(sie)h (de\014nierten)870 3018 y(Riemannsc)n(hen)f(Fl)1495 3022 y(\177)1495 3018 y(ac)n(hen")427 3117 y(1998-2001)79 b(Angestellter)27 b(am)h(Mathematisc)n(hen)f(Institut)i(der)e(LMU)870 3217 y(\(Sekretariatsarb)r(eiten\))427 3317 y(seit)h(2000)122 b(Angestellter)27 b(am)h(Institut)g(f)1802 3321 y(\177)1800 3317 y(ur)g(Biomathematik)f(der)g(GSF)h(Neuherb)r(erg)870 3416 y(\(Sekretariatsarb)r(eiten\))427 3516 y(2001-2004)79 b(Wissensc)n(haftlic)n(her)27 b(Angestellter)g(am)g(Mathematisc)n(hen)g (Institut)i(der)e(LMU)870 3616 y(\(mit)i(semesterw)n(eisen)d(Un)n (terbrec)n(h)n(ungen\))p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF