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4444 y(Coulom)n(b-Dirac)d(op)r(erator)g([5,)i(11)o(].)g(Suc)n(h)f (transformations)f(allo)n(w)h(for)g(the)h(decoupling)g(of)386 4544 y(the)j(electon)f(and)h(p)r(ositron)f(sp)r(ectral)g(subspaces)g (to)g(arbitrary)f(order)g Fv(n)i Fx(in)g(the)g(p)r(oten)n(tial)p 386 4652 499 4 v 486 4746 a Fq(Date)5 b Fz(:)22 b(No)n(v)n(em)n(b)r(er) i(23,)f(2005.)1846 4845 y Ft(1)p eop %%Page: 2 2 2 1 bop 386 259 a Ft(2)937 b(D.)22 b(H.)g(JAKUBASSA-AMUNDSEN)386 459 y Fx(strength.)40 b(The)h(con)n(v)n(ergence)d(of)i(the)h(resulting) f(series)f(of)i(op)r(erators)d(w)n(as)h(sho)n(wn)h(only)386 558 y(recen)n(tly)27 b([19)o(].)575 658 y(Let)h(\(in)g(relativistic)f (units,)h Fp(~)23 b Fx(=)f Fv(c)h Fx(=)g(1\))1441 876 y Fv(H)53 b Fx(=)1717 772 y Ft(2)1674 797 y Fs(X)1673 976 y Fo(k)q Ft(=1)1794 876 y Fx(\()p Fv(D)1897 833 y Ft(\()p Fo(k)q Ft(\))1895 898 y(0)2009 876 y Fx(+)18 b Fv(V)2159 842 y Ft(\()p Fo(k)q Ft(\))2251 876 y Fx(\))885 b(\(1.1\))386 1119 y(b)r(e)33 b(the)f(Dirac)g(op)r(erator)f(for)h(t)n (w)n(o)f(nonin)n(teracting)g(electrons,)h(where)f Fv(D)2760 1076 y Ft(\()p Fo(k)q Ft(\))2758 1141 y(0)2884 1119 y Fx(:=)g Fn(\013)3066 1089 y Ft(\()p Fo(k)q Ft(\))3159 1119 y FA(p)3212 1131 y Fo(k)3274 1119 y Fx(+)386 1223 y Fv(\014)437 1193 y Ft(\()p Fo(k)q Ft(\))530 1223 y Fv(m)54 b Fx(\(with)30 b FA(p)25 b Fx(:=)g Fu(\000)p Fv(i)p Fu(r)1234 1235 y Fm(x)1277 1223 y Fx(\))k(is)g(the)g(free)g (Dirac)g(op)r(erator)e([22)o(],)i(and)g Fv(V)2701 1193 y Ft(\()p Fo(k)q Ft(\))2819 1223 y Fx(:=)c Fu(\000)3025 1186 y Fo(\015)p 3007 1204 74 4 v 3007 1252 a(x)3045 1261 y Fl(k)3145 1223 y Fx(\(with)386 1336 y Fv(\015)32 b Fx(=)27 b Fv(Z)6 b(e)655 1306 y Ft(2)691 1336 y Fv(;)69 b(e)822 1306 y Ft(2)886 1336 y Fu(\031)27 b Fx(1)p Fv(=)p Fx(137)p Fv(:)p Fx(04)g(the)j(\014ne-structure)g(constan)n(t)f(and)h Fv(x)e Fx(:=)f Fu(j)p FA(x)p Fu(j)p Fx(\))58 b(is)30 b(the)h(cen)n(tral)386 1436 y(Coulom)n(b)f(p)r(oten)n(tial)h(for)f (electron)g Fv(k)s Fx(.)h Fv(H)37 b Fx(acts)30 b(in)h(the)h(Hilb)r(ert) f(space)f Fu(A)p Fx(\()p Fv(L)2839 1448 y Ft(2)2876 1436 y Fx(\()p Fp(R)2962 1406 y Ft(3)3006 1436 y Fx(\))21 b Fu(\012)f Fp(C)3198 1406 y Ft(4)3241 1436 y Fx(\))3273 1406 y Ft(2)386 1590 y Fx(\(w)n(e)37 b(shall)g(use)g(this)g(notation)g (as)g(shorthand)f(for)g Fu(A)p Fx(\()2213 1511 y Ft(2)2183 1527 y Fs(N)2168 1665 y Fo(k)q Ft(=1)2290 1590 y Fx(\()p Fv(L)2379 1602 y Ft(2)2416 1590 y Fx(\()p Fp(R)2502 1560 y Ft(3)2545 1590 y Fx(\))25 b Fu(\012)g Fp(C)2746 1560 y Ft(4)2789 1590 y Fx(\))2821 1560 y Ft(\()p Fo(k)q Ft(\))2914 1590 y Fx(\))77 b(where)36 b Fu(A)386 1739 y Fx(denotes)29 b(an)n(tisymmetrization)f(of)h(the)g(t)n(w)n(o-electron)e(function,)j (and)f(the)g(form)g(domain)g(is)386 1838 y(the)f(subspace)f Fu(A)p Fx(\()p Fv(H)1043 1853 y Ft(1)p Fo(=)p Ft(2)1148 1838 y Fx(\()p Fp(R)1234 1808 y Ft(3)1277 1838 y Fx(\))19 b Fu(\012)f Fp(C)1465 1808 y Ft(4)1508 1838 y Fx(\))1540 1808 y Ft(2)1578 1838 y Fv(:)575 1961 y Fx(Let)36 b Fv(V)799 1931 y Ft(\(12\))959 1961 y Fx(:=)1180 1928 y Fo(e)1211 1903 y Fk(2)p 1094 1942 236 4 v 1094 1989 a Fr(j)p Fm(x)1154 1997 y Fk(1)1186 1989 y Fr(\000)p Fm(x)1278 1997 y Fk(2)1310 1989 y Fr(j)1413 1961 y Fx(b)r(e)h(the)g(electron-electron)d(in)n (teraction.)h(Then)i(the)g(\(t)n(w)n(o-)386 2075 y(particle\))27 b(Bro)n(wn-Ra)n(v)n(enhall)e(op)r(erator)h(is)h(de\014ned)h(b)n(y)1274 2221 y Fv(H)1350 2187 y Fo(B)s(R)1504 2221 y Fx(=)45 b(\003)1672 2233 y Ft(+)p Fo(;)p Ft(2)1803 2221 y Fx(\()p Fv(H)39 b Fx(+)32 b Fv(V)2107 2187 y Ft(\(12\))2229 2221 y Fx(\))24 b(\003)2343 2233 y Ft(+)p Fo(;)p Ft(2)3168 2221 y Fx(\(1.2\))543 2427 y(=)663 2371 y(1)18 b(+)g Fv(\014)857 2341 y Ft(\(1\))p 663 2408 284 4 v 784 2484 a Fx(2)981 2371 y(1)g(+)g Fv(\014)1175 2341 y Ft(\(2\))p 981 2408 V 1102 2484 a Fx(2)1297 2427 y Fv(U)1363 2384 y Ft(\(1\))1354 2449 y(0)1452 2427 y Fv(U)1518 2384 y Ft(\(2\))1509 2449 y(0)1630 2427 y Fx(\()p Fv(H)25 b Fx(+)18 b Fv(V)1906 2393 y Ft(\(12\))2029 2427 y Fx(\))23 b(\()p Fv(U)2182 2384 y Ft(\(1\))2173 2449 y(0)2271 2427 y Fv(U)2337 2384 y Ft(\(2\))2328 2449 y(0)2426 2427 y Fx(\))2458 2393 y Fr(\000)p Ft(1)2571 2371 y Fx(1)18 b(+)g Fv(\014)2765 2341 y Ft(\(1\))p 2571 2408 V 2692 2484 a Fx(2)2888 2371 y(1)g(+)g Fv(\014)3082 2341 y Ft(\(2\))p 2888 2408 V 3009 2484 a Fx(2)386 2601 y(where)28 b(\003)685 2613 y Ft(+)p Fo(;)p Ft(2)817 2601 y Fx(=)23 b(\003)963 2558 y Ft(\(1\))963 2622 y(+)1052 2601 y Fx(\003)1110 2558 y Ft(\(2\))1110 2622 y(+)1252 2601 y Fx(\(as)28 b(short-hand)f(for)h(\003)1994 2558 y Ft(\(1\))1994 2622 y(+)2101 2601 y Fu(\012)19 b Fx(\003)2243 2558 y Ft(\(2\))2243 2622 y(+)2332 2601 y Fx(\))53 b(is)28 b(the)g(\(tensor\))h(pro)r(duct)f (of)386 2724 y(the)d(single-particle)f(pro)5 b(jectors)23 b(\003)1501 2681 y Ft(\()p Fo(k)q Ft(\))1501 2744 y(+)1618 2724 y Fx(on)n(to)h(the)i(p)r(ositiv)n(e)e(sp)r(ectral)h(subspace)f(of) h Fv(D)3064 2681 y Ft(\()p Fo(k)q Ft(\))3062 2746 y(0)3157 2724 y Fv(;)g Fx(and)386 2823 y(the)37 b(second)e(equalit)n(y)h 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Fu(\012)g Fp(C)2661 4322 y Ft(2)2704 4353 y Fx(\))2736 4322 y Ft(2)2773 4353 y Fv(;)51 b Fx(th)n(us)27 b(reducing)386 4453 y(the)e(single-particle)f(spinor)g(degrees)f(of)i(freedom)g(from)f (4)h(to)g(2.)f(One)h(obtains)f(\(see)h(e.g.)g([6)o(]\))1250 4671 y Fv(h)1298 4637 y Fo(B)s(R)1452 4671 y Fx(=)1606 4567 y Ft(2)1563 4592 y Fs(X)1562 4771 y Fo(k)q Ft(=1)1683 4671 y Fx(\()p Fv(T)1776 4637 y Ft(\()p Fo(k)q Ft(\))1900 4671 y Fx(+)32 b Fv(b)2033 4628 y Ft(\()p Fo(k)q Ft(\))2033 4693 y(1)p Fo(m)2129 4671 y Fx(\))42 b(+)f Fv(v)2352 4637 y Ft(\(12\))3168 4671 y Fx(\(1.5\))p eop %%Page: 3 3 3 2 bop 1606 259 a Ft(HVZ)30 b(THEOREM)1187 b(3)433 496 y Fv(T)494 462 y Ft(\()p Fo(k)q Ft(\))608 496 y Fx(:=)46 b Fv(E)803 508 y Fo(p)837 517 y Fl(k)878 496 y Fv(;)97 b(b)1034 453 y Ft(\()p Fo(k)q Ft(\))1034 518 y(1)p Fo(m)1153 496 y Fx(:=)46 b Fu(\000)22 b Fv(\015)1450 379 y Fs(\022)1511 496 y Fv(A)p Fx(\()p Fv(p)1647 508 y Fo(k)1688 496 y Fx(\))1767 440 y(1)p 1744 477 89 4 v 1744 553 a Fv(x)1791 565 y Fo(k)1856 496 y Fv(A)p Fx(\()p Fv(p)1992 508 y Fo(k)2033 496 y Fx(\))33 b(+)f Fn(\033)2255 462 y Ft(\()p Fo(k)q Ft(\))2347 496 y FA(p)2400 508 y Fo(k)2455 496 y Fv(g)s Fx(\()p Fv(p)2572 508 y Fo(k)2613 496 y Fx(\))2692 440 y(1)p 2669 477 V 2669 553 a Fv(x)2716 565 y Fo(k)2781 496 y Fv(g)s Fx(\()p Fv(p)2898 508 y Fo(k)2938 496 y Fx(\))14 b Fn(\033)3044 462 y Ft(\()p Fo(k)q Ft(\))3137 496 y FA(p)3190 508 y Fo(k)3231 379 y Fs(\023)386 707 y Fx(where)32 b Fn(\033)691 677 y Ft(\()p Fo(k)q Ft(\))816 707 y Fx(is)h(the)f(v)n(ector)g(of)g(P)n(auli)g(matrices,)f(while)i Fv(v)2252 677 y Ft(\(12\))2407 707 y Fx(results)f(from)g(the)h (electron-)386 807 y(electron)27 b(in)n(teraction)f(term)i(and)f(is)h (sp)r(eci\014ed)g(later)f(\(section)g(5\).)575 907 y(W)-7 b(e)27 b(note)g(that)g Fv(h)1128 877 y Fo(B)s(R)1262 907 y Fx(is)f(a)g(w)n(ell-de\014ned)h(\(in)g(the)g(form)g(sense\),)f(p) r(ositiv)n(e)g(op)r(erator)f(for)386 1006 y(p)r(oten)n(tial)35 b(strengths)f Fv(\015)40 b Fu(\024)35 b Fv(\015)1338 1018 y Fo(B)s(R)1481 1006 y Fx(=)1708 974 y Ft(2)p 1591 988 268 4 v 1591 1035 a Fo(\031)r(=)p Ft(2+2)p Fo(=\031)1903 1006 y Fu(\031)g Fx(0)p Fv(:)p Fx(906)69 b(whic)n(h)35 b(relies)f(on)h(the)g(estimates)386 1144 y(\()p Fv(u;)503 1082 y Fs(P)590 1103 y Ft(2)590 1169 y Fo(k)q Ft(=1)715 1144 y Fx(\()p Fv(T)808 1114 y Ft(\()p Fo(k)q Ft(\))911 1144 y Fx(+)11 b Fv(b)1023 1101 y Ft(\()p Fo(k)q Ft(\))1023 1167 y(1)p Fo(m)1119 1144 y Fx(\))j Fv(u)p Fx(\))37 b Fu(\025)22 b Fx(2)p Fv(m)p Fx(\(1)11 b Fu(\000)g Fv(\015)5 b Fx(\)\()p Fv(u;)14 b(u)p Fx(\))46 b([24)o(])24 b(and)g Fv(V)2346 1114 y Ft(\(12\))2491 1144 y Fu(\025)f Fx(0)p Fv(:)46 b Fx(In)24 b(particular)f(from)386 1244 y(the)35 b(resp)r(ectiv)n(e)f(prop)r(ert)n(y)g(of)g(the)h(single-particle)f(op)r (erator)f([6)o(])i(and)g(using)f([3,)h(11)o(])f(that)386 1360 y(\()p Fv(\036)467 1372 y Ft(+)523 1360 y Fv(;)14 b(V)627 1330 y Ft(\(12\))749 1360 y Fv(\036)798 1372 y Ft(+)853 1360 y Fx(\))41 b Fu(\024)1075 1327 y Fo(e)1106 1302 y Fk(2)p 1027 1341 160 4 v 1027 1388 a Ft(2)p Fo(\015)1095 1396 y Fl(B)r(R)1211 1360 y Fx(\()p Fv(\036)1292 1372 y Ft(+)1348 1360 y Fv(;)1385 1298 y Fs(P)1472 1318 y Ft(2)1472 1385 y Fo(k)q Ft(=1)1611 1360 y Fv(T)1672 1330 y Ft(\()p Fo(k)q Ft(\))1764 1360 y Fv(\036)1813 1372 y Ft(+)1869 1360 y Fx(\))p Fv(;)56 b Fx(the)30 b(total)g(p)r(oten)n (tial)g Fv(V)45 b Fx(:=)26 b Fv(b)2922 1317 y Ft(\(1\))2922 1382 y(1)p Fo(m)3038 1360 y Fx(+)19 b Fv(b)3158 1317 y Ft(\(2\))3158 1382 y(1)p Fo(m)3274 1360 y Fx(+)386 1482 y Fv(v)429 1452 y Ft(\(12\))581 1482 y Fx(is)29 b(\()p Fv(T)759 1452 y Ft(\(1\))867 1482 y Fx(+)19 b Fv(T)1012 1452 y Ft(\(2\))1100 1482 y Fx(\)-form)28 b(b)r(ounded)i (with)g(form)e(b)r(ound)i(less)f(than)g(one)g(for)f Fv(\015)i(<)c(\015) 3183 1494 y Fo(B)s(R)3290 1482 y Fv(:)386 1581 y Fx(Th)n(us)33 b Fv(h)650 1551 y Fo(B)s(R)791 1581 y Fx(is)h(a)f(self-adjoin)n(t)g(op) r(erator)f(b)n(y)h(means)g(of)h(the)g(F)-7 b(riedric)n(hs)33 b(extension)g(of)g(the)386 1681 y(restriction)27 b(of)g Fv(h)926 1651 y Fo(B)s(R)1061 1681 y Fx(to)g Fu(A)p Fx(\()p Fv(C)1325 1651 y Fr(1)1319 1702 y Ft(0)1397 1681 y Fx(\()p Fp(R)1483 1651 y Ft(3)1526 1681 y Fx(\))19 b Fu(\012)f Fp(C)1714 1651 y Ft(2)1757 1681 y Fx(\))1789 1651 y Ft(2)1827 1681 y Fv(:)832 1945 y Fx(2.)41 b Fw(The)32 b(HVZ)e(theorem)i(and)f (the)h(stra)-6 b(tegy)32 b(of)g(pr)n(oof)575 2094 y Fx(W)-7 b(e)28 b(in)n(tro)r(duce)f(the)h(three)f(t)n(w)n(o-cluster)g(decomp)r (ositions)f(of)i(our)f(op)r(erator,)1143 2262 y Fv(h)1191 2227 y Fo(B)s(R)1344 2262 y Fx(=)46 b Fv(T)d Fx(+)32 b Fv(a)1688 2274 y Fo(j)1755 2262 y Fx(+)g Fv(r)1889 2274 y Fo(j)1925 2262 y Fv(;)263 b(j)28 b Fx(=)22 b(0)p Fv(;)14 b Fx(1)p Fv(;)g Fx(2)p Fv(;)585 b Fx(\(2.1\))386 2435 y(where)21 b Fv(T)34 b Fx(:=)23 b Fv(T)875 2405 y Ft(\(1\))970 2435 y Fx(+)7 b Fv(T)1103 2405 y Ft(\(2\))1213 2435 y Fx(and)22 b Fv(a)1413 2447 y Fo(j)1470 2435 y Fx(collects)f(all)g(in)n(teractions)g(not)h(in)n(v)n(olving)e(particle) i Fv(j)51 b Fx(\()p Fv(j)28 b Fx(=)386 2535 y(1)p Fv(;)14 b Fx(2)25 b(denotes)h(the)g(t)n(w)n(o)g(electrons)f(and)h Fv(j)i Fx(=)22 b(0)k(refers)f(to)h(the)g(n)n(ucleus)g(whic)n(h)g(is)g (\014xed)g(at)g(the)386 2634 y(origin\).)20 b(The)i(remainder)e Fv(r)1258 2646 y Fo(j)1315 2634 y Fx(is)h(supp)r(osed)g(to)g(v)-5 b(anish)21 b(when)h(particle)e Fv(j)27 b Fx(is)21 b(mo)n(v)n(ed)f(to)h (in\014nit)n(y)386 2734 y(\(resp)r(ectiv)n(ely)33 b(b)r(oth)h (electrons)f(are)f(mo)n(v)n(ed)h(to)g(in\014nit)n(y)h(in)g(the)g(case)f Fv(j)38 b Fx(=)33 b(0\).)g(De\014ne)h(for)386 2834 y Fv(j)28 b Fx(=)23 b(0)p Fv(;)14 b Fx(1)p Fv(;)g Fx(2)p Fv(;)589 3013 y Fx(\006)649 3025 y Ft(0)709 3013 y Fx(:=)23 b(min)874 3065 y Fo(j)986 3013 y Fx(inf)e Fv(\033)s Fx(\()p Fv(T)30 b Fx(+)18 b Fv(a)1389 3025 y Fo(j)1424 3013 y Fx(\))46 b(=)g(min)14 b Fu(f)p Fx(inf)21 b Fv(\033)s Fx(\()p Fv(T)30 b Fx(+)18 b Fv(b)2202 2969 y Ft(\(1\))2202 3035 y(1)p Fo(m)2298 3013 y Fx(\))p Fv(;)c Fx(inf)21 b Fv(\033)s Fx(\()p Fv(T)30 b Fx(+)18 b Fv(v)2769 2978 y Ft(\(12\))2891 3013 y Fx(\))p Fu(g)203 b Fx(\(2.2\))386 3247 y(\(note)38 b(that)f(the)h(t)n(w)n(o)f(electrons)f(mo)n(v)n(e)h (in)g(the)h(same)f(p)r(oten)n(tial,)h(suc)n(h)f(that)h Fv(b)2987 3204 y Ft(\(1\))2987 3270 y(1)p Fo(m)3082 3247 y Fx(\()p FA(x)3164 3259 y Ft(1)3202 3247 y Fx(\))i(=)386 3365 y Fv(b)422 3321 y Ft(\(2\))422 3387 y(1)p Fo(m)518 3365 y Fx(\()p FA(x)600 3377 y Ft(2)638 3365 y Fx(\)\).)28 b(Then)g(w)n(e)f(ha)n(v)n(e)386 3498 y FA(Theorem)j(1)i Fx(\(HVZ)c(theorem\))p FA(.)386 3614 y Fi(L)l(et)34 b Fv(h)582 3584 y Fo(B)s(R)722 3614 y Fx(=)f Fv(T)g Fx(+)22 b Fv(b)1025 3571 y Ft(\(1\))1025 3636 y(1)p Fo(m)1143 3614 y Fx(+)g Fv(b)1266 3571 y Ft(\(2\))1266 3636 y(1)p Fo(m)1384 3614 y Fx(+)g Fv(v)1514 3584 y Ft(\(12\))1671 3614 y Fi(b)l(e)35 b(the)g(two-ele)l(ctr)l(on)g(Br)l(own-R)l(avenhal)t (l)i(op)l(er)l(ator)386 3713 y(with)e(p)l(otential)g(str)l(ength)f Fv(\015)j(<)31 b(\015)1454 3725 y Fo(B)s(R)1562 3713 y Fv(;)j Fi(and)i(let)e(\(2.1\))i(b)l(e)f(its)f(two-cluster)g(de)l(c)l (omp)l(ositions.)386 3813 y(Then)c(the)g(essential)h(sp)l(e)l(ctrum)e (of)h Fv(h)1572 3783 y Fo(B)s(R)1709 3813 y Fi(is)g(given)h(by)1469 3981 y Fv(\033)1516 3993 y Fo(ess)1615 3981 y Fx(\()p Fv(h)1695 3946 y Fo(B)s(R)1803 3981 y Fx(\))37 b(=)23 b([\006)2043 3993 y Ft(0)2080 3981 y Fv(;)14 b Fu(1)p Fx(\))p Fv(:)913 b Fx(\(2.3\))575 4148 y(Ph)n(ysically)27 b(this)j(means)e(that)h(the)h(b)r(ottom)f(of)g(the)h(essen)n(tial)e(sp) r(ectrum)h(is)g(giv)n(en)f(b)n(y)386 4248 y(the)g(ground)e(state)h(of)h (the)f(one-electron)f(ion,)i(increased)e(b)n(y)h(the)h(rest)f(mass)f (of)i(the)f(second)386 4347 y(electron.)575 4447 y(The)40 b(strategy)e(of)h(pro)r(of)g(is)h(based)f(on)g(Simon)h([20)o(])f(\(see) h(also)e([4]\))i(as)f(w)n(ell)g(as)g(on)386 4547 y(Lieb)33 b(and)g(Y)-7 b(au)34 b([14)o(])f(and)g(Lewis)g(et)g(al)g([13)o(].)h(W) -7 b(e)34 b(start)e(b)n(y)h(c)n(haracterizing)e(the)j(essen)n(tial)386 4646 y(sp)r(ectrum)k(b)n(y)g(means)f(of)h(a)f(W)-7 b(eyl)38 b(sequence,)g(lo)r(cated)f(outside)h(a)g(ball)f Fv(B)2840 4658 y Fo(n)2886 4646 y Fx(\(0\))j Fu(\032)g Fp(R)3191 4616 y Ft(6)3272 4646 y Fx(of)386 4746 y(radius)27 b Fv(n)g Fx(cen)n(tered)g(at)h(the)g(origin.)p eop %%Page: 4 4 4 3 bop 386 259 a Ft(4)937 b(D.)22 b(H.)g(JAKUBASSA-AMUNDSEN)386 459 y FA(Lemma)30 b(1.)40 b Fi(L)l(et)30 b Fv(h)1032 429 y Fo(B)s(R)1162 459 y Fx(=)23 b Fv(T)29 b Fx(+)18 b Fv(V)5 b(;)30 b Fi(let)g Fv(V)49 b Fi(b)l(e)29 b(r)l(elatively)j (form)e(b)l(ounde)l(d)g(with)h(r)l(esp)l(e)l(ct)e(to)h Fv(T)12 b Fi(.)386 558 y(Then)23 b Fv(\025)g Fu(2)g Fv(\033)791 570 y Fo(ess)890 558 y Fx(\()p Fv(h)970 528 y Fo(B)s(R)1078 558 y Fx(\))f Fi(i\013)g(ther)l(e)h(exists)e(a)i(se)l(quenc)l(e)e(of)i (functions)f Fv(')2540 570 y Fo(n)2609 558 y Fu(2)h(A)p Fx(\()p Fv(C)2850 528 y Fr(1)2844 579 y Ft(0)2921 558 y Fx(\()p Fp(R)3007 528 y Ft(6)3051 558 y Fu(n)p Fv(B)3156 570 y Fo(n)3200 558 y Fx(\(0\)\))386 658 y Fu(\012)18 b Fp(C)523 628 y Ft(4)566 658 y Fx(\))30 b Fi(with)g Fu(k)p Fv(')904 670 y Fo(n)949 658 y Fu(k)23 b Fx(=)g(1)29 b Fi(such)h(that)1134 800 y Fu(k)p Fx(\()p Fv(h)1256 765 y Fo(B)s(R)1382 800 y Fu(\000)18 b Fv(\025)p Fx(\))23 b Fv(')1622 812 y Fo(n)1668 800 y Fu(k)46 b(\000)-15 b(!)46 b Fx(0)199 b Fi(as)53 b Fv(n)23 b Fu(!)g(1)p Fv(:)578 b Fx(\(2.4\))575 938 y(Recall)35 b(that)g Fv(')1074 950 y Fo(n)1155 938 y Fx(is)g(a)g(t)n(w)n(o-particle)e(function,)j(eac)n(h) e(particle)h(b)r(eing)g(describ)r(ed)g(b)n(y)386 1038 y(a)j(t)n(w)n(o-spinor.)f(F)-7 b(or)38 b(Sc)n(hr\177)-42 b(odinger)36 b(op)r(erators,)h(Lemma)h(1)g(is)h(pro)n(v)n(en)e(in)i([4) o(,)g(Thm)g(3.11].)386 1140 y(W)-7 b(e)33 b(note)f(that)g(Lemma)g(1)g (holds)g(also)f(for)h(the)h(single-particle)e(op)r(erators,)f Fv(h)h Fx(:=)f Fv(T)3144 1110 y Ft(\(1\))3265 1140 y Fx(or)386 1252 y Fv(h)23 b Fx(:=)g Fv(T)629 1221 y Ft(\(1\))735 1252 y Fx(+)18 b Fv(b)854 1208 y Ft(\(1\))854 1274 y(1)p Fo(m)950 1252 y Fv(;)28 b Fx(with)g(a)f(pro)r(of)g(closely)g(follo)n (wing)f(the)i(one)f(giv)n(en)g(in)h(section)f(7.)386 1370 y FA(Lemma)i(2)j Fx(\(P)n(ersson's)26 b(theorem\))p FA(.)386 1470 y Fi(L)l(et)38 b Fv(h)586 1440 y Fo(B)s(R)733 1470 y Fx(=)g Fv(T)e Fx(+)25 b Fv(V)19 b Fi(,)39 b(let)f Fv(V)58 b Fi(b)l(e)39 b(r)l(elatively)h(form)f(b)l(ounde)l(d)g(with)g (r)l(esp)l(e)l(ct)g(to)f Fv(T)12 b Fi(,)39 b(and)g(let)386 1570 y Fv(')23 b Fu(2)h(A)p Fx(\()p Fv(C)705 1540 y Fr(1)699 1590 y Ft(0)776 1570 y Fx(\()p Fp(R)862 1540 y Ft(6)905 1570 y Fu(n)p Fv(B)1010 1582 y Fo(R)1064 1570 y Fx(\(0\)\))19 b Fu(\012)f Fp(C)1358 1540 y Ft(4)1401 1570 y Fx(\))p Fv(:)53 b Fi(Then)1146 1711 y Fx(inf)21 b Fv(\033)1308 1723 y Fo(ess)1407 1711 y Fx(\()p Fv(h)1487 1677 y Fo(B)s(R)1595 1711 y Fx(\))46 b(=)79 b(lim)1784 1765 y Fo(R)p Fr(!1)2028 1711 y Fx(inf)1980 1769 y Fr(k)p Fo(')p Fr(k)p Ft(=1)2176 1711 y Fx(\()p Fv(';)14 b(h)2347 1677 y Fo(B)s(R)2469 1711 y Fv(')p Fx(\))p Fv(:)590 b Fx(\(2.5\))575 1901 y(The)36 b(pro)r(of)f(giv)n(en)f(in)i([4,)f(Thm)h(3.12])f(relies)g(on)g (Lemma)g(1)g(and)h(on)f(the)h(min-max)386 2001 y(principle)27 b([18,)g(XI)r(I)r(I.1],)h([12,)f(p.60].)g(It)h(also)f(holds)g(in)h(our) f(case.)575 2100 y(Let)18 b(us)h(no)n(w)e(in)n(tro)r(duce)h(the)h (Ruelle-Simon)g(partition)e(of)i(unit)n(y)f(\()p Fv(\036)2679 2112 y Fo(j)2715 2100 y Fx(\))2747 2112 y Fo(j)s Ft(=0)p Fo(;)p Ft(1)p Fo(;)p Ft(2)2995 2100 y Fu(2)23 b Fv(C)3138 2070 y Fr(1)3209 2100 y Fx(\()p Fp(R)3295 2070 y Ft(6)3339 2100 y Fx(\))386 2200 y(whic)n(h)29 b(is)h(sub)r(ordinate)f(to)g(the)h (cluster)f(decomp)r(osition)g(\(2.1\),)h(see)f(e.g.)g([4,)h(p.33],)f ([21)o(].)g(It)386 2299 y(is)e(de\014ned)h(on)g(the)g(unit)g(sphere)f (and)g(has)g(the)h(follo)n(wing)f(prop)r(erties,)856 2408 y Ft(2)813 2433 y Fs(X)815 2610 y Fo(j)s Ft(=0)947 2512 y Fv(\036)996 2478 y Ft(2)996 2533 y Fo(j)1079 2512 y Fx(=)46 b(1)p Fv(;)179 b(\036)1483 2524 y Fo(j)1519 2512 y Fx(\()p Fv(\025)p FA(x)p Fx(\))47 b(=)f Fv(\036)1888 2524 y Fo(j)1924 2512 y Fx(\()p FA(x)p Fx(\))111 b(for)27 b Fv(x)d Fx(=)e(1)51 b(and)27 b Fv(\025)d Fu(\025)e Fx(1)p Fv(;)393 2752 y Fx(supp)14 b Fv(\036)627 2764 y Fo(j)695 2752 y Fu(\\)32 b Fp(R)836 2718 y Ft(6)879 2752 y Fu(n)p Fv(B)984 2764 y Ft(1)1021 2752 y Fx(\(0\))37 b Fu(\022)23 b(f)p FA(x)g Fu(2)g Fp(R)1499 2718 y Ft(6)1542 2752 y Fu(n)p Fv(B)1647 2764 y Ft(1)1684 2752 y Fx(\(0\))g(:)37 b Fu(j)p FA(x)1946 2764 y Ft(1)2002 2752 y Fu(\000)18 b FA(x)2135 2764 y Ft(2)2173 2752 y Fu(j)23 b(\025)g Fv(C)6 b(x)28 b Fx(and)g Fv(x)2656 2764 y Fo(j)2714 2752 y Fu(\025)23 b Fv(C)6 b(x)p Fu(g)p Fv(;)60 b(j)28 b Fx(=)23 b(1)p Fv(;)14 b Fx(2)p Fv(;)601 2881 y Fx(supp)g Fv(\036)835 2893 y Ft(0)905 2881 y Fu(\\)33 b Fp(R)1047 2847 y Ft(6)1090 2881 y Fu(n)p Fv(B)1195 2893 y Ft(1)1231 2881 y Fx(\(0\))k Fu(\022)g(f)p FA(x)23 b Fu(2)g Fp(R)1723 2847 y Ft(6)1766 2881 y Fu(n)p Fv(B)1871 2893 y Ft(1)1908 2881 y Fx(\(0\))g(:)37 b Fv(x)2144 2893 y Fo(k)2209 2881 y Fu(\025)22 b Fv(C)6 b(x)47 b Fu(8)22 b Fv(k)k Fu(2)e(f)p Fx(1)p Fv(;)14 b Fx(2)p Fu(g)g(g)p Fv(;)212 b Fx(\(2.6\))386 3002 y(where)27 b FA(x)c Fx(=)g(\()p FA(x)869 3014 y Ft(1)907 3002 y Fv(;)14 b FA(x)994 3014 y Ft(2)1032 3002 y Fx(\))p Fv(;)60 b(x)23 b Fx(=)g Fu(j)p FA(x)p Fu(j)51 b Fx(and)28 b Fv(C)34 b Fx(is)27 b(a)g(p)r(ositiv)n(e)g(constan)n(t.)575 3101 y(According)19 b(to)i(Lemma)f(2,)g(w)n(e)g(can)g(alw)n(a)n(ys)f(assume) h Fv(')j Fu(2)g(A)p Fx(\()p Fv(C)2548 3071 y Fr(1)2542 3122 y Ft(0)2620 3101 y Fx(\()p Fp(R)2706 3071 y Ft(6)2749 3101 y Fu(n)p Fv(B)2854 3113 y Fo(R)2908 3101 y Fx(\(0\)\))t Fu(\012)t Fp(C)3173 3071 y Ft(4)3217 3101 y Fx(\))d(in)386 3201 y(the)j(follo)n(wing.)g(F)-7 b(or)22 b(later)g(use)h(w)n(e)g(in)n (tro)r(duce)g(a)g(smo)r(oth)f(auxiliary)g(function)i Fv(\037)f Fu(2)g Fv(C)3083 3171 y Fr(1)3154 3201 y Fx(\()p Fp(R)3240 3171 y Ft(6)3283 3201 y Fx(\))p Fv(;)386 3301 y Fx(ran)i Fv(\037)e Fx(=)f([0)p Fv(;)14 b Fx(1])p Fv(;)25 b Fx(whic)n(h)h(is)f(unit)n(y)h(on)f(the)i(supp)r(ort)e(of)h Fv(':)g Fx(Then,)f Fv(\037')f Fx(=)e Fv(':)50 b Fx(Ha)n(ving)24 b(in)i(mind)386 3400 y(\(2.2\))h(w)n(e)g(aim)h(at)f(a)h(lo)r (calization)e(form)n(ula)h(for)g(our)g(op)r(erator.)f(W)-7 b(e)27 b(write)1227 3617 y(\()p Fv(';)14 b(h)1398 3583 y Fo(B)s(R)1506 3617 y Fv(')p Fx(\))47 b(=)f(\()1825 3514 y Ft(2)1782 3539 y Fs(X)1785 3715 y Fo(j)s Ft(=0)1916 3617 y Fv(\036)1965 3629 y Fo(j)2014 3617 y Fv(\037\036)2115 3629 y Fo(j)2164 3617 y Fv(';)14 b(h)2303 3583 y Fo(B)s(R)2411 3617 y Fv(')p Fx(\))972 3935 y(=)1126 3831 y Ft(2)1082 3856 y Fs(X)1085 4033 y Fo(j)s Ft(=0)1202 3935 y Fx(\()p Fv(\036)1283 3947 y Fo(j)1319 3935 y Fv(';)g(h)1458 3901 y Fo(B)s(R)1566 3935 y Fv(\037\036)1667 3947 y Fo(j)1716 3935 y Fv(')p Fx(\))42 b Fu(\000)1993 3831 y Ft(2)1950 3856 y Fs(X)1952 4033 y Fo(j)s Ft(=0)2070 3935 y Fx(\()p Fv(\036)2151 3947 y Fo(j)2187 3935 y Fv(';)14 b Fx([)p Fv(h)2349 3901 y Fo(B)s(R)2456 3935 y Fv(;)g(\037\036)2594 3947 y Fo(j)2629 3935 y Fx(])g Fv(')p Fx(\))416 b(\(2.7\))386 4145 y(where)31 b([)p Fv(B)t(;)14 b(A)p Fx(])29 b(=)f Fv(B)t(A)22 b Fu(\000)e Fv(AB)36 b Fx(denotes)30 b(the)i(comm)n (utator.)e(One)h(can)f(sho)n(w)h(that)g(not)g(only)386 4244 y(the)26 b(con)n(tribution)g(of)g Fv(r)1129 4256 y Fo(j)1190 4244 y Fx(to)g(the)g(\014rst)g(term)g(of)g(\(2.7\))g(v)-5 b(anishes)25 b(uniformly)h(as)f Fv(R)f Fu(!)f(1)p Fv(;)j Fx(but)386 4344 y(also)g(the)i(second)f(term)h(con)n(taining)f(the)h (comm)n(utator.)e(More)h(precisely)-7 b(,)26 b(one)i(has)386 4463 y FA(Lemma)39 b(3.)45 b Fi(L)l(et)37 b Fv(h)1053 4433 y Fo(B)s(R)1197 4463 y Fx(=)f Fv(T)f Fx(+)24 b Fv(a)1515 4475 y Fo(j)1573 4463 y Fx(+)g Fv(r)1699 4475 y Fo(j)1734 4463 y Fv(;)87 b Fx(\()p Fv(\036)1925 4475 y Fo(j)1961 4463 y Fx(\))1993 4475 y Fo(j)s Ft(=0)p Fo(;)p Ft(1)p Fo(;)p Ft(2)2255 4463 y Fi(the)38 b(R)n(uel)t(le-Simon)f(p)l(artition)h (of)386 4562 y(unity)29 b(and)h Fv(')24 b Fu(2)f(A)p Fx(\()p Fv(C)1080 4532 y Fr(1)1074 4583 y Ft(0)1151 4562 y Fx(\()p Fp(R)1238 4532 y Ft(6)1281 4562 y Fu(n)p Fv(B)1386 4574 y Fo(R)1440 4562 y Fx(\(0\)\))c Fu(\012)f Fp(C)1734 4532 y Ft(4)1777 4562 y Fx(\))30 b Fi(with)g Fv(R)24 b(>)f Fx(1)p Fv(:)52 b Fi(Then)1056 4722 y Fu(j)p Fx(\()p Fv(\036)1160 4734 y Fo(j)1196 4722 y Fv(';)14 b(r)1324 4734 y Fo(j)1374 4722 y Fv(\036)1423 4734 y Fo(j)1458 4722 y Fv(')p Fx(\))p Fu(j)47 b(\024)1748 4666 y Fv(c)p 1734 4703 64 4 v 1734 4779 a(R)1831 4722 y Fu(k)p Fv(')p Fu(k)1969 4688 y Ft(2)2006 4722 y Fv(;)268 b(j)28 b Fx(=)23 b(0)p Fv(;)14 b Fx(1)p Fv(;)g Fx(2)p Fv(;)498 b Fx(\(2.8\))p eop %%Page: 5 5 5 4 bop 1606 259 a Ft(HVZ)30 b(THEOREM)1187 b(5)386 459 y Fi(wher)l(e)30 b Fv(c)g Fi(is)g(some)g(c)l(onstant.)575 576 y Fx(W)-7 b(e)31 b(call)g(an)g(op)r(erator)f Fu(O)1480 544 y Ft(1)p 1471 558 51 4 v 1471 605 a Fo(R)1531 576 y Fx(-b)r(ounded)i(if)f Fu(O)j Fx(is)d(b)r(ounded)h(b)n(y)2650 544 y Fo(c)p 2640 558 V 2640 605 a(R)2700 576 y Fv(:)60 b Fx(Th)n(us)31 b(Lemma)g(3)386 683 y(states)c(that)h Fv(\036)851 695 y Fo(j)886 683 y Fv(r)923 695 y Fo(j)959 683 y Fv(\036)1008 695 y Fo(j)1071 683 y Fx(is)1173 651 y Ft(1)p 1164 665 V 1164 712 a Fo(R)1225 683 y Fx(-b)r(ounded.)386 808 y FA(Lemma)h(4.)41 b Fi(Assume)29 b Fv(h)1198 778 y Fo(B)s(R)1305 808 y Fv(;)37 b(\036)1414 820 y Fo(j)1480 808 y Fi(and)30 b Fv(')g Fi(as)g(in)g(L)l(emma)g(3,)g Fv(R)24 b(>)f Fx(2)p Fv(:)52 b Fi(Then)1073 1000 y Fx(\()p Fv(a)p Fx(\))253 b Fu(j)1512 922 y Ft(2)1484 938 y Fs(P)1471 1073 y Fo(j)s Ft(=0)1586 1000 y Fx(\()p Fv(\036)1667 1012 y Fo(j)1702 1000 y Fv(';)14 b Fx([)p Fv(T)7 b(;)14 b(\036)1958 1012 y Fo(j)1993 1000 y Fx(])g Fv(')p Fx(\))p Fu(j)84 b(\024)2376 944 y Fv(c)p 2344 981 101 4 v 2344 1057 a(R)2408 1033 y Ft(2)2477 1000 y Fu(k)p Fv(')p Fu(k)2615 966 y Ft(2)1081 1218 y Fx(\()p Fv(b)p Fx(\))279 b Fu(j)p Fx(\()p Fv(\036)1564 1230 y Fo(j)1600 1218 y Fv(';)14 b Fx([)p Fv(b)1750 1175 y Ft(\()p Fo(k)q Ft(\))1750 1240 y(1)p Fo(m)1846 1218 y Fv(;)g(\036)1932 1230 y Fo(j)1967 1218 y Fx(])g Fv(')p Fx(\))p Fu(j)110 b(\024)2357 1162 y Fv(c)p 2344 1199 64 4 v 2344 1275 a(R)2440 1218 y Fu(k)p Fv(')p Fu(k)2578 1184 y Ft(2)3168 1218 y Fx(\(2.9\))1080 1400 y(\()p Fv(c)p Fx(\))263 b Fu(j)p Fx(\()p Fv(\036)1547 1412 y Fo(j)1583 1400 y Fv(';)14 b Fx([)p Fv(v)1740 1370 y Ft(\(12\))1863 1400 y Fv(;)g(\036)1949 1412 y Fo(j)1984 1400 y Fx(])g Fv(')p Fx(\))p Fu(j)93 b(\024)2357 1344 y Fv(c)p 2344 1381 V 2344 1457 a(R)2440 1400 y Fu(k)p Fv(')p Fu(k)2578 1366 y Ft(2)386 1559 y Fi(wher)l(e)30 b Fv(c)g Fi(is)g(a)g(generic)h(c)l(onstant.)575 1676 y Fx(The)d(pro)r(of)f(of)g(\()p Fv(a)p Fx(\))h(in)g(Lemma)f(4)h(is)f (pro)n(vided)g(in)h([13)o(].)575 1776 y(With)g(Lemmata)g(3)f(and)g(4,)g (\(2.7\))h(turns)f(in)n(to)h(the)g(lo)r(calization)e(form)n(ula)913 1990 y(\()p Fv(';)14 b(h)1084 1956 y Fo(B)s(R)1192 1990 y Fv(')p Fx(\))46 b(=)1478 1887 y Ft(2)1435 1912 y Fs(X)1438 2088 y Fo(j)s Ft(=0)1555 1990 y Fx(\()p Fv(\036)1636 2002 y Fo(j)1672 1990 y Fv(';)14 b Fx(\()p Fv(T)30 b Fx(+)18 b Fv(a)2001 2002 y Fo(j)2036 1990 y Fx(\))c Fv(\036)2131 2002 y Fo(j)2166 1990 y Fv(')p Fx(\))42 b(+)f Fv(O)r Fx(\()2519 1934 y(1)p 2507 1971 V 2507 2047 a Fv(R)2582 1990 y Fx(\))23 b Fu(k)p Fv(')p Fu(k)2775 1956 y Ft(2)3127 1990 y Fx(\(2.10\))386 2214 y(for)k Fv(R)d(>)e Fx(2)p Fv(:)51 b Fx(Using)27 b(P)n(ersson's)e(theorem)i(\(Lemma)h(2\),)g(w)n (e)f(obtain)899 2429 y(inf)35 b Fv(\033)1075 2441 y Fo(ess)1174 2429 y Fx(\()p Fv(h)1254 2395 y Fo(B)s(R)1361 2429 y Fx(\))47 b(=)79 b(lim)1551 2482 y Fo(R)p Fr(!1)1808 2429 y Fx(inf)1761 2486 y Fr(k)p Fo(')p Fr(k)p Ft(=1)2028 2325 y(2)1984 2350 y Fs(X)1987 2527 y Fo(j)s Ft(=0)2104 2429 y Fx(\()p Fv(\036)2185 2441 y Fo(j)2221 2429 y Fv(';)14 b Fx(\()p Fv(T)30 b Fx(+)18 b Fv(a)2550 2441 y Fo(j)2585 2429 y Fx(\))c Fv(\036)2680 2441 y Fo(j)2715 2429 y Fv(')p Fx(\))p Fv(:)303 b Fx(\(2.11\))386 2653 y(Recalling)24 b(the)i(de\014nition)f(\(2.2\))g(of)g(\006)1599 2665 y Ft(0)1661 2653 y Fx(in)g(terms)g(of)g(the)g(smallest)g(in\014m)n(um)h (of)f Fv(\033)s Fx(\()p Fv(T)g Fx(+)13 b Fv(a)3226 2665 y Fo(j)3260 2653 y Fx(\))p Fv(;)386 2752 y Fx(w)n(e)27 b(can)g(estimate)863 2946 y(inf)815 3003 y Fr(k)p Fo(')p Fr(k)p Ft(=1)1091 2842 y(2)1048 2867 y Fs(X)1050 3044 y Fo(j)s Ft(=0)1168 2946 y Fx(\()p Fv(\036)1249 2958 y Fo(j)1285 2946 y Fv(';)14 b Fx(\()p Fv(T)29 b Fx(+)19 b Fv(a)1614 2958 y Fo(j)1648 2946 y Fx(\))14 b Fv(\036)1743 2958 y Fo(j)1779 2946 y Fv(')p Fx(\))47 b Fu(\025)2066 2842 y Ft(2)2022 2867 y Fs(X)2025 3044 y Fo(j)s Ft(=0)2156 2946 y Fx(\006)2216 2958 y Ft(0)2276 2946 y Fx(\()p Fv(\036)2357 2958 y Fo(j)2393 2946 y Fv(';)14 b(\036)2533 2958 y Fo(j)2569 2946 y Fv(')p Fx(\))47 b(=)e(\006)2872 2958 y Ft(0)3127 2946 y Fx(\(2.12\))386 3232 y(since)630 3153 y Ft(2)603 3170 y Fs(P)589 3305 y Fo(j)s Ft(=0)718 3232 y Fv(\036)767 3202 y Ft(2)767 3254 y Fo(j)828 3232 y Fx(=)22 b(1)37 b(=)23 b Fu(k)p Fv(')p Fu(k)p Fv(:)50 b Fx(This)27 b(pro)n(v)n(es)f (the)i(inclusion)f Fv(\033)2275 3244 y Fo(ess)2374 3232 y Fx(\()p Fv(h)2454 3202 y Fo(B)s(R)2562 3232 y Fx(\))37 b Fu(\032)22 b Fx([\006)2801 3244 y Ft(0)2839 3232 y Fv(;)14 b Fu(1)p Fx(\))p Fv(:)575 3403 y Fx(F)-7 b(or)33 b(the)h(remaining)f(inclusion,)h([\006)1732 3415 y Ft(0)1769 3403 y Fv(;)14 b Fu(1)p Fx(\))33 b Fu(\032)g Fv(\033)2099 3415 y Fo(ess)2198 3403 y Fx(\()p Fv(h)2278 3373 y Fo(B)s(R)2386 3403 y Fx(\))p Fv(;)67 b Fx(in)34 b(the)g(literature)f(called)386 3503 y(the)38 b('easy)f(part')g(of)h(the)g(pro)r(of)g(of)f(the)i(HVZ)f (theorem,)f(w)n(e)g(use)h(the)g(strategy)f(of)g(W)-7 b(eyl)386 3602 y(sequences)27 b([4)o(,)h(21)o(].)575 3702 y(Let)j Fv(\025)d Fu(2)g Fx([\006)969 3714 y Ft(0)1007 3702 y Fv(;)14 b Fu(1)p Fx(\))p Fv(:)59 b Fx(Consider)29 b(the)i(case)f(that)h(\006)2158 3714 y Ft(0)2223 3702 y Fx(=)c(inf)21 b Fv(\033)s Fx(\()p Fv(T)32 b Fx(+)20 b Fv(a)2722 3714 y Fo(j)2757 3702 y Fx(\))31 b(for)f Fv(j)j Fx(=)28 b(1)p Fv(;)i Fx(and)386 3802 y(assume)22 b(that)h Fv(\025)h Fu(2)f Fv(\033)s Fx(\()p Fv(T)d Fx(+)9 b Fv(a)1262 3814 y Ft(1)1299 3802 y Fx(\))p Fv(:)46 b Fx(\(This)23 b(assumption)g(is)f(pro)n(v)n(en)f(in)i(\(2.16\))f(where)h (it)g(is)g(sho)n(wn)386 3917 y(that)31 b Fv(\033)s Fx(\()p Fv(T)i Fx(+)20 b Fv(a)862 3929 y Ft(1)899 3917 y Fx(\))30 b(=)e Fv(\033)1101 3929 y Fo(ess)1200 3917 y Fx(\()p Fv(T)k Fx(+)20 b Fv(a)1442 3929 y Ft(1)1480 3917 y Fx(\))p Fv(:)p Fx(\))60 b(Since)32 b Fv(T)f Fx(+)21 b Fv(a)2058 3929 y Ft(1)2124 3917 y Fx(=)28 b Fv(T)2278 3887 y Ft(\(1\))2401 3917 y Fx(+)34 b(\()p Fv(T)2593 3887 y Ft(\(2\))2703 3917 y Fx(+)20 b Fv(b)2824 3874 y Ft(\(2\))2824 3939 y(1)p Fo(m)2920 3917 y Fx(\))60 b(describ)r(es)386 4021 y(t)n(w)n(o)32 b(indep)r(enden)n(t)h(particles,)f(w)n(e)h(can)f(decomp) r(ose)g Fv(\025)g Fx(=)f Fv(\025)2316 4033 y Ft(1)2376 4021 y Fx(+)21 b Fv(\025)2510 4033 y Ft(2)2581 4021 y Fx(with)33 b Fv(\025)2823 4033 y Ft(1)2892 4021 y Fu(2)f Fv(\033)3026 4033 y Fo(ess)3125 4021 y Fx(\()p Fv(T)3218 3991 y Ft(\(1\))3306 4021 y Fx(\))386 4137 y(and)27 b Fv(\025)595 4149 y Ft(2)656 4137 y Fu(2)c Fv(\033)s Fx(\()p Fv(T)877 4107 y Ft(\(2\))985 4137 y Fx(+)18 b Fv(b)1104 4094 y Ft(\(2\))1104 4159 y(1)p Fo(m)1200 4137 y Fx(\))p Fv(:)575 4254 y Fx(Let)27 b(\()p Fv(')809 4211 y Ft(\(1\))809 4264 y Fo(n)898 4254 y Fx(\))930 4266 y Fo(n)p Fr(2)p Fh(N)1089 4254 y Fx(b)r(e)g(a)g(W)-7 b(eyl)27 b(sequence)f(corresp)r (onding)f(to)h Fv(\025)2503 4266 y Ft(1)2541 4254 y Fv(;)h Fx(i.e.)f Fv(')2777 4211 y Ft(\(1\))2777 4264 y Fo(n)2894 4254 y Fx(is)g(c)n(haracter-)386 4354 y(ized)i(b)n(y)601 4495 y Fu(k)p Fv(')697 4460 y Ft(\(1\))697 4515 y Fo(n)785 4495 y Fu(k)37 b Fx(=)23 b(1)p Fv(;)96 b(')1167 4460 y Ft(\(1\))1167 4515 y Fo(n)1296 4448 y(w)1279 4495 y Fv(*)24 b Fx(0)p Fv(;)179 b Fu(k)p Fx(\()p Fv(T)1765 4460 y Ft(\(1\))1871 4495 y Fu(\000)18 b Fv(\025)2002 4507 y Ft(1)2040 4495 y Fx(\))c Fv(')2140 4460 y Ft(\(1\))2140 4515 y Fo(n)2230 4495 y Fu(k)36 b(!)23 b Fx(0)46 b(as)k Fv(n)23 b Fu(!)g(1)p Fv(:)215 b Fx(\(2.13\))386 4646 y(F)-7 b(rom)38 b(Lemma)h(1)g(w)n(e)f(can)h(require)f Fv(')1646 4603 y Ft(\(1\))1646 4656 y Fo(n)1777 4646 y Fu(2)43 b Fv(C)1940 4616 y Fr(1)1934 4667 y Ft(0)2010 4646 y Fx(\()p Fp(R)2097 4616 y Ft(3)2140 4646 y Fu(n)p Fv(B)2245 4658 y Fo(n)2290 4646 y Fx(\(0\)\))26 b Fu(\012)g Fp(C)2599 4616 y Ft(2)2642 4646 y Fv(:)81 b Fx(Let)39 b(\()p Fv(\036)2987 4603 y Ft(\(2\))2987 4656 y Fo(n)3077 4646 y Fx(\))3109 4658 y Fo(n)p Fr(2)p Fh(N)3283 4646 y Fu(2)386 4746 y Fv(C)451 4716 y Fr(1)445 4766 y Ft(0)522 4746 y Fx(\()p Fp(R)608 4716 y Ft(3)651 4746 y Fx(\))26 b Fu(\012)g Fp(C)854 4716 y Ft(2)937 4746 y Fx(b)r(e)39 b(a)g(de\014ning)g(sequence)g(for)f Fv(\025)2009 4758 y Ft(2)2086 4746 y Fx(according)g(to)h([27)o(,)g(Thm)g(7.22])f(with)p eop %%Page: 6 6 6 5 bop 386 259 a Ft(6)937 b(D.)22 b(H.)g(JAKUBASSA-AMUNDSEN)386 462 y Fu(k)p Fv(\036)477 419 y Ft(\(2\))477 472 y Fo(n)566 462 y Fu(k)40 b Fx(=)27 b(1)p Fv(:)56 b Fx(Since)44 b Fu(k)p Fx(\()p Fv(T)1229 432 y Ft(\(2\))1337 462 y Fx(+)19 b Fv(b)1457 419 y Ft(\(2\))1457 485 y(1)p Fo(m)1573 462 y Fu(\000)h Fv(\025)1706 474 y Ft(2)1743 462 y Fx(\))14 b Fv(\036)1838 419 y Ft(\(2\))1838 472 y Fo(n)1928 462 y Fu(k)40 b(!)27 b Fx(0)56 b(as)30 b Fv(n)23 b Fu(!)g(1)p Fv(;)30 b Fx(for)f(an)n(y)g(giv)n(en)g Fv(\017)e(>)f Fx(0)386 562 y(there)h(exists)h Fv(N)j Fu(2)24 b Fp(N)37 b Fx(suc)n(h)28 b(that)1321 702 y Fu(k)p Fx(\()p Fv(T)1456 667 y Ft(\(2\))1562 702 y Fx(+)18 b Fv(b)1681 658 y Ft(\(2\))1681 724 y(1)p Fo(m)1795 702 y Fu(\000)g Fv(\025)1926 714 y Ft(2)1964 702 y Fx(\))c Fv(\036)2059 658 y Ft(\(2\))2059 726 y Fo(N)2149 702 y Fu(k)45 b Fv(<)h(\017:)723 b Fx(\(2.14\))575 861 y(W)-7 b(e)32 b(de\014ne)g(the)h(sequence)e Fv( )1516 873 y Fo(n)1592 861 y Fx(:=)e Fv(')1763 818 y Ft(\(1\))1763 871 y Fo(n)1874 861 y Fu(\001)22 b Fv(\036)1968 818 y Ft(\(2\))1968 885 y Fo(N)2119 861 y Fx(for)32 b Fv(n)e Fu(2)g Fp(N)42 b Fx(and)32 b(the)g(an)n(tisymmetric)386 980 y(sequence)c Fu(A)14 b Fv( )865 992 y Fo(n)936 980 y Fx(:=)1086 948 y Ft(1)p 1059 962 88 4 v 1059 970 a Fr(p)p 1113 970 34 3 v 1113 1018 a Ft(2)1170 980 y Fx(\()p Fv(')1256 937 y Ft(\(1\))1256 990 y Fo(n)1346 980 y Fv(\036)1395 937 y Ft(\(2\))1395 1005 y Fo(N)1504 980 y Fu(\000)19 b Fv(')1642 937 y Ft(\(2\))1642 990 y Fo(n)1731 980 y Fv(\036)1780 937 y Ft(\(1\))1780 1005 y Fo(N)1870 980 y Fx(\))54 b(and)29 b(claim)f(that)i(a)e(subsequence)g(of)h Fu(A)14 b Fv( )3293 992 y Fo(n)386 1104 y Fx(is)27 b(a)h(W)-7 b(eyl)27 b(sequence)h(for)f Fv(\025)c Fu(2)g Fv(\033)1416 1116 y Fo(ess)1515 1104 y Fx(\()p Fv(h)1595 1073 y Fo(B)s(R)1703 1104 y Fx(\))p Fv(:)546 1233 y Fx(\(i\))42 b(The)27 b(w)n(eak)g(con)n (v)n(ergence,)e Fu(A)14 b Fv( )1669 1245 y Fo(n)1754 1186 y(w)1738 1233 y Fv(*)23 b Fx(0)p Fv(;)50 b Fx(follo)n(ws)27 b(from)455 1316 y Fu(p)p 525 1316 42 4 v 525 1389 a Fx(2)22 b Fu(j)p Fx(\()p Fu(A)14 b Fv( )778 1401 y Fo(n)824 1389 y Fv(;)g(f)911 1354 y Ft(\(1\))999 1389 y Fv(g)1042 1354 y Ft(\(2\))1131 1389 y Fx(\))p Fu(j)37 b(\024)g(j)p Fx(\()p Fv(')1434 1354 y Ft(\(1\))1434 1409 y Fo(n)1524 1389 y Fv(;)14 b(f)1611 1354 y Ft(\(1\))1699 1389 y Fx(\))p Fu(j)19 b(\001)g(j)p Fx(\()p Fv(\036)1919 1346 y Ft(\(2\))1919 1413 y Fo(N)2009 1389 y Fv(;)14 b(g)2089 1354 y Ft(\(2\))2177 1389 y Fx(\))p Fu(j)33 b Fx(+)f Fu(j)p Fx(\()p Fv(\036)2466 1346 y Ft(\(1\))2466 1413 y Fo(N)2556 1389 y Fv(;)14 b(f)2643 1354 y Ft(\(1\))2731 1389 y Fx(\))p Fu(j)19 b(\001)g(j)p Fx(\()p Fv(')2956 1354 y Ft(\(2\))2956 1409 y Fo(n)3045 1389 y Fv(;)14 b(g)3125 1354 y Ft(\(2\))3214 1389 y Fx(\))p Fu(j)643 1553 y(\024)37 b(j)p Fx(\()p Fv(')854 1519 y Ft(\(1\))854 1574 y Fo(n)944 1553 y Fv(;)14 b(f)1031 1519 y Ft(\(1\))1119 1553 y Fx(\))p Fu(j)24 b(k)p Fv(\036)1289 1510 y Ft(\(2\))1289 1577 y Fo(N)1378 1553 y Fu(k)e(k)p Fv(g)1527 1519 y Ft(\(2\))1616 1553 y Fu(k)31 b Fx(+)18 b Fu(j)p Fx(\()p Fv(')1881 1519 y Ft(\(2\))1881 1574 y Fo(n)1971 1553 y Fv(;)c(g)2051 1519 y Ft(\(2\))2140 1553 y Fx(\))p Fu(j)g(k)p Fv(\036)2300 1510 y Ft(\(1\))2300 1577 y Fo(N)2389 1553 y Fu(k)23 b(k)p Fv(f)2546 1519 y Ft(\(1\))2634 1553 y Fu(k)45 b(!)24 b Fx(0)257 b(\(2.15\))675 1695 y(for)27 b(all)g Fv(f)967 1665 y Ft(\(1\))1056 1695 y Fv(;)14 b(g)1136 1665 y Ft(\(2\))1247 1695 y Fu(2)24 b Fv(L)1383 1707 y Ft(2)1420 1695 y Fx(\()p Fp(R)1506 1665 y Ft(3)1549 1695 y Fx(\))19 b Fu(\012)f Fp(C)1737 1665 y Ft(2)1780 1695 y Fv(;)51 b Fx(since)27 b(b)n(y)h(\(2.13\))e Fv(')2466 1652 y Ft(\(1\))2466 1705 y Fo(n)2596 1648 y(w)2579 1695 y Fv(*)d Fx(0)p Fv(:)522 1795 y Fx(\(ii\))66 b(\()p Fv( )784 1807 y Fo(n)829 1795 y Fx(\))861 1807 y Fo(n)p Fr(2)p Fh(N)1021 1795 y Fx(ob)r(eys)27 b(the)h(W)-7 b(eyl)28 b(criterion)f(for)g Fv(\025)c Fu(2)h Fv(\033)2262 1807 y Fo(ess)2360 1795 y Fx(\()p Fv(T)30 b Fx(+)18 b Fv(a)2598 1807 y Ft(1)2635 1795 y Fx(\))28 b(since)1224 1950 y Fu(k)p Fx(\()p Fv(T)1359 1916 y Ft(\(1\))1465 1950 y Fu(\000)18 b Fv(\025)1596 1962 y Ft(1)1652 1950 y Fx(+)g Fv(T)1796 1916 y Ft(\(2\))1903 1950 y Fx(+)g Fv(b)2022 1907 y Ft(\(2\))2022 1972 y(1)p Fo(m)2136 1950 y Fu(\000)g Fv(\025)2267 1962 y Ft(2)2305 1950 y Fx(\))23 b Fv( )2414 1962 y Fo(n)2459 1950 y Fu(k)626 b Fx(\(2.16\))615 2112 y Fu(\024)45 b(k)p Fx(\()p Fv(T)860 2078 y Ft(\(1\))967 2112 y Fu(\000)18 b Fv(\025)1098 2124 y Ft(1)1135 2112 y Fx(\))c Fv(')1235 2078 y Ft(\(1\))1235 2133 y Fo(n)1325 2112 y Fu(k)23 b(k)p Fv(\036)1481 2069 y Ft(\(2\))1481 2137 y Fo(N)1570 2112 y Fu(k)41 b Fx(+)g Fu(k)p Fx(\()p Fv(T)1894 2078 y Ft(\(2\))2000 2112 y Fx(+)18 b Fv(b)2119 2069 y Ft(\(2\))2119 2134 y(1)p Fo(m)2233 2112 y Fu(\000)h Fv(\025)2365 2124 y Ft(2)2402 2112 y Fx(\))14 b Fv(\036)2497 2069 y Ft(\(2\))2497 2137 y Fo(N)2587 2112 y Fu(k)22 b(k)p Fv(')2747 2078 y Ft(\(1\))2747 2133 y Fo(n)2836 2112 y Fu(k)46 b Fv(<)g Fx(2)p Fv(\017)675 2234 y Fx(for)27 b(arbitrary)e Fv(\017)j Fx(and)f Fv(n)h Fx(su\016cien)n(tly)f(large)f (b)r(ecause)h(of)h(\(2.13\))f(and)g(\(2.14\).)499 2334 y(\(iii\))43 b(Using)27 b(that)h Fv(h)1136 2304 y Fo(B)s(R)1271 2334 y Fx(is)f(symmetric)h(up)r(on)f(particle)h(exc)n(hange,)e(w)n(e)h (ha)n(v)n(e)987 2513 y Fu(k)p Fx(\()p Fv(h)1109 2478 y Fo(B)s(R)1234 2513 y Fu(\000)18 b Fv(\025)p Fx(\))c Fu(A)g Fv( )1545 2525 y Fo(n)1591 2513 y Fu(k)46 b(\024)f Fx(2)14 b Fu(k)p Fx(\()p Fv(h)1967 2478 y Fo(B)s(R)2092 2513 y Fu(\000)k Fv(\025)p Fx(\))2314 2456 y(1)p 2280 2494 111 4 v 2280 2510 a Fu(p)p 2349 2510 42 4 v 69 x Fx(2)2414 2513 y Fv(')2468 2478 y Ft(\(1\))2468 2533 y Fo(n)2558 2513 y Fv(\036)2607 2470 y Ft(\(2\))2607 2537 y Fo(N)2696 2513 y Fu(k)386 2730 y(\024)497 2657 y(p)p 566 2657 V 73 x Fx(2)13 b Fu(k)p Fx(\()p Fv(T)27 b Fx(+)15 b Fv(a)895 2742 y Ft(1)948 2730 y Fu(\000)g Fv(\025)p Fx(\))f Fv( )1176 2742 y Fo(n)1222 2730 y Fu(k)38 b Fx(+)1406 2657 y Fu(p)p 1475 2657 V 73 x Fx(2)13 b Fu(k)p Fv(r)1609 2742 y Ft(1)1646 2730 y Fv( )1700 2742 y Fo(n)1746 2730 y Fu(k)45 b Fv(<)h Fx(2)1986 2657 y Fu(p)p 2055 2657 V 73 x Fx(2)13 b Fv(\017)38 b Fx(+)2286 2657 y Fu(p)p 2355 2657 V 73 x Fx(2)13 b Fu(k)p Fv(b)2488 2687 y Ft(\(1\))2488 2752 y(1)p Fo(m)2584 2730 y Fv( )2638 2742 y Fo(n)2683 2730 y Fu(k)38 b Fx(+)2866 2657 y Fu(p)p 2936 2657 V 2936 2730 a Fx(2)13 b Fu(k)p Fv(v)3076 2696 y Ft(\(12\))3198 2730 y Fv( )3252 2742 y 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Fp(R)2095 4616 y Ft(3)2139 4646 y Fx(\))12 b Fu(\012)g Fp(C)2313 4616 y Ft(2)2356 4646 y Fx(\))2388 4616 y Ft(2)2450 4646 y Fx(and)24 b(using)g(Lemma)g(6,)g(this)386 4746 y(pro)n(v)n(es)i(that) h(the)h(r.h.s.)g(of)f(\(2.17\))g(is)h(smaller)e(than)i(4)p Fv(\017)f Fx(for)g(su\016cien)n(tly)g(large)g Fv(n)p Fx(.)p eop %%Page: 7 7 7 6 bop 1606 259 a Ft(HVZ)30 b(THEOREM)1187 b(7)502 459 y Fx(\(iv\))42 b(Concerning)26 b(the)i(normalizabilit)n(y)e(of)i Fu(A)14 b Fv( )2057 471 y Fo(n)2102 459 y Fx(,)28 b(w)n(e)f(ha)n(v)n(e) 435 639 y(2)c Fu(kA)14 b Fv( )676 651 y Fo(n)721 639 y Fu(k)763 605 y Ft(2)846 639 y Fx(=)45 b Fu(k)p Fv(')1052 605 y Ft(\(1\))1052 659 y Fo(n)1141 639 y Fu(k)23 b(k)p Fv(\036)1297 596 y Ft(\(2\))1297 663 y Fo(N)1386 639 y Fu(k)41 b Fx(+)g(2)27 b(Re)h(\()p Fv(')1856 605 y Ft(\(1\))1856 659 y Fo(n)1946 639 y Fv(\036)1995 596 y Ft(\(2\))1995 663 y Fo(N)2084 639 y Fv(;)14 b(')2175 605 y Ft(\(2\))2175 659 y Fo(n)2265 639 y Fv(\036)2314 596 y Ft(\(1\))2314 663 y Fo(N)2403 639 y 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Fo(n)1137 1177 y Fv(\036)1186 1134 y Ft(\(1\))1186 1201 y Fo(N)1276 1177 y Fx(\))46 b(=)1465 1056 y Fs(\014)1465 1106 y(\014)1465 1156 y(\014)1465 1206 y(\014)1492 1064 y(Z)1539 1252 y Fh(R)1586 1236 y Fk(3)1631 1177 y Fv(d)p FA(x)1724 1189 y Ft(1)p 1785 1080 144 4 v 1785 1177 a Fv(')1839 1134 y Ft(\(1\))1839 1186 y Fo(n)1942 1177 y Fv(\036)1991 1134 y Ft(\(1\))1991 1201 y Fo(N)2081 1056 y Fs(\014)2081 1106 y(\014)2081 1156 y(\014)2081 1206 y(\014)2108 1077 y Ft(2)2192 1177 y Fx(=)f(0)166 b(if)28 b Fv(n)23 b(>)g(R)2810 1189 y Ft(0)2847 1177 y Fv(:)257 b Fx(\(2.22\))675 1391 y(Th)n(us)21 b Fu(kA)14 b Fv( )1055 1403 y Fo(n)1100 1391 y Fu(k)36 b Fx(=)23 b(1)44 b(for)21 b(su\016cien)n(tly)h(large)e Fv(n)p Fx(.)h(Therefore)g(a)g(subsequence)g(of)g Fu(A)14 b Fv( )3293 1403 y Fo(n)675 1490 y Fx(is)27 b(a)g(W)-7 b(eyl)28 b(sequence)f(for)g Fv(\025)p Fx(,)h(resulting)g(in)f Fv(\025)d Fu(2)f Fv(\033)2243 1502 y Fo(ess)2342 1490 y Fx(\()p Fv(h)2422 1460 y Fo(B)s(R)2530 1490 y Fx(\))p Fv(:)575 1622 y Fx(Due)38 b(to)f(the)h(symmetry)f(up)r(on)g(particle)g (exc)n(hange,)f(this)i(pro)n(v)n(es)e(the)h(case)g Fv(j)44 b Fx(=)39 b(2)386 1722 y(as)e(w)n(ell.)h(Consideration)e(of)i(the)h (case)e Fv(j)45 b Fx(=)40 b(0)d(can)h(b)r(e)g(omitted,)g(since)g Fv(V)2858 1691 y Ft(\(12\))3021 1722 y Fu(\025)h Fx(0)f(and)386 1833 y(hence)c Fv(v)666 1803 y Ft(\(12\))821 1833 y Fu(\025)f Fx(0)67 b(\(whereas)33 b Fv(V)1447 1803 y Ft(\(1\))1570 1833 y Fu(\024)g Fx(0)g(and)h(so)f Fv(b)2055 1790 y Ft(\(1\))2055 1855 y(1)p Fo(m)2184 1833 y Fu(\024)g Fx(0\))p Fv(:)67 b Fx(Th)n(us)33 b(inf)21 b Fv(\033)s Fx(\()p Fv(T)34 b Fx(+)22 b Fv(v)3072 1803 y Ft(\(12\))3195 1833 y Fx(\))47 b Fu(\025)386 1950 y Fx(inf)21 b Fv(\033)s Fx(\()p Fv(T)30 b Fx(+)18 b Fv(b)781 1907 y Ft(\(1\))781 1972 y(1)p Fo(m)877 1950 y Fx(\))51 b(suc)n(h)27 b(that)h(one)f(has)g(\006)1687 1962 y Ft(0)1747 1950 y Fx(=)c(inf)e Fv(\033)s Fx(\()p Fv(T)30 b Fx(+)18 b Fv(b)2230 1907 y Ft(\(1\))2230 1972 y(1)p Fo(m)2326 1950 y Fx(\))p Fv(:)882 2204 y Fx(3.)41 b Fw(Ingredients)32 b(f)n(or)g(the)f(pr)n(oofs)h(of)g(the)g(lemma)-6 b(t)g(a)575 2353 y Fx(The)27 b(main)h(di\013erence)f(in)g(the)h(pro)r (ofs)e(of)i(Lemmata)e(1,)h(3,)g(5)g(and)g(6)g(as)g(con)n(trasted)f(to) 386 2453 y(the)31 b(pro)r(of)e(of)i(Lemma)f(4)g(lies)g(in)h(the)f(fact) h(that)g(the)f(momen)n(tum)h(represen)n(tation)e(is)h(used)386 2552 y(for)d(the)h(former,)f(whereas)f(the)i(pro)r(of)f(of)h(Lemma)f(4) g(is)h(carried)e(out)h(in)h(co)r(ordinate)f(space.)575 2652 y(An)j(imp)r(ortan)n(t)f(estimate)g(whic)n(h)g(holds)g(in)h (either)f(space)g(is)g(the)h(Lieb)f(and)g(Y)-7 b(au)30 b(for-)386 2752 y(m)n(ula)c(\(whic)n(h)h(is)f(related)g(to)g(the)h(Sc)n (h)n(ur)f(test)h(for)f(the)h(b)r(oundedness)f(of)h(in)n(tegral)e(op)r (erators)386 2851 y(and)h(is)g(easily)g(deriv)n(ed)f(from)h(the)g(Sc)n (h)n(w)n(arz)f(inequalit)n(y\),)h(generalized)f(to)h(the)g(t)n(w)n (o-particle)386 2951 y(case)h([14)o(,)g(11)o(].)386 3083 y FA(Lemma)i(7)j Fx(\(Generalized)27 b(Lieb)h(and)f(Y)-7 b(au)28 b(form)n(ula\))p FA(.)386 3182 y Fi(L)l(et)j Fv(A)h Fi(b)l(e)g(an)f(essential)t(ly)i(self-adjoint)h(inte)l(gr)l(al)e (op)l(er)l(ator)g(and)g Fv(k)2519 3194 y Fo(A)2573 3182 y Fx(\()p Fn(\030)s Fv(;)14 b Fn(\030)2732 3145 y Fr(0)2755 3182 y Fx(\))32 b Fi(its)g(kernel,)g Fn(\030)d Fu(2)386 3282 y Fp(R)440 3252 y Ft(3)p Fo(l)534 3282 y Fi(with)i Fv(l)24 b Fu(2)f(f)p Fx(1)p Fv(;)14 b Fx(2)p Fu(g)28 b Fi(denoting)i(the)g(numb)l(er)f(of)i(p)l(articles.)g(Then)g(for)f Fv( )s(;)14 b(')24 b Fu(2)f(D)r Fx(\()p Fv(A)p Fx(\))p Fv(;)1009 3489 y Fu(j)p Fx(\()p Fv( )s(;)14 b(A)g(')p Fx(\))p Fu(j)47 b Fx(=)f Fu(j)1538 3376 y Fs(Z)1635 3489 y Fv(d)p Fn(\030)p 1746 3416 167 4 v 26 w Fv( )s Fx(\()p Fn(\030)s Fx(\))1940 3376 y Fs(Z)2037 3489 y Fv(d)p Fn(\030)2125 3451 y Fr(0)2171 3489 y Fv(k)2214 3501 y Fo(A)2268 3489 y Fx(\()p Fn(\030)s Fv(;)14 b Fn(\030)2427 3451 y Fr(0)2450 3489 y Fx(\))23 b Fv(')p Fx(\()p Fn(\030)2637 3451 y Fr(0)2660 3489 y Fx(\))p Fu(j)930 3811 y(\024)1041 3694 y Fs(\022)1102 3698 y(Z)1199 3811 y Fv(d)p Fn(\030)j Fu(j)p Fv( )s Fx(\()p Fn(\030)s Fx(\))p Fu(j)1522 3777 y Ft(2)1583 3811 y Fv(I)7 b Fx(\()p Fn(\030)s Fx(\))1735 3694 y Fs(\023)1806 3689 y Fk(1)p 1806 3698 29 3 v 1806 3731 a(2)1867 3811 y Fu(\001)1909 3694 y Fs(\022)1970 3698 y(Z)2067 3811 y Fv(d)p Fn(\030)2154 3774 y Fr(0)2201 3811 y Fu(j)p Fv(')p Fx(\()p Fn(\030)2355 3774 y Fr(0)2378 3811 y Fx(\))p Fu(j)2433 3777 y Ft(2)2494 3811 y Fv(J)h Fx(\()p Fn(\030)2625 3774 y Fr(0)2648 3811 y Fx(\))2680 3694 y Fs(\023)2752 3689 y Fk(1)p 2752 3698 V 2752 3731 a(2)3168 3811 y Fx(\(3.1\))386 4003 y Fi(with)1301 4159 y Fv(I)f Fx(\()p Fn(\030)s Fx(\))23 b(:=)1610 4046 y Fs(Z)1707 4159 y Fv(d)p Fn(\030)1795 4122 y Fr(0)1841 4159 y Fu(j)p Fv(k)1907 4171 y Fo(A)1961 4159 y Fx(\()p Fn(\030)s Fv(;)14 b Fn(\030)2120 4122 y Fr(0)2143 4159 y Fx(\))p Fu(j)2243 4103 y Fv(f)9 b Fx(\()p Fn(\030)s Fx(\))p 2232 4140 183 4 v 2232 4220 a Fv(f)g Fx(\()p Fn(\030)2358 4183 y Fr(0)2382 4220 y Fx(\))1284 4451 y Fv(J)f Fx(\()p Fn(\030)1415 4414 y Fr(0)1438 4451 y Fx(\))23 b(:=)1627 4338 y Fs(Z)1724 4451 y Fv(d)p Fn(\030)j Fu(j)p Fv(k)1901 4463 y Fo(A)1955 4451 y Fx(\()p Fn(\030)s Fv(;)14 b Fn(\030)2114 4414 y Fr(0)2137 4451 y Fx(\))p Fu(j)2226 4395 y Fv(f)9 b Fx(\()p Fn(\030)2352 4358 y Fr(0)2376 4395 y Fx(\))p 2226 4432 V 2238 4508 a Fv(f)g Fx(\()p Fn(\030)r Fx(\))2418 4451 y Fv(;)727 b Fx(\(3.2\))386 4646 y Fi(wher)l(e)30 b Fv(f)9 b Fx(\()p Fn(\030)r Fx(\))30 b Fi(is)f(a)g(nonne)l(gative)h(c)l(onver)l(genc)l(e)f(gener)l(ating)h (function)f(and)g(al)t(l)h(inte)l(gr)l(als)f(run)386 4746 y(over)h Fp(R)623 4716 y Ft(3)p Fo(l)688 4746 y Fv(:)p eop %%Page: 8 8 8 7 bop 386 259 a Ft(8)937 b(D.)22 b(H.)g(JAKUBASSA-AMUNDSEN)575 459 y Fx(If)30 b Fu(j)p Fv(k)726 471 y Fo(A)780 459 y Fu(j)f Fx(is)g(symmetric)g(in)h Fn(\030)r Fv(;)14 b Fn(\030)1548 422 y Fr(0)1601 459 y Fx(then)29 b Fv(J)8 b Fx(\()p Fn(\030)1922 422 y Fr(0)1946 459 y Fx(\))26 b(=)f Fv(I)7 b Fx(\()p Fn(\030)2214 422 y Fr(0)2237 459 y Fx(\))p Fv(:)56 b Fx(Pro)n(vided)27 b(w)n(e)i(can)g(sho)n(w)f(that)386 558 y Fv(I)41 b Fx(and)34 b Fv(J)42 b Fx(are)883 526 y Ft(1)p 874 540 51 4 v 874 587 a Fo(R)935 558 y Fx(-b)r(ounded)34 b(functions)g(\(for)g(all)g(v)-5 b(alues)33 b(of)i Fv(\030)j Fx(and)c Fv(\030)2597 528 y Fr(0)2620 558 y Fx(\),)h(then)f(the)h (uniform)404 633 y Ft(1)p 396 647 V 396 694 a Fo(R)456 665 y Fx(-b)r(oundedness)27 b(of)h Fv(A)g Fx(follo)n(ws)e(from)i (\(3.1\))f(and)g(from)h(the)g(equiv)-5 b(alence)27 b(\(2.20\),)849 904 y Fu(j)p Fx(\()p Fv( )s(;)14 b(A)g(')p Fx(\))p Fu(j)47 b(\024)1341 787 y Fs(\022)1402 791 y(Z)1499 904 y Fv(d)p Fn(\030)26 b Fu(j)p Fv( )s Fx(\()p Fn(\030)s Fx(\))p Fu(j)1822 869 y Ft(2)1906 848 y Fv(c)p 1892 885 64 4 v 1892 961 a(R)1966 787 y Fs(\023)2037 782 y Fk(1)p 2037 791 29 3 v 2037 824 a(2)2093 787 y Fs(\022)2154 791 y(Z)2251 904 y Fv(d)p Fn(\030)2339 867 y Fr(0)2385 904 y Fu(j)p Fv(')p Fx(\()p Fn(\030)2540 867 y Fr(0)2563 904 y Fx(\))p Fu(j)2618 869 y Ft(2)2703 848 y Fv(c)p 2689 885 64 4 v 2689 961 a(R)2762 787 y Fs(\023)2833 782 y Fk(1)p 2833 791 29 3 v 2833 824 a(2)1592 1153 y Fx(=)1726 1097 y Fv(c)p 1713 1134 64 4 v 1713 1210 a(R)1809 1153 y Fu(k)p Fv( )s Fu(k)c(k)p Fv(')p Fu(k)p Fv(;)1035 b Fx(\(3.3\))386 1311 y(with)28 b Fv(c)g Fx(a)f(suitable)g(constan)n(t.)575 1410 y(Let)39 b(us)g(no)n(w)g(consider)f(the)h(prop)r(erties)f(of)h (the)h(smo)r(oth)f(auxiliary)e(function)j Fv(\037)i Fu(2)386 1510 y Fv(C)451 1480 y Fr(1)522 1510 y Fx(\()p Fp(R)608 1480 y Ft(3)p Fo(l)672 1510 y Fx(\))26 b(where)f Fv(l)g Fx(=)d(2)j(denotes)h(the)f(t)n(w)n(o-particle)f(and)h Fv(l)g Fx(=)e(1)i(the)h(one-particle)e(case.)g(F)-7 b(or)386 1609 y Fv(')23 b Fu(2)h Fv(C)607 1579 y Fr(1)601 1630 y Ft(0)677 1609 y Fx(\()p Fp(R)764 1579 y Ft(3)p Fo(l)828 1609 y Fu(n)p Fv(B)933 1621 y Fo(R)987 1609 y Fx(\(0\)\))19 b Fu(\012)f Fp(C)1281 1579 y Ft(2)p Fo(l)1373 1609 y Fx(w)n(e)27 b(de\014ne)h(for)f FA(x)d Fu(2)f Fp(R)2068 1579 y Ft(3)p Fo(l)1380 1811 y Fv(\037)p Fx(\()1481 1755 y FA(x)p 1474 1792 V 1474 1868 a Fv(R)1548 1811 y Fx(\))g(:=)1737 1694 y Fs(\032)1841 1761 y Fx(0)p Fv(;)82 b(x)24 b(<)e(R)q(=)p Fx(2)1841 1860 y(1)p Fv(;)124 b(x)23 b Fu(\025)g Fv(R)3168 1811 y Fx(\(3.4\))386 2008 y(suc)n(h)k(that)h Fv(\037)23 b Fx(=)g(1)k(on)g(supp)14 b Fv(':)51 b Fx(Moreo)n(v)n(er,)25 b(de\014ne)1471 2178 y Fv(\037)1523 2190 y Ft(0)1560 2178 y Fx(\()1609 2122 y FA(x)p 1602 2159 V 1602 2235 a Fv(R)1676 2178 y Fx(\))e(:=)46 b(1)41 b Fu(\000)g Fv(\037)p Fx(\()2155 2122 y FA(x)p 2148 2159 V 2148 2235 a Fv(R)2222 2178 y Fx(\))914 b(\(3.5\))386 2367 y(with)28 b Fv(\037)627 2379 y Ft(0)687 2367 y Fu(2)c(S)6 b Fx(\()p Fp(R)908 2336 y Ft(3)q Fo(l)973 2367 y Fx(\))p Fv(;)60 b Fu(S)34 b Fx(b)r(eing)28 b(the)g(Sc)n(h)n(w)n(artz)e(space.)575 2466 y(In)35 b(our)g(pro)r(ofs)f(w)n(e)h(shall)g(in)n(tro)r(duce)g (comm)n(utators)f(with)h Fv(\037)p Fx(,)h(suc)n(h)f(that)g Fv(\037)g Fx(can)g(b)r(e)386 2566 y(replaced)29 b(b)n(y)h Fv(\037)886 2578 y Ft(0)923 2566 y Fx(,)g(viz.)g([)p 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y(1)p 1672 3145 283 4 v 1672 3223 a(\(2)p Fv(\031)s Fx(\))1828 3199 y Ft(3)p Fo(l=)p Ft(2)1978 3051 y Fs(Z)2024 3240 y Fh(R)2071 3223 y Fk(3)p Fl(l)2136 3164 y Fv(d)p FA(x)24 b Fv(e)2292 3130 y Fr(\000)p Fo(i)p Fm(p)n(x)2473 3164 y Fv(\037)2525 3176 y Ft(0)2562 3164 y Fx(\()2611 3108 y FA(x)p 2604 3145 64 4 v 2604 3221 a Fv(R)2678 3164 y Fx(\))912 3435 y(=)1153 3379 y(1)p 1032 3416 283 4 v 1032 3493 a(\(2)p Fv(\031)s Fx(\))1188 3469 y Ft(3)p Fo(l=)p Ft(2)1347 3435 y Fv(R)1411 3400 y Ft(3)p Fo(l)1483 3322 y Fs(Z)1530 3510 y Fh(R)1577 3494 y Fk(3)p Fl(l)1642 3435 y Fv(d)p FA(z)g Fv(e)1790 3400 y Fr(\000)p Fo(i)p Fg(p)p Fo(R)p Fg(z)2021 3435 y Fv(\037)2073 3447 y Ft(0)2110 3435 y Fx(\()p FA(z)p Fx(\))47 b(=)f Fv(R)2438 3400 y Ft(3)p Fo(l)2529 3435 y Fx(^)-52 b Fv(\037)2571 3447 y Ft(0)2609 3435 y Fx(\()p FA(p)p Fv(R)q Fx(\))p Fv(:)355 b Fx(\(3.6\))386 3615 y(In)26 b(Lemma)f(4)g(the)h(comm)n(utator)e(with) i(the)g(partition)f(of)h(unit)n(y)-7 b(,)26 b Fv(\036)2501 3627 y Fo(j)2536 3615 y Fx(,)g(is)f(needed.)h(As)g(neither)386 3714 y Fv(\036)435 3726 y Fo(j)498 3714 y Fx(nor)h(1)18 b Fu(\000)h Fv(\036)839 3726 y Fo(j)902 3714 y Fx(is)28 b(a)f(Sc)n(h)n(w)n(artz)f(\(or)h(ev)n(en)h(an)f Fv(L)1904 3726 y Ft(2)1941 3714 y Fx(\))h(function,)h(its)f(F)-7 b(ourier)26 b(transform)h(is)h(not)386 3814 y(w)n(ell)21 b(de\014ned.)g(Therefore)f(w)n(e)h(w)n(ork)f(in)h(co)r(ordinate)f (space)g(instead.)h(The)h(strategy)d(w)n(e)i(apply)386 3914 y(is)27 b(to)g(construct)f(comm)n(utators)g([)p Fv(B)t(;)14 b(\036)1611 3926 y Fo(j)1646 3914 y Fx(])28 b(where)e Fv(B)32 b Fx(is)26 b(again)g(a)h(m)n(ultiplication)g(op)r (erator)e(in)386 4013 y(momen)n(tum)h(space.)g(Then)g(its)h(k)n(ernel)e Fv(k)1680 4025 y Fo(B)1764 4013 y Fx(factorizes,)g Fv(k)2196 4025 y Fo(B)2253 4013 y Fx(\()p FA(p)p Fv(;)14 b FA(p)2428 3983 y Fr(0)2452 4013 y Fx(\))23 b(=)g Fv(B)t Fx(\()p FA(p)p Fx(\))14 b Fv(\016)s Fx(\()p FA(p)i Fu(\000)f FA(p)3067 3983 y Fr(0)3091 4013 y Fx(\))p Fv(:)49 b Fx(Our)386 4113 y(aim)34 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Fm(x)1507 735 y Fx(\))1539 700 y Fo(l)1648 678 y Fx(4)p Fv(\031)p 1598 715 V 1598 792 a Fx(\(2)p Fv(\031)s Fx(\))1754 768 y Ft(3)1837 678 y Fx(1)p 1834 715 48 4 v 1839 792 a(~)-47 b Fv(x)1905 622 y Fs(Z)1988 642 y Fr(1)1952 810 y Ft(0)2073 735 y Fv(p)14 b(dp)36 b Fx(sin)14 b Fv(p)5 b Fx(~)-47 b Fv(x)23 b(g)s Fx(\()p Fv(p)p Fx(\))386 913 y(where)630 912 y(~)625 913 y FA(x)h Fx(:=)e FA(x)859 925 y Ft(1)914 913 y Fu(\000)16 b FA(x)1045 883 y Fr(0)1045 934 y Ft(1)1110 913 y Fx(is)27 b(in)n(tro)r(duced.)f(This)h(in)n(tegral)f(can)g(b)r(e)h (estimated)g(with)g(the)h(help)f(of)386 1013 y(complex)g(and)h (asymptotic)f(analysis)f([15)o(].)575 1112 y(In)21 b(the)h(follo)n (wing)e(w)n(e)h(pro)n(v)n(e)e(successiv)n(ely)h(Lemma)g(5)h(\(section)g (4\),)g(Lemma)g(6)g(\(section)386 1212 y(5\),)27 b(Lemma)h(3)f (\(section)g(6\),)h(Lemma)f(1)h(\(section)f(7\))g(and)h(Lemma)f(4)g (\(section)h(8\).)1422 1417 y(4.)41 b Fw(Pr)n(oof)31 b(of)h(Lemma)f(5)575 1566 y Fx(According)h(to)h(the)h(equiv)-5 b(alence)33 b(\(2.20\))g(w)n(e)g(pro)n(v)n(e)45 b Fu(j)p Fx(\()p Fv(\036;)14 b(b)2469 1523 y Ft(\(1\))2469 1588 y(1)p Fo(m)2566 1566 y Fv(')p Fx(\))p Fu(j)33 b(\024)2826 1534 y Fo(c)p 2815 1548 51 4 v 2815 1595 a(R)2889 1566 y Fu(k)p Fv(\036)p Fu(k)14 b(k)p Fv(')p Fu(k)65 b Fx(for)386 1671 y(all)27 b Fv(\036)d Fu(2)f Fv(C)717 1641 y Fr(1)711 1691 y Ft(0)788 1671 y Fx(\()p Fp(R)874 1641 y Ft(3)917 1671 y Fx(\))c Fu(\012)f Fp(C)1105 1641 y Ft(2)1148 1671 y Fv(:)575 1786 y Fx(The)23 b(op)r(erator)f Fv(b)1108 1743 y Ft(\(1\))1108 1809 y(1)p Fo(m)1204 1786 y Fv(;)i Fx(de\014ned)g(in)f(\(1.5\))h(with)g(\(1.3\),)f(is)g(a)g(sum)h(of)f (terms)h(eac)n(h)e(of)i(whic)n(h)386 1887 y(has)i(the)g(structure)g Fv(B)t Fx(\()p FA(p)p Fx(\))1225 1854 y Ft(1)p 1224 1868 38 4 v 1224 1916 a Fo(x)1271 1887 y Fv(B)t Fx(\()p FA(p)p Fx(\))41 b(where)26 b(the)h(indices)f(on)g FA(p)2315 1899 y Ft(1)2378 1887 y Fx(and)g FA(x)2588 1899 y Ft(1)2652 1887 y Fx(are)g(suppressed)f(and)386 1987 y Fv(B)f Fx(is)d(an)e 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Fv(c)2120 2565 y Ft(0)p 2028 2590 185 4 v 2028 2666 a Fx(1)18 b(+)g Fv(p)3168 2609 y Fx(\(4.2\))386 2782 y(with)23 b(some)f(constan)n(t)h Fv(c)1140 2794 y Ft(0)1177 2782 y Fv(:)46 b Fx(W)-7 b(e)23 b(in)n(tro)r(duce)f(the)h (auxiliary)f(function)h Fv(\037)g Fx(from)g(\(3.4\))f(\(with)i Fv(l)g Fx(=)386 2898 y(1\))h(and)h(estimate)f(\(b)n(y)h(means)f(of)g (the)h(triangle)f(inequalit)n(y\))g(eac)n(h)g(term)g(of)h Fv(b)2834 2855 y Ft(\(1\))2834 2920 y(1)p Fo(m)2955 2898 y Fx(separately)-7 b(,)555 3085 y Fu(j)p Fx(\()p Fv(\036;)14 b(B)790 3029 y Fx(1)p 787 3066 48 4 v 787 3142 a Fv(x)859 3085 y(B)k(')p Fx(\))p Fu(j)46 b Fx(=)g Fu(j)p Fx(\()p Fv(B)t(\036;)1428 3029 y Fx(1)p 1425 3066 V 1425 3142 a Fv(x)1496 3085 y(B)t(\037)14 b(')p Fx(\))p Fu(j)47 b(\024)f(j)p Fx(\()1962 3064 y(~)1951 3085 y Fv(\036;)2050 3029 y Fx(1)p 2047 3066 V 2047 3142 a Fv(x)2118 3085 y(\037)28 b Fx(~)-56 b Fv(')q Fx(\))p Fu(j)41 b Fx(+)g Fu(j)p Fx(\()2507 3064 y(~)2496 3085 y Fv(\036)q(;)2596 3029 y Fx(1)p 2593 3066 V 2593 3142 a Fv(x)2664 3085 y Fx([)p Fv(B)t(;)14 b(\037)p Fx(])24 b Fv(')p Fx(\))p Fu(j)169 b Fx(\(4.3\))386 3275 y(where)26 b(w)n(e)h(ha)n(v)n(e)f (abbreviated)1403 3254 y(~)1392 3275 y Fv(\036)d Fx(:=)g Fv(B)t(\036)28 b Fx(and)40 b(~)-55 b Fv(')23 b Fx(:=)g Fv(B)t(':)50 b Fx(Recalling)27 b(that)g Fv(\037)p Fx(\()2902 3243 y Fm(x)p 2896 3257 51 4 v 2896 3304 a Fo(R)2957 3275 y Fx(\))g(is)g(non)n(v)-5 b(a-)386 3375 y(nishing)27 b(only)g(if)h Fv(x)23 b Fu(\025)g Fv(R)q(=)p Fx(2)49 b(and)27 b(ran)14 b Fv(\037)22 b Fu(2)i Fx([0)p Fv(;)14 b Fx(1])p Fv(;)49 b Fx(the)28 b(\014rst)f(con)n(tribution)g(in)g (\(4.3\))g(is)h(easily)386 3475 y(estimated)g(b)n(y)1076 3625 y Fu(j)p Fx(\()1142 3603 y(~)1131 3625 y Fv(\036)q(;)1230 3568 y Fx(1)p 1227 3606 48 4 v 1227 3682 a Fv(x)1299 3625 y(\037)f Fx(~)-55 b Fv(')p Fx(\))p Fu(j)46 b(\024)1631 3512 y Fs(Z)1677 3700 y Fh(R)1724 3684 y Fk(3)1769 3625 y Fv(d)p FA(x)24 b Fu(j)1920 3603 y Fx(~)1909 3625 y Fv(\036)q Fx(\()p FA(x)p Fx(\))p Fu(j)2133 3568 y Fx(1)p 2130 3606 V 2130 3682 a Fv(x)2210 3625 y(\037)p Fx(\()2311 3568 y FA(x)p 2304 3606 64 4 v 2304 3682 a Fv(R)2378 3625 y Fx(\))f Fu(j)13 b Fx(~)-55 b Fv(')q Fx(\()p FA(x)p Fx(\))p Fu(j)1329 3861 y(\024)1461 3804 y Fx(2)p 1450 3841 V 1450 3918 a Fv(R)1546 3861 y Fu(k)1599 3839 y Fx(~)1588 3861 y Fv(\036)p Fu(k)23 b(k)13 b Fx(~)-55 b Fv(')p Fu(k)45 b(\024)2020 3804 y Fv(c)p 2006 3841 V 2006 3918 a(R)2103 3861 y Fu(k)p Fv(\036)p Fu(k)22 b(k)p Fv(')p Fu(k)772 b Fx(\(4.4\))386 4033 y(where)30 b Fu(k)682 4011 y Fx(~)671 4033 y Fv(\036)p Fu(k)41 b(\024)h(k)p Fv(B)t Fu(k)14 b(k)p Fv(\036)p Fu(k)40 b(\024)28 b Fv(c)1377 4045 y Ft(1)1414 4033 y Fu(k)p Fv(\036)p Fu(k)i Fx(w)n(as)f(used)45 b(\()p Fv(c)28 b Fx(:=)g(2)p Fv(c)2236 4003 y Ft(2)2236 4054 y(1)2272 4033 y Fx(\))p Fv(:)59 b Fx(In)31 b(the)g(second)f(term)h (w)n(e)f(set)386 4133 y Fv(\037)25 b Fx(=)f(1)19 b Fu(\000)f Fv(\037)748 4145 y Ft(0)814 4133 y Fx(and)29 b(note)f(that)1355 4100 y Ft(1)p 1353 4114 38 4 v 1353 4161 a Fo(x)1414 4133 y Fx([)p Fv(B)t(;)14 b(\037)p Fx(])39 b(=)25 b Fu(\000)1838 4100 y Ft(1)p 1819 4114 72 4 v 1819 4161 a Fo(px)1920 4133 y Fu(\001)19 b Fv(p)14 b Fx([)p Fv(B)t(;)g(\037)2197 4145 y Ft(0)2234 4133 y Fx(])54 b(with)2530 4100 y Ft(1)p 2511 4114 V 2511 4161 a Fo(px)2621 4133 y Fx(a)28 b(b)r(ounded)h(op)r (erator)386 4243 y([12)o(,)f(p.307].)e(T)-7 b(ransforming)26 b(in)n(to)h(F)-7 b(ourier)27 b(space,)g(w)n(e)g(get)h(with)g(\(3.6\)) 479 4340 y Fs(\020)529 4433 y Fv(p)641 4409 y Fp(\\)585 4433 y Fx([)p Fv(B)t(;)14 b(\037)764 4445 y Ft(0)800 4433 y Fx(])g Fv(')892 4340 y Fs(\021)955 4433 y Fx(\()p FA(p)p Fx(\))47 b(=)1323 4376 y(1)p 1239 4414 209 4 v 1239 4501 a(\(2)p Fv(\031)s Fx(\))1405 4450 y Fk(3)p 1406 4459 29 3 v 1406 4493 a(2)1481 4433 y Fv(p)1537 4320 y Fs(Z)1583 4508 y Fh(R)1630 4492 y Fk(3)1675 4433 y Fv(d)p FA(p)1771 4398 y Fr(0)1818 4433 y Fv(R)1882 4398 y Ft(3)1942 4433 y Fx(^)-51 b Fv(\037)1985 4445 y Ft(0)2022 4433 y Fx(\(\()p FA(p)19 b Fu(\000)f FA(p)2294 4398 y Fr(0)2317 4433 y Fx(\))p Fv(R)q Fx(\))24 b(\()p Fv(B)t Fx(\()p FA(p)p Fx(\))19 b Fu(\000)f Fv(B)t Fx(\()p FA(p)2939 4398 y Fr(0)2963 4433 y Fx(\)\))37 b(^)-56 b Fv(')q Fx(\()p FA(p)3190 4398 y Fr(0)3213 4433 y Fx(\))1297 4703 y(=:)1408 4590 y Fs(Z)1454 4779 y Fh(R)1501 4762 y Fk(3)1547 4703 y Fv(d)p FA(p)1643 4669 y Fr(0)1689 4703 y Fv(k)1732 4718 y Fo(p)p Ft([)p Fo(B)s(;\037)1899 4726 y Fk(0)1932 4718 y Ft(])1955 4703 y Fx(\()p FA(p)p Fv(;)14 b FA(p)2130 4669 y Fr(0)2154 4703 y Fx(\))36 b(^)-55 b Fv(')p Fx(\()p FA(p)2348 4669 y Fr(0)2372 4703 y Fx(\))p Fv(:)741 b Fx(\(4.5\))p eop %%Page: 10 10 10 9 bop 386 259 a Ft(10)904 b(D.)22 b(H.)g(JAKUBASSA-AMUNDSEN)386 459 y Fx(W)-7 b(e)26 b(de\014ne)f Fv( )i Fx(:=)985 426 y Ft(1)p 965 440 72 4 v 965 487 a Fo(px)1058 437 y Fx(~)1047 459 y Fv(\036)49 b Fx(and)25 b(apply)h(the)g(Lieb)f(and)g(Y)-7 b(au)26 b(form)n(ula)e(\(3.1\))i(with)g Fv(l)e Fx(=)f(1)p Fv(;)36 b Fn(\030)26 b Fx(:=)d FA(p)386 569 y Fx(and)k Fv(f)32 b Fx(=)23 b(1)p Fv(:)50 b Fx(Using)27 b(our)g(k)n(ernel)g(from) g(\(4.5\))g(w)n(e)h(can)f(estimate)g(with)i(\(4.1\))e(and)g(\(4.2\))792 764 y Fv(I)7 b Fx(\()p FA(p)p Fx(\))24 b(:=)1110 651 y Fs(Z)1156 839 y Fh(R)1203 823 y Fk(3)1248 764 y Fv(d)p FA(p)1344 729 y Fr(0)1455 707 y Fv(R)1519 677 y Ft(3)p 1401 744 209 4 v 1401 832 a Fx(\(2)p Fv(\031)s Fx(\))1567 781 y Fk(3)p 1567 790 29 3 v 1567 824 a(2)1643 764 y Fv(p)f Fu(j)10 b Fx(^)-52 b Fv(\037)1783 776 y Ft(0)1820 764 y Fx(\(\()p FA(p)19 b Fu(\000)f FA(p)2092 729 y Fr(0)2115 764 y Fx(\))p Fv(R)q Fu(j)23 b(j)p FA(p)c Fu(\000)f FA(p)2488 729 y Fr(0)2511 764 y Fu(j)23 b(jr)2649 776 y Fm(p)2695 764 y Fv(B)t Fx(\()p Fn(\030)2840 784 y Ft(0)2877 764 y Fx(\))p Fu(j)904 1023 y(\024)1092 967 y Fv(c)1128 979 y Ft(0)p 1024 1004 209 4 v 1024 1092 a Fx(\(2)p Fv(\031)s Fx(\))1190 1040 y Fk(3)p 1191 1049 29 3 v 1191 1083 a(2)1257 910 y Fs(Z)1303 1098 y Fh(R)1350 1082 y Fk(3)1395 1023 y Fv(d)p FA(y)i Fu(j)10 b Fx(^)-52 b Fv(\037)1588 1035 y Ft(0)1626 1023 y Fx(\()p FA(y)q Fx(\))p Fu(j)24 b Fv(y)1876 967 y Fx(1)p 1865 1004 64 4 v 1865 1080 a Fv(R)1957 1023 y Fu(\001)2366 967 y Fv(p)p 2008 1004 757 4 v 2008 1080 a Fx(1)18 b(+)g Fu(j)p FA(p)h Fu(\000)f Fx(\(1)g Fu(\000)g Fv(\025)p Fx(\))p FA(y)q Fv(=R)q Fu(j)2798 1023 y Fv(:)347 b Fx(\(4.6\))386 1202 y(The)33 b(last)f(factor)g(is)h(b)r(ounded)h(for) e(all)g Fv(p)g Fu(\025)f Fx(0)p Fv(;)i Fx(and)g(the)g(in)n(tegral)f(is) g(\014nite)i(b)r(ecause)42 b(^)-52 b Fv(\037)3214 1214 y Ft(0)3283 1202 y Fu(2)386 1302 y(S)6 b Fx(\()p Fp(R)529 1272 y Ft(3)572 1302 y Fx(\))p Fv(:)51 b Fx(Th)n(us)27 b Fv(I)7 b Fx(\()p FA(p)p Fx(\))29 b(is)e(1)p Fv(=R)q Fx(-b)r(ounded,)g(i.e.)g Fv(I)7 b Fx(\()p FA(p)p Fx(\))24 b Fu(\024)2122 1269 y Fo(c)p 2112 1283 51 4 v 2112 1331 a(R)2172 1302 y Fv(:)575 1402 y Fx(The)19 b(second)g(in)n(tegral)f Fv(J)8 b Fx(\()p FA(p)1430 1372 y Fr(0)1453 1402 y Fx(\))20 b(in)f(the)h(Lieb)f(and)g(Y)-7 b(au)19 b(form)n(ula)f(can)h(b)r(e)h (estimated)f(in)g(the)386 1501 y(same)h(w)n(a)n(y)-7 b(.)20 b(There,)g(the)h(last)f(factor)g(in)h(\(4.6\))g(is)f(replaced)g (b)n(y)34 b Fu(j)p FA(p)2415 1471 y Fr(0)2443 1501 y Fx(+)5 b FA(y)q Fv(=R)q Fu(j)p Fv(=)p Fx(\(1)g(+)g Fu(j)p FA(p)2960 1471 y Fr(0)2986 1501 y Fx(+)g Fv(\025)p FA(y)q Fv(=R)q Fu(j)p Fx(\))386 1601 y(whic)n(h)41 b(is)f(also)g(b)r(ounded)h (for)f(all)h Fv(p)1576 1571 y Fr(0)1644 1601 y Fu(\025)j Fx(0)p Fv(:)86 b Fx(\(In)41 b(the)g(limiting)g(case)f Fv(m)45 b Fx(=)g(0,)40 b(one)g(has)386 1701 y Fv(B)t Fx(\()p FA(p)p Fx(\))29 b(:=)f Fv(A)p Fx(\()p Fv(p)p Fx(\))g(=)1041 1668 y Ft(1)p 1014 1682 88 4 v 1014 1690 a Fr(p)p 1069 1690 34 3 v 49 x Ft(2)1142 1701 y Fx(whic)n(h)j(comm)n (utes)f(with)h Fv(\037)p Fx(,)g(and)g(for)f Fv(B)t Fx(\()p FA(p)p Fx(\))f(:=)f Fn(\033)s FA(p)p Fv(=p;)i Fx(one)g(should)386 1839 y(use)c(the)h(explicit)g(result)f Fv(B)t Fx(\()p FA(p)p Fx(\))17 b Fu(\000)f Fv(B)t Fx(\()p FA(p)1626 1809 y Fr(0)1650 1839 y Fx(\))37 b(=)1817 1802 y Fm(p)p Fr(\000)p Fm(p)1953 1777 y Fj(0)p 1817 1820 158 4 v 1879 1868 a Fo(p)1999 1839 y Fx(\()p Fn(\033)19 b Fu(\000)d Fx(\()p Fn(\033)s FA(p)2333 1809 y Fr(0)2357 1839 y Fv(=p)2441 1809 y Fr(0)2463 1839 y Fx(\)\()p FA(p)h Fx(+)f FA(p)2731 1809 y Fr(0)2755 1839 y Fx(\))p Fv(=)p Fx(\()p Fv(p)g Fx(+)g Fv(p)3042 1809 y Fr(0)3065 1839 y Fx(\)\))50 b(suc)n(h)386 1950 y(that)28 b(the)g(last)f(factor)g(in)h(\(4.6\))f(is)h(not)f (presen)n(t.\))575 2049 y(T)-7 b(ogether)26 b(with)i(\(3.3\),)g(this)g (pro)n(v)n(es)918 2232 y Fu(j)p Fx(\()984 2210 y(~)973 2232 y Fv(\036)q(;)1073 2175 y Fx(1)p 1070 2212 48 4 v 1070 2288 a Fv(x)1141 2232 y Fx([)p Fv(B)t(;)14 b(\037)p Fx(])g Fv(')p Fx(\))p Fu(j)47 b(\024)1647 2175 y Fv(c)p 1633 2212 64 4 v 1633 2288 a(R)1730 2232 y Fu(k)1805 2175 y Fx(1)p 1782 2212 90 4 v 1782 2288 a Fv(px)1895 2232 y(B)18 b(\036)p Fu(k)23 b(k)p Fv(')p Fu(k)45 b(\024)2409 2175 y Fx(~)-43 b Fv(c)p 2394 2212 64 4 v 2394 2288 a(R)2490 2232 y Fu(k)p Fv(\036)p Fu(k)23 b(k)p Fv(')p Fu(k)p Fv(;)361 b Fx(\(4.7\))386 2412 y(and)27 b(th)n(us)h(the)g(assertion)e(of)i (Lemma)f(5.)1422 2601 y(5.)41 b Fw(Pr)n(oof)31 b(of)h(Lemma)f(6)575 2750 y Fx(W)-7 b(e)32 b(ha)n(v)n(e)f(to)h(sho)n(w)f(that)46 b Fu(j)p Fx(\()p Fv(';)14 b(v)1621 2720 y Ft(\(12\))1744 2750 y Fv( )1798 2762 y Fo(n)1843 2750 y Fx(\))p Fu(j)44 b(\024)2053 2717 y Fo(c)p 2047 2731 42 4 v 2047 2779 a(n)2112 2750 y Fu(k)p Fv(')p Fu(k)14 b(k)p Fv( )2360 2762 y Fo(n)2404 2750 y Fu(k)62 b Fx(for)31 b Fv( )2693 2762 y Fo(n)2768 2750 y Fx(=)f Fv(')2917 2707 y Ft(\(1\))2917 2760 y Fo(n)3007 2750 y Fv(\036)3056 2707 y Ft(\(2\))3056 2774 y Fo(N)3177 2750 y Fx(with)386 2874 y Fv(')440 2830 y Ft(\(1\))440 2883 y Fo(n)552 2874 y Fu(2)24 b Fv(C)696 2843 y Fr(1)690 2894 y Ft(0)766 2874 y Fx(\()p Fp(R)853 2843 y Ft(3)896 2874 y Fu(n)p Fv(B)1001 2886 y Fo(n)1046 2874 y Fx(\(0\)\))18 b Fu(\012)g Fp(C)1339 2843 y Ft(2)1433 2874 y Fx(and)28 b Fv(\036)1644 2830 y Ft(\(2\))1644 2898 y Fo(N)1756 2874 y Fu(2)c Fv(C)1900 2843 y Fr(1)1894 2894 y Ft(0)1971 2874 y Fx(\()p Fp(R)2057 2843 y Ft(3)2100 2874 y Fx(\))19 b Fu(\012)f Fp(C)2288 2843 y Ft(2)2331 2874 y Fv(:)575 2973 y Fx(F)-7 b(or)28 b(the)g(de\014nition)h(of)g(the) f(auxiliary)f(function)i Fv(\037)g Fx(w)n(e)f(note)g(that)h Fu(9)14 b Fv(R)2843 2985 y Ft(0)2904 2973 y Fx(:)39 b(supp)13 b Fv( )3204 2985 y Fo(n)3274 2973 y Fu(\032)386 3073 y Fp(R)440 3043 y Ft(3)483 3073 y Fu(n)p Fv(B)588 3085 y Fo(n)633 3073 y Fx(\(0\))39 b Fu(\\)h Fv(B)936 3085 y Fo(R)986 3093 y Fk(0)1022 3073 y Fx(\(0\))p Fv(:)79 b Fx(Cho)r(ose)37 b Fv(n)h Fx(so)g(large)f(that)h Fv(R)2197 3085 y Ft(0)2275 3073 y Fv(<)i(n=)p Fx(2)p Fv(:)78 b Fx(Then,)38 b(on)g(supp)14 b Fv( )3230 3085 y Fo(n)3316 3073 y Fx(:)386 3172 y Fu(j)p FA(x)459 3184 y Ft(1)515 3172 y Fu(\000)k FA(x)648 3184 y Ft(2)686 3172 y Fu(j)37 b(\025)22 b Fv(x)880 3184 y Ft(1)936 3172 y Fu(\000)c Fv(x)1066 3184 y Ft(2)1141 3172 y Fv(>)23 b(n)18 b Fu(\000)1390 3140 y Fo(n)p 1390 3154 V 1394 3201 a Ft(2)1478 3172 y Fx(=)23 b Fv(n=)p Fx(2)p Fv(:)49 b Fx(De\014ne)1080 3371 y Fv(\037)1132 3383 y Ft(12)1203 3371 y Fx(\()1245 3315 y FA(x)1295 3327 y Ft(1)1351 3315 y Fu(\000)18 b FA(x)1484 3327 y Ft(2)p 1245 3352 277 4 v 1358 3428 a Fv(n)1532 3371 y Fx(\))23 b(:=)1721 3254 y Fs(\032)1825 3321 y Fx(0)p Fv(;)105 b Fu(j)p Fv(x)2065 3333 y Ft(1)2121 3321 y Fu(\000)18 b Fv(x)2251 3333 y Ft(2)2289 3321 y Fu(j)23 b Fv(<)g(n=)p Fx(4)1825 3420 y(1)p Fv(;)105 b Fu(j)p Fv(x)2065 3432 y Ft(1)2121 3420 y Fu(\000)18 b Fv(x)2251 3432 y Ft(2)2289 3420 y Fu(j)23 b(\025)g Fv(n=)p Fx(2)2621 3371 y Fv(;)524 b Fx(\(5.1\))386 3562 y(a)27 b(smo)r(oth)g(function)i(mapping)e(to)g([0)p Fv(;)14 b Fx(1])27 b(with)h(the)g(prop)r(ert)n(y)f Fv( )2436 3574 y Fo(n)2481 3562 y Fv(\037)2533 3574 y Ft(12)2626 3562 y Fx(=)c Fv( )2768 3574 y Fo(n)2813 3562 y Fv(:)575 3665 y Fx(The)32 b(op)r(erator)f Fv(v)1133 3635 y Ft(\(12\))1255 3665 y Fv(;)i Fx(de\014ned)f(in)h(\(1.2\))f(-)g(\(1.5\),)g(reads)f (explicitly)h(\(note)h(that)f(only)386 3768 y(terms)27 b(ev)n(en)g(in)h Fn(\013)966 3738 y Ft(\()p Fo(k)q Ft(\))1087 3768 y Fx(surviv)n(e)e(the)i(pro)5 b(jection)27 b(b)n(y)g(\(1)18 b(+)g Fv(\014)2250 3738 y Ft(\()p Fo(k)q Ft(\))2343 3768 y Fx(\))p Fv(=)p Fx(2\))1104 3962 y Fv(v)1147 3928 y Ft(\(12\))1316 3962 y Fx(=)45 b Fv(A)p Fx(\()p Fv(p)1562 3974 y Ft(1)1600 3962 y Fx(\))p Fv(A)p Fx(\()p Fv(p)1768 3974 y Ft(2)1806 3962 y Fx(\))1985 3906 y Fv(e)2024 3876 y Ft(2)p 1862 3943 323 4 v 1862 4019 a Fu(j)p FA(x)1935 4031 y Ft(1)1991 4019 y Fu(\000)18 b FA(x)2124 4031 y Ft(2)2162 4019 y Fu(j)2209 3962 y Fv(A)p Fx(\()p Fv(p)2345 3974 y Ft(1)2382 3962 y Fx(\))p Fv(A)p Fx(\()p Fv(p)2550 3974 y Ft(2)2588 3962 y Fx(\))548 b(\(5.2\))993 4220 y(+)14 b Fv(A)p Fx(\()p Fv(p)1208 4232 y Ft(1)1245 4220 y Fx(\))p Fv(g)s Fx(\()p Fv(p)1394 4232 y Ft(2)1431 4220 y Fx(\))g Fn(\033)1537 4186 y Ft(\(2\))1626 4220 y FA(p)1679 4232 y Ft(2)1864 4164 y Fv(e)1903 4134 y Ft(2)p 1740 4201 V 1740 4277 a Fu(j)p FA(x)1813 4289 y Ft(1)1869 4277 y Fu(\000)k FA(x)2002 4289 y Ft(2)2040 4277 y Fu(j)2087 4220 y Fv(A)p 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Fv(e)1903 4615 y Ft(2)p 1740 4682 V 1740 4758 a Fu(j)p FA(x)1813 4770 y Ft(1)1869 4758 y Fu(\000)k FA(x)2002 4770 y Ft(2)2040 4758 y Fu(j)2087 4701 y Fv(g)s Fx(\()p Fv(p)2204 4713 y Ft(1)2241 4701 y Fx(\))c Fn(\033)2347 4667 y Ft(\(1\))2436 4701 y FA(p)2489 4713 y Ft(1)2540 4701 y Fv(g)s Fx(\()p Fv(p)2657 4713 y Ft(2)2694 4701 y Fx(\))g Fn(\033)2800 4667 y Ft(\(2\))2889 4701 y FA(p)2942 4713 y Ft(2)p eop %%Page: 11 11 11 10 bop 1606 259 a Ft(HVZ)30 b(THEOREM)1153 b(11)386 459 y Fx(with)23 b Fv(A)g Fx(and)f Fv(g)k Fx(as)c(in)h(\(1.3\).)f(F)-7 b(or)22 b(the)h(presen)n(t)f(pro)r(of,)g(w)n(e)g(again)g(need)g(only)h (the)g(structure)f(of)386 572 y(eac)n(h)g(term)h(in)g Fv(v)897 542 y Ft(\(12\))1019 572 y Fv(;)60 b(B)t Fx(\()p FA(p)1254 584 y Ft(1)1292 572 y Fv(;)14 b FA(p)1382 584 y Ft(2)1419 572 y Fx(\))1561 540 y Fo(e)1592 515 y Fk(2)p 1475 554 236 4 v 1475 601 a Fr(j)p Fm(x)1535 609 y Fk(1)1567 601 y Fr(\000)p Fm(x)1659 609 y Fk(2)1691 601 y Fr(j)1735 572 y Fv(B)t Fx(\()p FA(p)1887 584 y 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Ft(2)1134 1821 y Fx(\))1166 1791 y Ft(2)1203 1821 y Fv(:)72 b Fx(Keeping)35 b FA(x)1679 1833 y Ft(2)1752 1821 y Fx(\014xed,)g(w)n(e)h(ha)n(v)n(e)e(with)i Fv(p)2552 1833 y Ft(1)2625 1821 y Fx(=)2726 1755 y Fs(p)p 2809 1755 210 4 v 66 x Fu(\000r)2943 1797 y Ft(2)2943 1842 y Fm(x)2983 1850 y Fk(1)3054 1821 y Fx(and)f(the)386 1921 y(substitution)28 b FA(y)898 1933 y Ft(1)959 1921 y Fx(:=)22 b FA(x)1119 1933 y Ft(1)1176 1921 y Fu(\000)c FA(x)1309 1933 y Ft(2)1346 1921 y Fv(;)386 2044 y Fs(\014)386 2093 y(\014)386 2143 y(\014)386 2193 y(\014)414 2051 y(Z)460 2240 y Fh(R)507 2223 y Fk(3)552 2164 y Fv(d)p FA(x)645 2176 y Ft(1)847 2108 y Fx(1)p 707 2145 323 4 v 707 2221 a Fu(j)p FA(x)780 2233 y Ft(1)836 2221 y Fu(\000)g FA(x)969 2233 y Ft(2)1007 2221 y Fu(j)p 1053 2092 334 4 v 1053 2164 a Fv( )s Fx(\()p FA(x)1192 2176 y Ft(1)1230 2164 y Fv(;)c FA(x)1317 2176 y Ft(2)1355 2164 y Fx(\))1430 2108 y(1)p 1411 2145 80 4 v 1411 2221 a Fv(p)1453 2233 y Ft(1)1514 2164 y Fv( )s Fx(\()p FA(x)1653 2176 y Ft(1)1691 2164 y Fv(;)g FA(x)1778 2176 y Ft(2)1815 2164 y Fx(\))1847 2044 y Fs(\014)1847 2093 y(\014)1847 2143 y(\014)1847 2193 y(\014)1912 2164 y Fx(=)2014 1994 y Fs(\014)2014 2044 y(\014)2014 2093 y(\014)2014 2143 y(\014)2014 2193 y(\014)2014 2243 y(\014)2041 2051 y(Z)2087 2240 y Fh(R)2134 2223 y Fk(3)2180 2164 y Fv(d)p FA(y)2273 2176 y Ft(1)2353 2108 y Fx(1)p 2334 2145 78 4 v 2334 2221 a Fv(y)2375 2233 y Ft(1)p 2436 2092 281 4 v 2436 2164 a Fv(')2490 2176 y Fo(x)2528 2184 y Fk(2)2565 2164 y Fx(\()p FA(y)2647 2176 y Ft(1)2685 2164 y Fx(\))2866 2108 y(1)p 2741 2145 294 4 v 2741 2162 a Fs(q)p 2824 2162 210 4 v 89 x Fu(\000r)2958 2228 y Ft(2)2958 2272 y Fm(y)2998 2280 y Fk(1)3058 2164 y Fv(')3112 2176 y Fo(x)3150 2184 y Fk(2)3186 2164 y Fx(\()p FA(y)3268 2176 y Ft(1)3306 2164 y Fx(\))3338 1994 y Fs(\014)3338 2044 y(\014)3338 2093 y(\014)3338 2143 y(\014)3338 2193 y(\014)3338 2243 y(\014)1454 2491 y Fu(\024)46 b Fv(c)1615 2378 y Fs(Z)1660 2567 y Fh(R)1707 2550 y Fk(3)1753 2491 y Fv(d)p FA(y)1846 2503 y Ft(1)1907 2491 y Fu(j)p Fv(')1984 2503 y Fo(x)2022 2511 y Fk(2)2058 2491 y Fx(\()p FA(y)2140 2503 y Ft(1)2178 2491 y Fx(\))p Fu(j)2233 2457 y Ft(2)3168 2491 y Fx(\(5.4\))386 2676 y(according)23 b(to)i(Kato's)e(inequalit)n(y)-7 b(.)25 b(Hence,)g Fu(j)p Fx(\()p Fv( )s(;)2089 2644 y Ft(1)p 1955 2658 303 4 v 1955 2705 a Fr(j)p Fm(x)2015 2713 y Fk(1)2047 2705 y Fr(\000)p Fm(x)2139 2713 y Fk(2)2171 2705 y Fr(j)p Fo(p)2225 2713 y Fk(1)2281 2676 y Fv( )s Fx(\))p Fu(j)37 b(\024)23 b Fv(c)2568 2609 y Fs(R)2607 2706 y Fh(R)2654 2689 y Fk(6)2699 2676 y Fv(d)p FA(x)2792 2688 y Ft(2)2830 2676 y Fv(d)p FA(y)2923 2688 y Ft(1)2975 2676 y Fu(j)p Fv(')3052 2688 y Fo(x)3090 2696 y Fk(2)3126 2676 y Fx(\()p FA(y)3208 2688 y Ft(1)3246 2676 y Fx(\))p Fu(j)3301 2646 y Ft(2)386 2795 y Fx(=)g Fv(c)14 b Fu(k)p Fv( )s Fu(k)665 2765 y Ft(2)701 2795 y Fv(:)575 2895 y Fx(F)-7 b(urtheron,)25 b(w)n(e)g(aim)h(at)g(a)f(reduction)g(of)h(the) g(comm)n(utator)f(to)g(the)h(one-particle)f(case)386 2995 y(suc)n(h)i(that)h(the)g(pro)r(of)f(of)h(Lemma)f(5)g(can)g(b)r(e)h (mimic)n(k)n(ed.)575 3094 y(First)f(w)n(e)f(note)h(that)g Fv(\037)1313 3106 y Ft(12)1410 3094 y Fx(dep)r(ends)g(only)g(on)f(the)i (di\013erence)f(of)f(v)-5 b(ariables)26 b(suc)n(h)g(that,)386 3194 y(de\014ning)k Fv(\037)754 3206 y Ft(12)p Fo(;)p Ft(0)905 3194 y Fx(:=)e(1)20 b Fu(\000)g Fv(\037)1220 3206 y Ft(12)1290 3194 y Fv(;)79 b Fx(^)-52 b Fv(\037)1434 3206 y Ft(12)p Fo(;)p Ft(0)1588 3194 y Fx(splits)31 b(o\013)f(a)g (Dirac)g Fv(\016)s Fx(-function.)h(Using)f Fv(l)f Fx(=)f(2)i(w)n(e)g (get)386 3293 y(up)r(on)e(substitution)g(of)f FA(z)1194 3263 y Fr(0)1194 3314 y Ft(1)1255 3293 y Fx(:=)c FA(z)1408 3305 y Ft(1)1464 3293 y Fu(\000)18 b FA(z)1589 3305 y Ft(2)1654 3293 y Fx(for)27 b FA(z)1823 3305 y Ft(1)1889 3293 y Fx(from)g(the)h(second)f(line)h(of)f(\(3.6\))404 3374 y Fs( )595 3450 y Fp(\\)470 3516 y Fv(\037)522 3528 y Ft(12)p Fo(;)p Ft(0)645 3516 y Fx(\()687 3460 y Fu(\016)18 b(\000)g(\003)p 687 3497 185 4 v 754 3573 a Fv(n)882 3516 y Fx(\))914 3374 y Fs(!)993 3516 y Fx(\()p FA(p)1078 3528 y Ft(1)1116 3516 y Fv(;)c FA(p)1206 3528 y Ft(2)1243 3516 y Fx(\))47 b(=)1518 3460 y(1)p 1442 3497 194 4 v 1442 3573 a(\(2)p Fv(\031)s Fx(\))1598 3549 y Ft(3)1669 3516 y Fv(n)1719 3482 y Ft(6)1770 3403 y Fs(Z)1816 3592 y Fh(R)1863 3575 y Fk(6)1908 3516 y Fv(d)p FA(z)1993 3528 y Ft(1)2031 3516 y Fv(d)p FA(z)2116 3528 y Ft(2)2177 3516 y Fv(e)2216 3482 y Fr(\000)p Fo(i)p Fg(p)2331 3497 y Fk(1)2363 3482 y Fo(n)p Fg(z)2444 3490 y Fk(1)2494 3516 y Fv(e)2533 3482 y Fr(\000)p Fo(i)p Fg(p)2648 3497 y Fk(2)2680 3482 y Fo(n)p Fg(z)2760 3490 y Fk(2)2820 3516 y Fv(\037)2872 3528 y Ft(12)p Fo(;)p Ft(0)2995 3516 y Fx(\()p FA(z)3069 3528 y Ft(1)3126 3516 y Fu(\000)18 b FA(z)3251 3528 y Ft(2)3288 3516 y Fx(\))3168 3691 y(\(5.5\))421 3846 y(\()p Fv(a)p Fx(\))70 b(=)22 b Fv(n)736 3812 y Ft(6)787 3733 y Fs(Z)773 3975 y Fh(R)820 3958 y Fk(3)884 3846 y Fv(d)p FA(z)969 3812 y Fr(0)969 3867 y Ft(1)1007 3846 y Fv(e)1046 3812 y Fr(\000)p Fo(i)p Fg(p)1161 3827 y Fk(1)1193 3812 y Fo(n)p Fg(z)1274 3787 y Fj(0)1274 3829 y Fk(1)1310 3846 y Fv(\037)1362 3858 y Ft(12)p Fo(;)p Ft(0)1485 3846 y Fx(\()p FA(z)1559 3812 y Fr(0)1559 3867 y Ft(1)1597 3846 y Fx(\))p Fv(\016)s Fx(\(\()p FA(p)1786 3858 y Ft(1)1843 3846 y Fx(+)c FA(p)1979 3858 y Ft(2)2016 3846 y Fx(\))p Fv(n)p Fx(\))24 b(=)e Fv(n)2291 3812 y Ft(3)2328 3846 y Fx(\(2)p Fv(\031)s Fx(\))2494 3790 y Fk(3)p 2495 3799 29 3 v 2495 3832 a(2)2547 3846 y Fx(^)-52 b Fv(\037)2589 3858 y Ft(12)p Fo(;)p Ft(0)2712 3846 y Fx(\()p FA(p)2797 3858 y Ft(1)2835 3846 y Fv(n)p Fx(\))p Fv(\016)s Fx(\()p FA(p)3042 3858 y Ft(1)3098 3846 y Fx(+)18 b FA(p)3234 3858 y Ft(2)3271 3846 y Fx(\))416 4120 y(\()p Fv(b)p Fx(\))1518 b(=)46 b Fv(n)2195 4085 y Ft(3)2232 4120 y Fx(\(2)p Fv(\031)s Fx(\))2398 4063 y Fk(3)p 2398 4072 V 2398 4105 a(2)2464 4120 y Fx(^)-52 b Fv(\037)2506 4132 y Ft(12)p Fo(;)p Ft(0)2629 4120 y Fx(\()p Fu(\000)p FA(p)2779 4132 y Ft(2)2816 4120 y Fv(n)p Fx(\))24 b Fv(\016)s Fx(\()p FA(p)3047 4132 y Ft(1)3103 4120 y Fx(+)18 b FA(p)3239 4132 y Ft(2)3276 4120 y Fx(\))386 4250 y(where)27 b(line)h(\(b\))g(is)g (obtained)f(from)g FA(p)1595 4262 y Ft(1)1656 4250 y Fx(=)22 b Fu(\000)p FA(p)1861 4262 y Ft(2)1898 4250 y Fv(:)575 4350 y Fx(No)n(w)27 b(w)n(e)g(mak)n(e)g(use)g(of)h(the)g (factorization)e(of)i Fv(B)t Fx(\()p FA(p)2213 4362 y Ft(1)2250 4350 y Fv(;)14 b FA(p)2340 4362 y Ft(2)2378 4350 y Fx(\))28 b(to)f(write)467 4498 y Fv(p)509 4510 y Fo(k)564 4498 y Fx([)p Fv(B)650 4510 y Ft(1)687 4498 y Fx(\()p FA(p)772 4510 y Ft(1)810 4498 y Fx(\))p Fv(B)905 4510 y Ft(2)942 4498 y Fx(\()p FA(p)1027 4510 y Ft(2)1065 4498 y Fx(\))p Fv(;)14 b(\037)1186 4510 y Ft(12)1256 4498 y Fx(])46 b(=)g Fu(\000)p Fv(p)1543 4510 y Fo(k)1597 4498 y Fx([)p Fv(B)1683 4510 y Ft(1)1720 4498 y Fx(\()p FA(p)1805 4510 y Ft(1)1843 4498 y Fx(\))p Fv(;)14 b(\037)1964 4510 y Ft(12)p Fo(;)p Ft(0)2087 4498 y Fx(])23 b Fv(B)2196 4510 y Ft(2)2266 4498 y Fu(\000)32 b Fv(B)2426 4510 y Ft(1)2477 4498 y Fv(p)2519 4510 y Fo(k)2573 4498 y Fx([)p Fv(B)2659 4510 y Ft(2)2697 4498 y Fx(\()p FA(p)2782 4510 y Ft(2)2819 4498 y Fx(\))p Fv(;)14 b(\037)2940 4510 y Ft(12)p Fo(;)p Ft(0)3063 4498 y Fx(])82 b(\(5.6\))386 4646 y(suc)n(h)31 b(that)g(the)g(second)g(summand)g(in)g(\(5.3\))g(can) g(b)r(e)g(split)h(\(via)e(the)i(triangle)e(inequalit)n(y\))386 4746 y(in)n(to)23 b(t)n(w)n(o)g(parts)g(for)g(eac)n(h)f(of)i(whic)n(h)f Fv(p)1583 4758 y Fo(k)1648 4746 y Fx(is)g(c)n(hosen)g(indep)r(enden)n (tly)h(\()p Fv(k)i Fx(=)d(1)g(for)g(the)h(\014rst)f(and)p eop %%Page: 12 12 12 11 bop 386 259 a Ft(12)904 b(D.)22 b(H.)g(JAKUBASSA-AMUNDSEN)386 459 y Fv(k)k Fx(=)c(2)k(for)e(the)i(second)f(term)h(in)f(\(5.6\)\).)h (Consider)f(the)g(k)n(ernel)g(of)g(the)h(\014rst)g(part)f(\(cf.)h (\(4.5\)\))1392 606 y Fv(k)1435 621 y Fo(p)1469 629 y Fk(1)1502 621 y Ft([)p Fo(B)1571 629 y Fk(1)1604 621 y Fo(;\037)1665 629 y Fk(12)p Fl(;)p Fk(0)1773 621 y Ft(])1796 606 y Fx(\()p FA(p)1881 618 y Ft(1)1918 606 y Fv(;)14 b FA(p)2008 618 y Ft(2)2045 606 y Fv(;)g FA(p)2135 572 y Fr(0)2135 627 y Ft(1)2173 606 y Fv(;)g FA(p)2263 572 y Fr(0)2263 627 y Ft(2)2300 606 y Fx(\))836 b(\(5.7\))575 830 y(=)46 b Fv(p)728 842 y Ft(1)849 774 y Fv(n)899 744 y Ft(3)p 788 811 209 4 v 788 899 a Fx(\(2)p Fv(\031)s Fx(\))954 848 y Fk(3)p 955 857 29 3 v 955 890 a(2)1031 830 y Fx(^)-52 b Fv(\037)1073 842 y Ft(12)p Fo(;)p Ft(0)1196 830 y Fx(\(\()p FA(p)1313 842 y Ft(1)1369 830 y Fu(\000)18 b FA(p)1505 796 y Fr(0)1505 851 y Ft(1)1543 830 y Fx(\))p Fv(n)p Fx(\))23 b Fv(\016)s Fx(\()p FA(p)1805 842 y Ft(1)1861 830 y Fu(\000)18 b FA(p)1997 796 y Fr(0)1997 851 y Ft(1)2053 830 y Fx(+)g FA(p)2189 842 y Ft(2)2245 830 y Fu(\000)g FA(p)2381 796 y Fr(0)2381 851 y Ft(2)2418 830 y Fx(\))23 b(\()p Fv(B)2568 842 y Ft(1)2606 830 y Fx(\()p FA(p)2691 842 y Ft(1)2728 830 y Fx(\))c Fu(\000)f Fv(B)2925 842 y Ft(1)2962 830 y Fx(\()p FA(p)3047 796 y Fr(0)3047 851 y Ft(1)3085 830 y Fx(\)\))386 1021 y(where)30 b(\()p Fv(a)p Fx(\))g(is)h(used)f(for)f(the)i(F)-7 b(ourier)29 b(transform)g(of)i Fv(\037)2145 1033 y Ft(12)p Fo(;)p Ft(0)2268 1021 y Fv(:)57 b Fx(Insertion)30 b(in)n(to)g(the)h(Lieb)f (and)386 1120 y(Y)-7 b(au)28 b(form)n(ula)e(\(for)i Fv(l)c Fx(=)f(2\))k(with)h Fv(f)k Fx(=)23 b(1)k(giv)n(es)926 1310 y Fv(I)7 b Fx(\()p FA(p)1054 1322 y Ft(1)1091 1310 y Fv(;)14 b FA(p)1181 1322 y Ft(2)1218 1310 y Fx(\))24 b(:=)1385 1197 y Fs(Z)1431 1386 y Fh(R)1478 1369 y Fk(6)1523 1310 y Fv(d)p FA(p)1619 1276 y Fr(0)1619 1331 y Ft(1)1657 1310 y Fv(d)p FA(p)1753 1276 y Fr(0)1753 1331 y Ft(2)1813 1310 y Fu(j)p Fv(k)1879 1325 y Fo(p)1913 1333 y Fk(1)1946 1325 y Ft([)p Fo(B)2015 1333 y Fk(1)2047 1325 y Fo(;\037)2108 1333 y Fk(12)p Fl(;)p Fk(0)2216 1325 y Ft(])2239 1310 y Fx(\()p FA(p)2324 1322 y Ft(1)2362 1310 y Fv(;)14 b FA(p)2452 1322 y Ft(2)2489 1310 y Fv(;)g FA(p)2579 1276 y Fr(0)2579 1331 y Ft(1)2616 1310 y Fv(;)g FA(p)2706 1276 y Fr(0)2706 1331 y Ft(2)2744 1310 y Fx(\))p Fu(j)810 1578 y Fx(=)46 b Fv(p)963 1590 y Ft(1)1085 1522 y Fv(n)1135 1492 y Ft(3)p 1024 1559 209 4 v 1024 1647 a Fx(\(2)p Fv(\031)s Fx(\))1190 1596 y Fk(3)p 1190 1605 29 3 v 1190 1638 a(2)1256 1465 y Fs(Z)1302 1654 y Fh(R)1349 1637 y Fk(3)1395 1578 y Fv(d)p FA(p)1491 1544 y Fr(0)1491 1599 y Ft(1)1551 1578 y Fu(j)10 b Fx(^)-52 b Fv(\037)1626 1590 y Ft(12)p Fo(;)p Ft(0)1749 1578 y Fx(\(\()p FA(p)1866 1590 y Ft(1)1923 1578 y Fu(\000)18 b FA(p)2059 1544 y Fr(0)2059 1599 y Ft(1)2096 1578 y Fx(\))p Fv(n)p Fx(\))p Fu(j)24 b(j)p Fv(B)2343 1590 y Ft(1)2380 1578 y Fx(\()p FA(p)2465 1590 y Ft(1)2502 1578 y Fx(\))19 b Fu(\000)f Fv(B)2699 1590 y Ft(1)2736 1578 y Fx(\()p FA(p)2821 1544 y Fr(0)2821 1599 y Ft(1)2859 1578 y Fx(\))p Fu(j)254 b Fx(\(5.8\))386 1769 y(whic)n(h)39 b(is)g(indep)r(enden)n(t)h(of)f FA(p)1371 1781 y Ft(2)1447 1769 y Fx(and)g(has)g(the)g(iden)n(tical)g (form)g(of)g(\(4.6\))g(\(note)g(that)g(the)386 1885 y(op)r(erators)g Fv(B)830 1897 y Ft(1)908 1885 y Fx(and)h Fv(B)1145 1897 y Ft(2)1223 1885 y Fx(are)g(the)h(same)f(as)g(o)r(ccurred)g(in)h Fv(b)2366 1842 y Ft(\(1\))2366 1907 y(1)p Fo(m)2461 1885 y Fv(;)g Fx(and)51 b(^)-52 b Fv(\037)2752 1897 y Ft(12)p Fo(;)p Ft(0)2920 1885 y Fu(2)45 b(S)6 b Fx(\()p Fp(R)3163 1855 y Ft(3)3206 1885 y Fx(\)\))p Fv(:)386 1987 y Fx(Therefore,)24 b Fv(I)33 b Fx(\(and)26 b(also)e Fv(J)8 b Fx(\))26 b(is)1416 1954 y Ft(1)p 1412 1968 42 4 v 1412 2016 a Fo(n)1463 1987 y Fx(-b)r(ounded)g(b)n(y)f(the)h(pro)r(of)f(of)h(Lemma)f(5.)g(The) h(same)f(holds)386 2087 y(true)g(for)g(the)h(second)f(part)g(of)g(the)h (second)f(summand)g(in)h(\(5.3\))f(whic)n(h)h(corresp)r(onds)d(to)i (the)386 2186 y(second)d(term)h(in)g(\(5.6\).)f(There,)g(expression)g (\()p Fv(b)p Fx(\))h(in)g(\(5.5\))f(has)g(to)h(b)r(e)g(used.)g(Th)n (us,)f(collecting)386 2286 y(results,)573 2468 y Fu(j)p Fx(\()p Fv(B)c(';)951 2411 y Fx(1)p 810 2449 323 4 v 810 2525 a Fu(j)p FA(x)883 2537 y Ft(1)940 2525 y Fu(\000)g FA(x)1073 2537 y Ft(2)1110 2525 y Fu(j)1157 2468 y Fx([)p Fv(B)t(;)c(\037)1336 2480 y Ft(12)1406 2468 y Fx(])g Fv( )1497 2480 y Fo(n)1543 2468 y Fx(\))p Fu(j)46 b(\024)g(j)p Fx(\()2000 2411 y(1)p 1820 2449 402 4 v 1820 2525 a Fv(p)1862 2537 y Ft(1)1899 2525 y Fu(j)p FA(x)1972 2537 y Ft(1)2028 2525 y Fu(\000)18 b FA(x)2161 2537 y Ft(2)2199 2525 y Fu(j)2246 2468 y Fv(B)g(';)c(p)2460 2480 y Ft(1)2511 2468 y Fx([)p Fv(B)2597 2480 y Ft(1)2634 2468 y Fv(;)g(\037)2723 2480 y Ft(12)p Fo(;)p Ft(0)2846 2468 y Fx(])g Fv(B)2946 2480 y Ft(2)2997 2468 y Fv( )3051 2480 y Fo(n)3096 2468 y Fx(\))p Fu(j)1120 2724 y Fx(+)23 b Fu(j)p Fx(\()1453 2668 y(1)p 1273 2705 V 1273 2781 a Fv(p)1315 2793 y Ft(2)1352 2781 y Fu(j)p FA(x)1425 2793 y Ft(1)1481 2781 y Fu(\000)18 b FA(x)1614 2793 y Ft(2)1652 2781 y Fu(j)1699 2724 y Fv(B)g(';)c(B)1934 2736 y Ft(1)1985 2724 y Fv(p)2027 2736 y Ft(2)2078 2724 y Fx([)p Fv(B)2164 2736 y Ft(2)2201 2724 y Fv(;)g(\037)2290 2736 y Ft(12)p Fo(;)p Ft(0)2413 2724 y Fx(])g Fv( )2504 2736 y Fo(n)2549 2724 y Fx(\))p Fu(j)564 b Fx(\(5.9\))405 2965 y Fu(\024)533 2909 y Fv(c)p 526 2946 50 4 v 526 3022 a(n)609 2965 y Fu(k)841 2909 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y(can)d(b)r(e)h(copied)f(to)h(obtain)1061 1532 y Fu(j)p Fx(\()p Fv(\036)1165 1544 y Ft(1)1217 1532 y Fv(';)14 b(b)1344 1489 y Ft(\(1\))1344 1554 y(1)p Fo(m)1454 1532 y Fv(\036)1503 1544 y Ft(1)1554 1532 y Fv(')p Fx(\))p Fu(j)47 b(\024)1846 1476 y Fx(~)-43 b Fv(c)p 1831 1513 64 4 v 1831 1589 a(R)1927 1532 y Fu(k)p Fv(\036)2018 1544 y Ft(1)2069 1532 y Fv(')p Fu(k)2165 1498 y Ft(2)2249 1532 y Fu(\024)2385 1476 y Fx(~)f Fv(c)p 2369 1513 V 2369 1589 a(R)2466 1532 y Fu(k)p Fv(')p Fu(k)2604 1498 y Ft(2)2641 1532 y Fv(:)504 b Fx(\(6.3\))575 1725 y(F)-7 b(or)38 b(the)h(estimate)g(of)g Fv(v)1382 1695 y Ft(\(12\))1504 1725 y Fx(,)g(w)n(e)g(de\014ne)g Fv(\037)g Fx(in)g(analogy)e(to)i (\(5.1\))f(b)n(y)h(noting)f(that)386 1825 y(additionally)-7 b(,)27 b(supp)14 b Fv(\036)1095 1837 y Ft(1)1133 1825 y Fv(')23 b Fu(\032)g Fp(R)1352 1795 y Ft(6)1395 1825 y Fu(n)p Fv(B)1500 1837 y Fo(R)1554 1825 y Fx(\(0\))32 b Fu(\\)h(f)p FA(x)23 b Fu(2)g Fp(R)2027 1795 y Ft(6)2093 1825 y Fx(:)37 b Fu(j)p FA(x)2226 1837 y Ft(1)2283 1825 y Fu(\000)18 b FA(x)2416 1837 y Ft(2)2453 1825 y Fu(j)37 b(\025)23 b Fv(C)6 b(x)p Fu(g)p Fv(:)51 b Fx(Therefore,)1061 2028 y Fv(\037)1113 2040 y Ft(12)1183 2028 y Fx(\()1225 1972 y FA(x)1275 1984 y Ft(1)1332 1972 y Fu(\000)18 b FA(x)1465 1984 y Ft(2)p 1226 2009 277 4 v 1332 2085 a Fv(R)1512 2028 y Fx(\))24 b(:=)1701 1911 y Fs(\032)1805 1978 y Fx(0)p Fv(;)83 b Fu(j)p FA(x)2026 1990 y Ft(1)2082 1978 y Fu(\000)18 b FA(x)2215 1990 y Ft(2)2253 1978 y Fu(j)36 b Fv(<)23 b(C)6 b(R)q(=)p Fx(2)1805 2077 y(1)p Fv(;)124 b Fu(j)p FA(x)2067 2089 y Ft(1)2123 2077 y Fu(\000)18 b FA(x)2256 2089 y Ft(2)2294 2077 y Fu(j)37 b(\025)23 b Fv(C)6 b(R)3168 2028 y Fx(\(6.4\))386 2237 y(suc)n(h)28 b(that)g(again,)f Fv(\036)1048 2249 y Ft(1)1085 2237 y Fv('\037)1191 2249 y Ft(12)1286 2237 y Fx(=)c Fv(\036)1423 2249 y Ft(1)1461 2237 y Fv(')28 b Fx(and)g Fv(\037)1757 2249 y Ft(12)p Fo(;)p Ft(0)1904 2237 y Fx(:=)23 b(1)18 b Fu(\000)g Fv(\037)2210 2249 y Ft(12)2304 2237 y Fu(2)24 b(S)6 b Fx(\()p 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3732 y Fu(\024)2045 3676 y Fx(2)p Fv(c)2123 3688 y Ft(1)p 2045 3713 V 2070 3789 a Fv(R)2192 3732 y Fu(k)p Fv(')p Fu(k)2330 3698 y Ft(2)2367 3732 y Fv(:)778 b Fx(\(6.6\))1422 3979 y(7.)41 b Fw(Pr)n(oof)31 b(of)h(Lemma)f(1)575 4128 y Fx(Assume)21 b(w)n(e)f(ha)n(v)n(e)f(a)i(normalized)e(sequence)h (of)h(functions)g(\()p Fv(')2517 4140 y Fo(n)2563 4128 y Fx(\))2595 4140 y Fo(n)p Fr(2)p Fh(N)2748 4128 y Fx(lo)r(calized)f (outside)386 4228 y Fv(B)449 4240 y Fo(n)494 4228 y Fx(\(0\))30 b(with)g(the)g(prop)r(ert)n(y)f(\(2.4\))h(for)f Fv(h)1686 4198 y Fo(B)s(R)1823 4228 y Fx(and)h(the)g Fv(\025)d Fu(2)g Fp(R)36 b Fx(under)29 b(consideration.)g(Since)386 4328 y(the)c(normalization)e(constan)n(t)g(of)i Fv(')1526 4340 y Fo(n)1596 4328 y Fx(tends)g(to)f(zero)f(as)h Fv(n)f Fu(!)g(1)p Fx(,)i(for)f(an)n(y)f Fv(\036)h Fu(2)f Fv(C)2987 4298 y Fr(1)2981 4348 y Ft(0)3058 4328 y Fx(\()p Fp(R)3144 4298 y Ft(6)3187 4328 y Fx(\))i(w)n(e)386 4437 y(ha)n(v)n(e)40 b Fu(j)p Fx(\()p Fv(\036;)14 b(')786 4449 y Fo(n)832 4437 y Fx(\))p Fu(j)38 b(!)23 b Fx(0)50 b(as)27 b Fv(n)c Fu(!)g(1)p Fv(;)28 b Fx(i.e.)f Fv(')1725 4449 y Fo(n)1811 4390 y(w)1794 4437 y Fv(*)c Fx(0)p Fv(:)50 b Fx(By)28 b(the)g(W)-7 b(eyl)27 b(criterion)g(it)h(follo)n(ws)e(that)386 4537 y Fv(\025)d Fu(2)h Fv(\033)583 4549 y Fo(ess)682 4537 y Fx(\()p Fv(h)762 4507 y Fo(B)s(R)869 4537 y Fx(\))p Fv(:)575 4636 y Fx(Con)n(v)n(ersely)-7 b(,)36 b(let)i Fv(\025)i Fu(2)g Fv(\033)1382 4648 y Fo(ess)1481 4636 y Fx(\()p Fv(h)1561 4606 y Fo(B)s(R)1669 4636 y Fx(\))p Fv(:)78 b Fx(Then)38 b(there)f(exists)h(a)f(W)-7 b(eyl)38 b(sequence)f Fv( )3198 4648 y Fo(n)3283 4636 y Fu(2)386 4746 y(A)p Fx(\()p Fv(C)549 4716 y Fr(1)543 4766 y Ft(0)620 4746 y Fx(\()p Fp(R)706 4716 y Ft(3)750 4746 y Fx(\))15 b Fu(\012)f Fp(C)930 4716 y Ft(2)973 4746 y Fx(\))1005 4716 y Ft(2)1043 4746 y Fv(;)60 b Fu(k)p Fv( )1222 4758 y Fo(n)1267 4746 y Fu(k)22 b Fx(=)h(1)p Fv(;)48 b Fx(with)27 b Fv( )1774 4758 y Fo(n)1859 4699 y(w)1842 4746 y Fv(*)c Fx(0)j(and)f Fu(k)p Fx(\()p Fv(h)2297 4716 y Fo(B)s(R)2419 4746 y Fu(\000)15 b Fv(\025)p Fx(\))f Fv( )2647 4758 y Fo(n)2692 4746 y Fu(k)23 b(!)g Fx(0)48 b(as)26 b Fv(n)d Fu(!)g(1)p Fv(:)p eop %%Page: 14 14 14 13 bop 386 259 a Ft(14)904 b(D.)22 b(H.)g(JAKUBASSA-AMUNDSEN)575 459 y Fx(De\014ne)27 b(a)f(smo)r(oth)h(symmetric)f(\(with)h(resp)r(ect) g(to)f(in)n(terc)n(hange)f(of)i FA(x)2785 471 y Ft(1)2849 459 y Fx(and)g FA(x)3060 471 y Ft(2)3097 459 y Fx(\))g(func-)386 558 y(tion)h Fv(\037)609 570 y Ft(0)669 558 y Fu(2)23 b Fv(C)812 528 y Fr(1)806 579 y Ft(0)883 558 y Fx(\()p Fp(R)969 528 y Ft(6)1012 558 y Fx(\))28 b(mapping)g(to)f([0)p Fv(;)14 b Fx(1])27 b(b)n(y)g(means)g(of)1396 753 y Fv(\037)1448 765 y Ft(0)1485 753 y Fx(\()1527 697 y FA(x)p 1527 734 51 4 v 1527 810 a Fv(n)1587 753 y Fx(\))47 b(=)1777 636 y Fs(\032)1880 702 y Fx(1)p Fv(;)104 b(x)23 b Fu(\024)g Fv(n)1880 802 y Fx(0)p Fv(;)83 b(x)23 b(>)g Fx(2)p Fv(n)3168 753 y Fx(\(7.1\))386 953 y(and)43 b(let)g Fv(\037)750 965 y Fo(n)795 953 y Fx(\()p FA(x)p Fx(\))49 b(:=)f(1)28 b Fu(\000)g Fv(\037)1309 965 y Ft(0)1347 953 y Fx(\()1390 920 y Fm(x)p 1389 934 42 4 v 1389 981 a Fo(n)1440 953 y Fx(\))p Fv(;)111 b(\037)1658 965 y Fo(n)1751 953 y Fu(2)49 b Fv(C)1920 922 y Fr(1)1991 953 y Fx(\()p Fp(R)2077 922 y Ft(6)2120 953 y Fu(n)p Fv(B)2225 965 y Fo(n)2270 953 y Fx(\(0\)\))p Fv(:)92 b Fx(Then)43 b Fv(')2809 965 y Fo(n)2902 953 y Fx(:=)49 b Fv( )3093 965 y Fo(n)3138 953 y Fv(\037)3190 965 y Fo(n)3283 953 y Fu(2)386 1057 y(A)p Fx(\()p Fv(C)549 1027 y Fr(1)543 1078 y Ft(0)620 1057 y Fx(\()p Fp(R)706 1027 y Ft(6)750 1057 y Fu(n)p Fv(B)855 1069 y Fo(n)899 1057 y Fx(\(0\)\))27 b Fu(\012)f Fp(C)1209 1027 y Ft(4)1252 1057 y Fx(\))40 b(and)f(w)n(e)g(claim)h (that)g(a)f(subsequence)g(of)g(\()p Fv(')2811 1069 y Fo(n)2857 1057 y Fx(\))2889 1069 y Fo(n)p Fr(2)p Fh(N)3061 1057 y Fx(satis\014es)386 1157 y(the)28 b(requiremen)n(ts)e(of)i(Lemma) f(1.)386 1327 y Fi(a\))83 b Fu(k)p Fx(\()p Fv(h)667 1297 y Fo(B)s(R)793 1327 y Fu(\000)18 b Fv(\025)p Fx(\))c Fv(')1024 1339 y Fo(n)1070 1327 y Fu(k)36 b(!)23 b Fx(0)51 b(as)27 b Fv(n)22 b Fu(!)i(1)f Fx(:)575 1450 y(W)-7 b(e)28 b(decomp)r(ose)769 1595 y Fu(k)p Fx(\()p Fv(h)891 1561 y Fo(B)s(R)1017 1595 y Fu(\000)18 b Fv(\025)p Fx(\))c Fv(\037)1246 1607 y Fo(n)1291 1595 y Fv( )1345 1607 y Fo(n)1391 1595 y Fu(k)45 b(\024)h(k)p Fv(\037)1683 1607 y Fo(n)1741 1595 y Fx(\()p Fv(h)1821 1561 y Fo(B)s(R)1947 1595 y Fu(\000)18 b Fv(\025)p Fx(\))c Fv( )2178 1607 y Fo(n)2224 1595 y Fu(k)32 b Fx(+)g Fu(k)14 b Fx([)p Fv(h)2522 1561 y Fo(B)s(R)2629 1595 y Fv(;)g(\037)2718 1607 y Ft(0)2755 1595 y Fx(])g Fv( )2846 1607 y Fo(n)2891 1595 y Fu(k)p Fv(;)212 b Fx(\(7.2\))386 1740 y(and)29 b(use)h(the)g(equiv)-5 b(alence)29 b(\(2.20\))g(again.)g(Concerning)f (the)i(\014rst)g(term)f(in)h(\(7.2\),)g(w)n(e)f(ha)n(v)n(e)386 1839 y(for)e(an)n(y)g Fv(\036)c Fu(2)h Fv(C)886 1809 y Fr(1)880 1860 y Ft(0)956 1839 y Fx(\()p Fp(R)1043 1809 y Ft(6)1086 1839 y Fx(\))19 b Fu(\012)f Fp(C)1274 1809 y Ft(4)1317 1839 y Fv(;)467 1987 y Fu(j)p Fx(\()p Fv(\036;)c(\037)660 1999 y Fo(n)720 1987 y Fx(\()p Fv(h)800 1953 y Fo(B)s(R)926 1987 y Fu(\000)k Fv(\025)p Fx(\))c Fv( )1157 1999 y Fo(n)1203 1987 y Fx(\))p Fu(j)46 b(\024)g(k)p Fv(\037)1509 1999 y Fo(n)1568 1987 y Fv(\036)p Fu(k)23 b(k)p Fx(\()p Fv(h)1804 1953 y Fo(B)s(R)1929 1987 y Fu(\000)18 b Fv(\025)p Fx(\))c Fv( )2160 1999 y Fo(n)2206 1987 y Fu(k)36 b(\000)-14 b(!)46 b Fx(0)194 b(as)50 b Fv(n)22 b Fu(!)i(1)81 b Fx(\(7.3\))386 2132 y(b)n(y)27 b(assumption,)g(since)h Fu(k)p Fv(\037)1261 2144 y Fo(n)1305 2132 y Fv(\036)p Fu(k)37 b(\024)23 b(k)p Fv(\036)p Fu(k)f Fv(<)h Fu(1)p Fv(:)575 2232 y Fx(In)k(order)e(to)i (treat)f(the)h(single-particle)e(con)n(tribution)h(to)h(the)g(second)f (term)h(in)g(\(7.2\),)386 2347 y Fv(T)447 2317 y Ft(\()p Fo(k)q Ft(\))569 2347 y Fx(and)i Fv(b)768 2304 y Ft(\()p Fo(k)q Ft(\))768 2369 y(1)p Fo(m)864 2347 y Fv(;)h Fx(w)n(e)g(c)n (hange)e(again)h(to)h(F)-7 b(ourier)29 b(space)g(and)g(in)n(tro)r(duce) h(the)g(6-dimensional)386 2451 y(F)-7 b(ourier)23 b(transform)h(of)g Fv(\037)1192 2463 y Ft(0)1254 2451 y Fx(according)f(to)h(\(3.6\))g (with)h Fv(l)g Fx(=)d(2)p Fv(:)j Fx(Then)f(the)h(k)n(ernel)f(of)g([)p Fv(T)3101 2421 y Ft(\(1\))3190 2451 y Fv(;)14 b(\037)3279 2463 y Ft(0)3316 2451 y Fx(])386 2551 y(in)28 b(momen)n(tum)g(space)e (reads)442 2744 y Fv(k)485 2763 y Ft([)p Fo(T)552 2747 y Fk(\(1\))629 2763 y Fo(;\037)690 2771 y Fk(0)723 2763 y Ft(])746 2744 y Fx(\()p FA(p)831 2756 y Ft(1)868 2744 y Fv(;)14 b FA(p)958 2756 y Ft(2)995 2744 y Fv(;)g FA(p)1085 2710 y Fr(0)1085 2765 y Ft(1)1123 2744 y Fv(;)g FA(p)1213 2710 y Fr(0)1213 2765 y Ft(2)1250 2744 y Fx(\))46 b(=)1502 2688 y Fv(n)1552 2658 y Ft(6)p 1449 2725 194 4 v 1449 2801 a Fx(\(2)p Fv(\031)s Fx(\))1605 2777 y Ft(3)1685 2744 y Fx(^)-51 b Fv(\037)1728 2756 y Ft(0)1765 2744 y Fx(\(\()p FA(p)1882 2756 y Ft(1)1938 2744 y Fu(\000)18 b FA(p)2074 2710 y Fr(0)2074 2765 y Ft(1)2111 2744 y Fx(\))p Fv(n;)c Fx(\()p FA(p)2315 2756 y Ft(2)2371 2744 y Fu(\000)k FA(p)2507 2710 y Fr(0)2507 2765 y Ft(2)2545 2744 y Fx(\))p Fv(n)p Fx(\))23 b(\()p Fv(E)2775 2756 y Fo(p)2809 2764 y Fk(1)2865 2744 y Fu(\000)18 b Fv(E)3009 2758 y Fo(p)3043 2739 y Fj(0)3043 2777 y Fk(1)3080 2744 y Fx(\))56 b(\(7.4\))386 2955 y(and)27 b(b)n(y)h(the)g(mean)f(v)-5 b(alue)28 b(theorem)f(\(4.1\),)g(using)g Fu(r)2072 2967 y Fm(p)2118 2955 y Fv(E)2179 2967 y Fo(p)2255 2955 y Fx(=)c Fu(r)2412 2967 y Fm(p)2471 2882 y Fs(p)p 2554 2882 291 4 v 73 x Fv(p)2596 2931 y Ft(2)2652 2955 y Fx(+)18 b Fv(m)2808 2931 y Ft(2)2882 2955 y Fx(=)23 b FA(p)p Fv(=E)3126 2967 y Fo(p)3164 2955 y Fv(;)1067 3152 y Fu(j)p Fv(E)1151 3164 y Fo(p)1185 3172 y Fk(1)1240 3152 y Fu(\000)18 b Fv(E)1384 3166 y Fo(p)1418 3147 y Fj(0)1418 3185 y Fk(1)1455 3152 y Fu(j)46 b Fx(=)g Fu(j)p FA(p)1711 3164 y Ft(1)1767 3152 y Fu(\000)18 b FA(p)1903 3118 y Fr(0)1903 3173 y Ft(1)1940 3152 y Fu(j)2000 3032 y Fs(\014)2000 3082 y(\014)2000 3131 y(\014)2000 3181 y(\014)2064 3096 y Fn(\030)p 2038 3133 98 4 v 2038 3209 a Fv(E)2099 3221 y Fo(\030)2145 3032 y Fs(\014)2145 3082 y(\014)2145 3131 y(\014)2145 3181 y(\014)2219 3152 y Fu(\024)46 b(j)p FA(p)2406 3164 y Ft(1)2462 3152 y Fu(\000)18 b FA(p)2598 3118 y Fr(0)2598 3173 y Ft(1)2635 3152 y Fu(j)510 b Fx(\(7.5\))386 3349 y(for)37 b(all)h Fn(\030)43 b Fx(:=)d Fv(\025)p FA(p)963 3361 y Ft(1)1027 3349 y Fx(+)25 b(\(1)g Fu(\000)g Fv(\025)p Fx(\))p FA(p)1439 3318 y Fr(0)1439 3369 y Ft(1)1515 3349 y Fx(with)38 b Fv(\025)j Fu(2)g Fx([0)p Fv(;)14 b Fx(1])p Fv(:)78 b Fx(F)-7 b(or)37 b(the)i(in)n(tegral)e Fv(I)45 b Fx(app)r(earing)36 b(in)386 3448 y(the)d(Lieb)f(and)h(Y)-7 b(au)32 b(form)n(ula)g(\(3.1\))g(w)n(e)g(ha)n(v)n(e)g(with)h Fv(f)39 b Fx(=)31 b(1)h(and)h(the)f(substitution)i FA(y)3179 3460 y Fo(k)3251 3448 y Fx(:=)386 3548 y(\()p FA(p)471 3560 y Fo(k)531 3548 y Fu(\000)18 b FA(p)667 3518 y Fr(0)667 3571 y Fo(k)707 3548 y Fx(\))p Fv(n;)61 b(k)25 b Fx(=)e(1)p Fv(;)14 b Fx(2)p Fv(;)561 3748 y(I)7 b Fx(\()p FA(p)689 3760 y Ft(1)727 3748 y Fv(;)14 b FA(p)817 3760 y Ft(2)854 3748 y Fx(\))23 b(:=)1020 3635 y Fs(Z)1066 3824 y Fh(R)1113 3807 y Fk(6)1158 3748 y Fv(d)p FA(p)1254 3714 y Fr(0)1254 3769 y Ft(1)1306 3748 y Fv(d)p FA(p)1402 3714 y Fr(0)1402 3769 y Ft(2)1526 3692 y Fv(n)1576 3662 y Ft(6)p 1472 3729 194 4 v 1472 3805 a Fx(\(2)p Fv(\031)s Fx(\))1628 3781 y Ft(3)1699 3748 y Fu(j)10 b Fx(^)-52 b Fv(\037)1774 3760 y Ft(0)1811 3748 y Fx(\(\()p FA(p)1928 3760 y Ft(1)1985 3748 y Fu(\000)18 b FA(p)2121 3714 y Fr(0)2121 3769 y Ft(1)2158 3748 y Fx(\))p Fv(n;)c Fx(\()p FA(p)2362 3760 y Ft(2)2418 3748 y Fu(\000)k FA(p)2554 3714 y Fr(0)2554 3769 y Ft(2)2591 3748 y Fx(\))p Fv(n)p Fx(\))p Fu(j)24 b(j)p Fv(E)2836 3760 y Fo(p)2870 3768 y Fk(1)2925 3748 y Fu(\000)18 b Fv(E)3069 3762 y Fo(p)3103 3743 y Fj(0)3103 3781 y Fk(1)3140 3748 y Fu(j)1040 4002 y(\024)1237 3946 y Fx(1)p 1161 3983 V 1161 4059 a(\(2)p Fv(\031)s Fx(\))1317 4035 y Ft(3)1378 3889 y Fs(Z)1424 4078 y Fh(R)1471 4062 y Fk(6)1517 4002 y Fv(d)p FA(y)1610 4014 y Ft(1)1661 4002 y Fv(d)p FA(y)1754 4014 y Ft(2)1815 4002 y Fu(j)10 b Fx(^)-52 b Fv(\037)1890 4014 y Ft(0)1927 4002 y Fx(\()p FA(y)2009 4014 y Ft(1)2047 4002 y Fv(;)14 b FA(y)2134 4014 y Ft(2)2172 4002 y Fx(\))p Fu(j)23 b Fv(y)2291 4014 y Ft(1)2347 4002 y Fu(\001)2402 3946 y Fx(1)p 2398 3983 50 4 v 2398 4059 a Fv(n)2504 4002 y Fu(\024)2632 3946 y Fv(c)p 2625 3983 V 2625 4059 a(n)3168 4002 y Fx(\(7.6\))386 4184 y(since)37 b(^)-52 b Fv(\037)641 4196 y Ft(0)702 4184 y Fu(2)23 b(S)6 b Fx(\()p Fp(R)923 4154 y Ft(6)966 4184 y Fx(\))p Fv(:)51 b Fx(In)28 b(a)f(similar)g(w)n(a)n(y)-7 b(,)26 b Fv(J)8 b Fx(\()p FA(p)1841 4154 y Fr(0)1841 4204 y Ft(1)1879 4184 y Fv(;)14 b FA(p)1969 4154 y Fr(0)1969 4204 y Ft(2)2006 4184 y Fx(\))23 b Fu(\024)g Fv(c=n;)50 b Fx(and)28 b(hence)1253 4350 y Fu(j)p Fx(\()p Fv(\036;)14 b Fx([)p Fv(T)1478 4315 y Ft(\(1\))1567 4350 y Fv(;)g(\037)1656 4362 y Ft(0)1693 4350 y Fx(])g Fv( )1784 4362 y Fo(n)1829 4350 y Fx(\))p Fu(j)46 b(\024)2058 4293 y Fv(c)p 2051 4330 V 2051 4407 a(n)2134 4350 y Fu(k)p Fv(\036)p Fu(k)23 b(k)p Fv( )2386 4362 y Fo(n)2430 4350 y Fu(k)696 b Fx(\(7.7\))386 4547 y(for)32 b(all)g Fv(\036)f Fu(2)g Fv(C)869 4516 y Fr(1)863 4567 y Ft(0)940 4547 y Fx(\()p Fp(R)1026 4516 y Ft(6)1069 4547 y Fx(\))22 b Fu(\012)f Fp(C)1263 4516 y Ft(4)1307 4547 y Fv(:)63 b Fx(F)-7 b(or)32 b(the)g(op)r(erator)f Fv(b)2070 4503 y Ft(\(1\))2070 4569 y(1)p Fo(m)2198 4547 y Fx(w)n(e)h(can)g(pro)r(ceed)f(as)h(in)h(the)f(pro)r(of)386 4646 y(of)f(Lemma)f(5,)h(b)r(ecause)f(\(according)g(to)h(\(7.6\)\))f (the)i(t)n(w)n(o-particle)d(nature)h(of)h Fv(\037)2960 4658 y Ft(0)3028 4646 y Fx(do)r(es)g(not)386 4746 y(a\013ect)d(the)g (con)n(v)n(ergence)d(of)i(the)h(single-particle)e(in)n(tegrals.)p eop %%Page: 15 15 15 14 bop 1606 259 a Ft(HVZ)30 b(THEOREM)1153 b(15)575 459 y Fx(F)-7 b(or)41 b(the)g(estimate)h(of)f(the)h(remaining)e(comm)n (utator,)h([)p Fv(v)2479 429 y Ft(\(12\))2601 459 y Fv(;)14 b(\037)2690 471 y Ft(0)2727 459 y Fx(],)42 b(w)n(e)f(follo)n(w)f(sec-) 386 558 y(tion)27 b(5)h(to)f(split)h(it)g(in)n(to)f(comm)n(utators)f (of)h Fv(\037)1810 570 y Ft(0)1875 558 y Fx(with)h(single-particle)e (\(b)r(ounded\))j(op)r(erators)386 658 y Fv(B)449 670 y Ft(1)486 658 y Fx(\()p FA(p)571 670 y Ft(1)609 658 y Fx(\))p Fv(;)37 b(B)764 670 y Ft(2)801 658 y Fx(\()p FA(p)886 670 y Ft(2)924 658 y Fx(\))p Fv(:)49 b Fx(The)25 b(only)g(di\013erence)g(as)g(compared)f(to)h(the)h(pro)r(of)e(of)h (Lemma)g(6)g(lies)g(in)386 758 y(the)k(t)n(w)n(o-particle)d(nature)i (of)g Fv(\037)1398 770 y Ft(0)1488 758 y Fx(\(cf.)h(\(7.4\))f(in)h (place)f(of)g(\(5.7\)\),)g(but)h(again)e(this)i(do)r(es)f(not)386 857 y(a\013ect)g(the)g(con)n(v)n(ergence)d(of)i(the)h(in)n(tegrals.)e (Th)n(us)i(w)n(e)f(get)659 1111 y Fu(j)p Fx(\()p Fv(\036;)14 b Fx([)p Fv(h)871 1077 y Fo(B)s(R)979 1111 y Fv(;)g(\037)1068 1123 y Ft(0)1105 1111 y Fx(])g Fv( )1196 1123 y Fo(n)1241 1111 y Fx(\))p Fu(j)47 b(\024)1497 1007 y Ft(2)1454 1032 y Fs(X)1453 1211 y Fo(k)q Ft(=1)1588 1019 y Fs(\020)1638 1111 y Fu(j)p Fx(\()p Fv(\036;)14 b Fx([)p Fv(T)1863 1077 y Ft(\()p Fo(k)q Ft(\))1955 1111 y Fv(;)g(\037)2044 1123 y Ft(0)2081 1111 y Fx(])g Fv( )2172 1123 y Fo(n)2217 1111 y Fx(\))p Fu(j)33 b Fx(+)f Fu(j)p Fx(\()p Fv(\036;)14 b Fx([)p Fv(b)2602 1068 y Ft(\()p Fo(k)q Ft(\))2602 1133 y(1)p Fo(m)2698 1111 y Fv(;)g(\037)2787 1123 y Ft(0)2825 1111 y Fx(])g Fv( )2916 1123 y Fo(n)2961 1111 y Fx(\))p Fu(j)3016 1019 y Fs(\021)1201 1444 y Fx(+)22 b Fu(j)p Fx(\()p Fv(\036;)14 b Fx([)p Fv(v)1495 1410 y Ft(\(12\))1619 1444 y Fv(;)g(\037)1708 1456 y Ft(0)1745 1444 y Fx(])g Fv( )1836 1456 y Fo(n)1881 1444 y Fx(\))p Fu(j)46 b(\024)2110 1388 y Fv(c)p 2103 1425 50 4 v 2103 1501 a(n)2186 1444 y Fu(k)p Fv(\036)p Fu(k)22 b(k)p Fv( )2437 1456 y Fo(n)2482 1444 y Fu(k)644 b Fx(\(7.8\))386 1630 y(with)28 b(the)g(generic)f (constan)n(t)f Fv(c)p Fx(.)386 1800 y Fi(b\))199 b Fu(k)p Fv(')753 1812 y Fo(n)799 1800 y Fu(k)36 b(6)p Fx(=)g(0)23 b(:)575 1924 y(W)-7 b(e)26 b(sho)n(w)e(that)i(for)f(an)n(y)g Fv(\017)e(>)f Fx(0)k(there)f(is)g(an)h Fv(N)2061 1936 y Ft(0)2121 1924 y Fu(2)d Fp(N)36 b Fx(suc)n(h)25 b(that)h Fu(k)p Fv(\037)2742 1936 y Ft(0)2778 1924 y Fv( )2832 1936 y Fo(n)2878 1924 y Fu(k)36 b Fv(<)23 b(\017)48 b Fx(for)25 b(all)386 2023 y Fv(n)e Fu(\025)f Fv(N)613 2035 y Ft(0)650 2023 y Fv(:)49 b Fx(Then)39 b Fu(k)p Fx(\(1)13 b Fu(\000)g Fv(\037)1209 2035 y Fo(n)1253 2023 y Fx(\))h Fv( )1353 2035 y Fo(n)1399 2023 y Fu(k)36 b Fx(=)g Fu(k)p Fv( )1674 2035 y Fo(n)1732 2023 y Fu(\000)13 b Fv(')1864 2035 y Fo(n)1910 2023 y Fu(k)36 b Fv(<)23 b(\017:)48 b Fx(As)25 b(a)f(consequence,)g Fu(k)p Fv(')2952 2035 y Fo(n)2997 2023 y Fu(k)f(6)p Fx(=)g(0)47 b(for)386 2123 y(su\016cien)n(tly)28 b(large)f Fv(n)c(>)h(n)1218 2135 y Ft(0)1279 2123 y Fu(\025)f Fv(N)1434 2135 y Ft(0)1500 2123 y Fx(since)k Fu(k)p Fv( )1799 2135 y Fo(n)1844 2123 y Fu(k)d Fx(=)f(1)52 b(\(i.e.)28 b(one)g(can)g(c)n(ho)r(ose)f(a)g (subsequence)386 2222 y(of)g(\()p Fv(')566 2234 y Fo(n)612 2222 y Fx(\))644 2234 y Fo(n)p Fr(2)p Fh(N)804 2222 y Fx(with)h(normalizable)e(elemen)n(ts\).)575 2322 y(Since)33 b Fv(h)845 2292 y Fo(B)s(R)984 2322 y Fx(is)g(a)f(self-adjoin)n(t)g(p)r (ositiv)n(e)g(op)r(erator,)f Fv(h)2302 2292 y Fo(B)s(R)2431 2322 y Fx(+)21 b(1)31 b Fv(>)f Fx(0)i(with)h(a)f(b)r(ounded)386 2422 y(in)n(v)n(erse.)26 b(Then)i(follo)n(wing)e([4])i(w)n(e)f (estimate)854 2605 y Fu(k)p Fv(\037)948 2617 y Ft(0)998 2605 y Fv( )1052 2617 y Fo(n)1097 2605 y Fu(k)46 b Fx(=)g Fu(k)p Fv(\037)1390 2617 y Ft(0)1440 2605 y Fx(\()p Fv(h)1520 2570 y Fo(B)s(R)1646 2605 y Fx(+)18 b(1\))1803 2570 y Fr(\000)p Ft(1)1915 2605 y Fx([\()p Fv(h)2018 2570 y Fo(B)s(R)2144 2605 y Fu(\000)g Fv(\025)p Fx(\))33 b(+)f(\(1)19 b(+)f Fv(\025)p Fx(\)])c Fv( )2784 2617 y Fo(n)2829 2605 y Fu(k)517 2833 y(\024)46 b(k)p Fv(\037)722 2845 y Ft(0)772 2833 y Fx(\()p Fv(h)852 2798 y Fo(B)s(R)978 2833 y Fx(+)18 b(1\))1135 2798 y Fr(\000)p Ft(1)1224 2833 y Fu(k)23 b(k)p Fx(\()p Fv(h)1411 2798 y Fo(B)s(R)1536 2833 y Fu(\000)18 b Fv(\025)p Fx(\))c Fv( )1767 2845 y Fo(n)1813 2833 y Fu(k)41 b Fx(+)g Fu(j)p Fx(1)18 b(+)g Fv(\025)p Fu(j)24 b(k)p Fv(\037)2357 2845 y Ft(0)2407 2833 y Fx(\()p Fv(h)2487 2798 y Fo(B)s(R)2613 2833 y Fx(+)18 b(1\))2770 2798 y Fr(\000)p Ft(1)2873 2833 y Fv( )2927 2845 y Fo(n)2972 2833 y Fu(k)p Fv(:)131 b Fx(\(7.9\))386 2995 y(Since)30 b Fv(\037)657 3007 y Ft(0)721 2995 y Fu(2)e Fv(C)869 2965 y Fr(1)863 3015 y Ft(0)939 2995 y Fx(\()p Fp(R)1025 2965 y Ft(6)1069 2995 y Fx(\))i(and)g(b)n(y)f(assumption)h Fu(9)14 b Fv(N)1981 3007 y Ft(1)2045 2995 y Fx(:)54 b Fu(k)p Fx(\()p Fv(h)2244 2965 y Fo(B)s(R)2371 2995 y Fu(\000)19 b Fv(\025)p Fx(\))14 b Fv( )2603 3007 y Fo(n)2649 2995 y Fu(k)41 b Fv(<)27 b Fx(~)-43 b Fv(\017)57 b Fx(for)29 b Fv(n)e Fu(\025)g Fv(N)3279 3007 y Ft(1)3316 2995 y Fv(;)386 3094 y Fx(the)h(\014rst)f(term)h(is)f(b)r(ounded)h(b)n(y)-7 b(,)28 b(sa)n(y)-7 b(,)26 b Fv(c)1648 3106 y Ft(1)1686 3094 y Fx(~)-42 b Fv(\017)27 b Fx(for)g Fv(n)c Fu(\025)f Fv(N)2101 3106 y Ft(1)2138 3094 y Fv(:)575 3194 y Fx(F)-7 b(or)40 b(the)h(second)f(term)g(w)n(e)g(sho)n(w)g(that)h(the)g(b)r (ounded)g(function)g Fv(\037)2830 3206 y Ft(0)2908 3194 y Fx(is)f(relativ)n(ely)386 3294 y(compact)27 b(with)h(resp)r(ect)f(to) h Fv(h)1341 3263 y Fo(B)s(R)1448 3294 y Fv(:)g Fx(W)-7 b(e)28 b(write)855 3481 y Fv(\037)907 3493 y Ft(0)958 3481 y Fx(\()p Fv(h)1038 3447 y Fo(B)s(R)1164 3481 y Fx(+)18 b(1\))1321 3447 y Fr(\000)p Ft(1)1456 3481 y Fx(=)46 b Fv(\037)1619 3493 y Ft(0)1679 3481 y Fx(\()p Fv(T)30 b Fx(+)18 b(1\))1947 3447 y Fr(\000)2009 3425 y Fk(1)p 2009 3434 29 3 v 2009 3467 a(2)2074 3481 y Fx(\()p Fv(T)30 b Fx(+)18 b(1\))2352 3425 y Fk(1)p 2352 3434 V 2352 3467 a(2)2417 3481 y Fx(\()p Fv(h)2497 3447 y Fo(B)s(R)2623 3481 y Fx(+)g(1\))2780 3447 y Fr(\000)p Ft(1)3127 3481 y Fx(\(7.10\))386 3669 y(and)i(note)g(that)g(\()p Fv(T)15 b Fx(+)s(1\))1137 3617 y Fk(1)p 1136 3626 V 1136 3659 a(2)1192 3669 y Fx(\()p Fv(h)1272 3639 y Fo(B)s(R)1383 3669 y Fx(+)s(1\))1525 3639 y Fr(\000)1587 3617 y Fk(1)p 1587 3626 V 1587 3659 a(2)1633 3669 y Fu(\001)s Fx(\()p Fv(h)1739 3639 y Fo(B)s(R)1850 3669 y Fx(+)s(1\))1992 3639 y Fr(\000)2053 3617 y Fk(1)p 2053 3626 V 2053 3659 a(2)2139 3669 y Fx(is)20 b(a)f(pro)r(duct)h(of)g(b)r(ounded)h(op)r (erators)386 3769 y(since)27 b Fv(V)47 b Fx(is)27 b Fv(T)12 b Fx(-from)26 b(b)r(ounded)i(\(with)h(form)e(b)r(ound)h Fv(<)23 b Fx(1)k(for)g Fv(\015)h(<)22 b(\015)2549 3781 y Fo(B)s(R)2657 3769 y Fx(\))p Fv(:)575 3877 y Fx(F)-7 b(or)32 b(the)h(compactness)f(of)h Fv(\037)1515 3889 y Ft(0)1566 3877 y Fx(\()p Fv(T)h Fx(+)21 b(1\))1841 3847 y Fr(\000)1903 3825 y Fk(1)p 1903 3834 V 1903 3867 a(2)1978 3877 y Fx(w)n(e)33 b(apply)f(a)h(theorem)f(\([22,)h(p.115],)f ([21)o(,)386 3977 y(Lemma)18 b(6.9])g(stating)h(that)g(for)f(b)r (ounded)h(functions)g Fv(f)9 b Fx(\()p Fv(x)p Fx(\))20 b(and)e Fv(g)s Fx(\()p Fv(p)p Fx(\))h(with)g Fv(f)t(;)14 b(g)26 b Fx(:)d([0)p Fv(;)14 b Fu(1)p Fx(\))23 b Fu(!)g Fp(C)386 4076 y Fx(and)55 b(lim)547 4126 y Fo(x)p Fr(!1)731 4076 y Fv(f)9 b Fx(\()p Fv(x)p Fx(\))24 b(=)f(0)f(=)48 b(lim)1156 4126 y Fo(p)p Fr(!1)1336 4076 y Fv(g)s Fx(\()p Fv(p)p Fx(\))j(the)28 b(pro)r(duct)f Fv(K)i Fx(:=)23 b Fv(g)s(f)35 b Fx(is)28 b(compact.)575 4233 y(Clearly)-7 b(,)37 b(b)r(oth)i Fv(\037)1149 4245 y Ft(0)1225 4233 y Fx(and)f(\()p Fv(T)f Fx(+)26 b(1\))1680 4203 y Fr(\000)1741 4181 y Fk(1)p 1741 4190 V 1741 4223 a(2)1825 4233 y Fx(=)41 b(\()1963 4161 y Fs(p)p 2046 4161 291 4 v 72 x Fv(p)2088 4205 y Ft(2)2088 4256 y(1)2144 4233 y Fx(+)18 b Fv(m)2300 4209 y Ft(2)2376 4233 y Fx(+)2466 4161 y Fs(p)p 2550 4161 V 2550 4233 a Fv(p)2592 4205 y Ft(2)2592 4256 y(2)2647 4233 y Fx(+)g Fv(m)2803 4209 y Ft(2)2880 4233 y Fx(+)25 b(1\))3044 4203 y Fr(\000)3106 4181 y Fk(1)p 3106 4190 29 3 v 3106 4223 a(2)3228 4233 y Fx(are)386 4345 y(b)r(ounded)f (functions,)f(going)g(to)g(zero)f(as)h Fv(x)g Fu(!)g(1)h Fx(and)f Fv(p)g Fx(:=)2296 4272 y Fs(p)p 2379 4272 260 4 v 73 x Fv(p)2421 4316 y Ft(2)2421 4367 y(1)2477 4345 y Fx(+)18 b Fv(p)2602 4316 y Ft(2)2602 4367 y(2)2662 4345 y Fu(!)23 b(1)p Fv(;)h Fx(resp)r(ectiv)n(ely)-7 b(.)386 4447 y(Therefore,)29 b Fv(\037)840 4459 y Ft(0)891 4447 y Fx(\()p Fv(h)971 4417 y Fo(B)s(R)1099 4447 y Fx(+)20 b(1\))1258 4417 y Fr(\000)p Ft(1)1377 4447 y Fx(is)30 b(compact)g(and)g(maps)g(the)g(w)n(eakly)f(con)n(v)n(ergen)n(t)f (sequence)386 4547 y(\()p Fv( )472 4559 y Fo(n)517 4547 y Fx(\))549 4559 y Fo(n)p Fr(2)p Fh(N)712 4547 y Fx(in)n(to)h(a)h (strongly)f(con)n(v)n(ergen)n(t)e(sequence.)j(So)f(the)i(second)e(term) h(in)g(\(7.9\))g(can)f(b)r(e)386 4646 y(estimated)37 b(b)n(y)f Fv(c)934 4658 y Ft(2)972 4646 y Fx(~)-42 b Fv(\017)36 b Fx(for,)h(sa)n(y)-7 b(,)36 b Fv(n)i Fu(\025)g Fv(N)1629 4658 y Ft(2)1666 4646 y Fv(:)76 b Fx(This)36 b(pro)n(v)n(es)f(the)j(assertion)49 b Fu(k)p Fv(\037)2846 4658 y Ft(0)2883 4646 y Fv( )2937 4658 y Fo(n)2982 4646 y Fu(k)j Fv(<)38 b(\017)g Fx(:=)386 4746 y(\()p Fv(c)454 4758 y Ft(1)510 4746 y Fx(+)18 b Fv(c)629 4758 y Ft(2)666 4746 y Fx(\))q(~)-43 b Fv(\017)51 b Fx(for)27 b Fv(n)c Fu(\025)f Fv(N)1137 4758 y Ft(0)1197 4746 y Fx(:=)h(max)p Fu(f)p Fv(N)1572 4758 y Ft(1)1608 4746 y Fv(;)14 b(N)1712 4758 y Ft(2)1749 4746 y Fu(g)p Fv(:)p eop %%Page: 16 16 16 15 bop 386 259 a Ft(16)904 b(D.)22 b(H.)g(JAKUBASSA-AMUNDSEN)1422 459 y Fx(8.)41 b Fw(Pr)n(oof)31 b(of)h(Lemma)f(4)575 608 y Fx(W)-7 b(e)19 b(ha)n(v)n(e)f(to)h(sho)n(w)f(that)h(for)g Fv(')k Fu(2)h(A)p Fx(\()p Fv(C)1791 578 y Fr(1)1785 629 y Ft(0)1862 608 y Fx(\()p Fp(R)1948 578 y Ft(6)1991 608 y Fu(n)p Fv(B)2096 620 y Fo(R)2150 608 y Fx(\(0\)\))q Fu(\012)q Fp(C)2410 578 y Ft(4)2453 608 y Fx(\))42 b(and)19 b Fv(\037)p Fx(\()2780 575 y Fm(x)p 2774 589 51 4 v 2774 637 a Fo(R)2834 608 y Fx(\))h(the)f(auxiliary)386 732 y(function)26 b(from)f(\(3.4\))g(with)h Fv(l)f Fx(=)e(2)p Fv(;)59 b Fx(\()p Fv(\036)1629 744 y Fo(j)1665 732 y Fv(';)14 b Fx([)p Fv(b)1815 689 y Ft(\(1\))1815 754 y(1)p Fo(m)1911 732 y Fv(;)g(\036)1997 744 y Fo(j)2032 732 y Fv(\037)p Fx(])g Fv(')p Fx(\))49 b(as)25 b(w)n(ell)h(as)e(\()p Fv(\036)2703 744 y Fo(j)2739 732 y Fv(';)14 b Fx([)p Fv(v)2896 702 y Ft(\(12\))3019 732 y Fv(;)g(\036)3105 744 y Fo(j)3140 732 y Fv(\037)p Fx(])g Fv(')p Fx(\))386 831 y(are)27 b(uniformly)g(1)p Fv(=R)q Fx(-b)r(ounded.)575 931 y(Let)37 b(us)f(denote)h Fv(\036)1174 943 y Fo(j)1209 931 y Fv(\037)h Fx(=:)g Fv( )1479 943 y Fo(j)1514 931 y Fv(:)75 b Fx(W)-7 b(e)37 b(ha)n(v)n(e)f(supp)14 b Fv(\036)2199 943 y Fo(j)2234 931 y Fv(\037)38 b Fu(\022)g Fx(supp)14 b Fv(\037)75 b Fx(whic)n(h)37 b(is)f(lo)r(cated)386 1031 y(outside)27 b Fv(B)735 1046 y Fo(R=)p Ft(2)857 1031 y Fx(\(0\))h(suc)n(h)f(that)h(the)g(scaling)e(holds,)1132 1209 y Fv( )1186 1221 y Fo(j)1221 1209 y Fx(\()p FA(x)p Fx(\))47 b(=)f Fv(\036)1542 1221 y Fo(j)1577 1209 y Fx(\()p FA(x)p Fx(\))14 b Fv(\037)p Fx(\()1807 1153 y FA(x)p 1799 1190 64 4 v 1799 1266 a Fv(R)1874 1209 y Fx(\))46 b(=)g Fv(\036)2112 1221 y Fo(j)2147 1209 y Fx(\()2238 1153 y FA(x)p 2189 1190 147 4 v 2189 1266 a Fv(R)q(=)p Fx(2)2346 1209 y(\))14 b Fv(\037)p Fx(\()2493 1153 y FA(x)p 2486 1190 64 4 v 2486 1266 a Fv(R)2560 1209 y Fx(\))576 b(\(8.1\))386 1400 y(for)24 b Fv(R)g Fu(\025)e Fx(2)p Fv(:)47 b Fx(Since)25 b Fv(\036)1059 1412 y Fo(j)1119 1400 y Fx(and)f Fv(\037)h Fx(are)e(analytic)h(functions)h(in)f(supp)14 b Fv(\037)25 b Fx(w)n(e)f(can)g(apply)g(the)h(mean)386 1500 y(v)-5 b(alue)27 b(theorem,)h(with)g FA(x)23 b Fx(=)g(\()p FA(x)1379 1512 y Ft(1)1417 1500 y Fv(;)14 b FA(x)1504 1512 y Ft(2)1541 1500 y Fx(\))p Fv(;)868 1647 y Fu(j)p Fv( )945 1659 y Fo(j)980 1647 y Fx(\()p FA(x)1062 1659 y Ft(1)1100 1647 y Fv(;)g FA(x)1187 1659 y Ft(2)1225 1647 y Fx(\))32 b Fu(\000)g Fv( )1440 1659 y Fo(j)1475 1647 y Fx(\()p FA(x)1557 1612 y Fr(0)1557 1667 y Ft(1)1595 1647 y Fv(;)14 b FA(x)1682 1659 y Ft(2)1720 1647 y Fx(\))p Fu(j)46 b Fx(=)g Fu(j)p FA(x)2005 1659 y Ft(1)2061 1647 y Fu(\000)18 b FA(x)2194 1612 y Fr(0)2194 1667 y Ft(1)2231 1647 y Fu(j)24 b(j)p Fx(\()p Fu(r)2402 1659 y Fm(x)2442 1667 y Fk(1)2478 1647 y Fv( )2532 1659 y Fo(j)2567 1647 y Fx(\)\()p Fn(\030)t Fv(;)14 b FA(x)2764 1659 y Ft(2)2801 1647 y Fx(\))p Fu(j)312 b Fx(\(8.2\))386 1794 y(with)30 b Fn(\030)j Fx(some)c(v)-5 b(alue)29 b(on)g(the)i(line)e(b)r(et)n(w)n (een)h FA(x)1872 1806 y Ft(1)1939 1794 y Fx(and)g FA(x)2153 1764 y Fr(0)2153 1814 y Ft(1)2190 1794 y Fv(:)57 b Fx(Since)30 b(from)f(\(3.5\),)g Fv(\037)2962 1764 y Fr(0)3012 1794 y Fx(=)d Fu(\000)p Fv(\037)3220 1764 y Fr(0)3220 1814 y Ft(0)3283 1794 y Fu(2)386 1893 y(S)6 b Fx(\()p Fp(R)529 1863 y Ft(6)572 1893 y Fx(\))33 b(is)g(a)g(b)r(ounded)g(function,)g(as) g(is)f Fv(\036)1743 1863 y Fr(0)1743 1915 y Fo(j)1844 1893 y Fx(\(b)r(ecause)h Fv(\036)2238 1905 y Fo(j)2305 1893 y Fu(2)f Fv(C)2457 1863 y Fr(1)2528 1893 y Fx(\()p Fp(R)2614 1863 y Ft(6)2657 1893 y Fx(\))i(is)e(de\014ned)i(on)e(the)386 1997 y(compact)26 b(unit)i(sphere)e(and)g(is)h(homogeneous)e(of)i (degree)e(zero)h(outside)h(the)g(unit)g(ball\),)g(w)n(e)386 2097 y(can)g(estimate)386 2273 y Fu(j)423 2156 y Fs(\022)484 2273 y Fu(r)553 2285 y Fm(x)593 2293 y Fk(1)643 2273 y Fv(\036)692 2285 y Fo(j)728 2273 y Fx(\()818 2217 y FA(x)p 770 2254 147 4 v 770 2330 a Fv(R)q(=)p Fx(2)927 2273 y(\))c Fv(\037)p Fx(\()1083 2217 y FA(x)p 1076 2254 64 4 v 1076 2330 a Fv(R)1150 2273 y Fx(\))1168 2156 y Fs(\023)1229 2273 y Fx(\()p Fn(\030)s Fv(;)14 b FA(x)1393 2285 y Ft(2)1431 2273 y Fx(\))p Fu(j)46 b Fx(=)g Fu(j)p Fv(\036)1715 2285 y Fo(j)1751 2273 y Fu(j)1808 2217 y Fx(1)p 1797 2254 V 1797 2330 a Fv(R)1894 2273 y Fu(j)p Fv(\037)1969 2238 y Fr(0)1992 2273 y Fx(\()2044 2217 y Fn(\030)p 2034 2254 V 2034 2330 a Fv(R)2108 2273 y(;)2155 2217 y FA(x)2205 2229 y Ft(2)p 2155 2254 88 4 v 2167 2330 a Fv(R)2253 2273 y Fx(\))p Fu(j)30 b Fx(+)f Fu(j)p Fv(\037)p Fu(j)2606 2217 y Fx(1)p 2554 2254 147 4 v 2554 2330 a Fv(R)q(=)p Fx(2)2734 2273 y Fu(j)p Fv(\036)2806 2238 y Fr(0)2806 2293 y Fo(j)2841 2273 y Fx(\()p Fn(\030)s Fv(;)14 b FA(x)3005 2285 y Ft(2)3043 2273 y Fx(\))p Fu(j)46 b(\024)3256 2217 y Fv(c)3292 2229 y Ft(0)p 3256 2254 74 4 v 3261 2330 a Fv(R)3168 2422 y Fx(\(8.3\))386 2522 y(\(the)23 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y(desired)i(result)g(\(8.5\).)575 3779 y(It)h(remains)g(to)g(sho)n(w)f(the)h(estimate)h(\(8.6\))e(for)h(the)h (k)n(ernel)e(of)h(our)f(op)r(erators.)g(Let)h(us)386 3878 y(\014rst)f(consider)g(the)h(limiting)g(case)f Fv(m)c Fx(=)f(0)27 b(as)g(this)h(is)g(v)n(ery)e(simple.)575 3978 y(F)-7 b(or)28 b Fv(m)c Fx(=)g(0)p Fv(;)62 b Fn(\033)1098 3948 y Ft(\(1\))1187 3978 y FA(p)1240 3990 y Ft(1)1277 3978 y Fv(g)41 b Fx(=)1484 3945 y Ft(1)p 1457 3959 88 4 v 1457 3968 a Fr(p)p 1512 3968 34 3 v 48 x Ft(2)1555 3978 y Fn(\033)1615 3948 y Ft(\(1\))1714 3941 y Fm(p)1756 3949 y Fk(1)p 1714 3959 75 4 v 1718 4007 a Fo(p)1752 4015 y Fk(1)1798 3978 y Fv(;)53 b Fx(while)28 b Fv(A)h Fx(is)g(a)e(constan)n(t)h(whic)n(h)g(need)h(not)f(b)r(e)386 4094 y(considered.)f(F)-7 b(rom)27 b(\(3.7\),)g(using)g(the)h(F)-7 b(ourier)27 b(transform)f(of)i(the)g(Coulom)n(b)f(p)r(oten)n(tial,)917 4260 y(\024)915 4282 y Fv(k)958 4301 y Fg(\033)1005 4285 y Fk(\(1\))1082 4301 y Fm(p)1124 4309 y Fk(1)1157 4301 y Fo(g)1195 4282 y Fx(\()p FA(x)1277 4294 y Ft(1)1315 4282 y Fv(;)14 b FA(x)1402 4248 y Fr(0)1402 4303 y Ft(1)1440 4282 y Fx(\))46 b(=)g Fu(\000)p Fv(i)p Fn(\033)1782 4248 y Ft(\(1\))1871 4282 y Fu(r)1940 4294 y Fm(x)1980 4302 y Fk(1)2084 4226 y Fx(1)p 2049 4263 111 4 v 2049 4280 a Fu(p)p 2119 4280 42 4 v 2119 4348 a Fx(2)2254 4226 y(4)p Fv(\031)p 2203 4263 194 4 v 2203 4339 a Fx(\(2)p Fv(\031)s Fx(\))2359 4315 y Ft(3)2599 4226 y Fx(1)p 2440 4263 361 4 v 2440 4339 a Fu(j)p FA(x)2513 4351 y Ft(1)2569 4339 y Fu(\000)18 b FA(x)2702 4310 y Fr(0)2702 4361 y Ft(1)2740 4339 y Fu(j)2763 4315 y Ft(2)1485 4564 y Fx(=)1616 4508 y Fv(i)p Fn(\033)1704 4478 y Ft(\(1\))p 1605 4545 199 4 v 1605 4561 a Fu(p)p 1675 4561 42 4 v 1675 4630 a Fx(2)o Fv(\031)1766 4606 y Ft(2)1888 4508 y FA(x)1938 4520 y Ft(1)1994 4508 y Fu(\000)g FA(x)2127 4478 y Fr(0)2127 4528 y Ft(1)p 1847 4545 361 4 v 1847 4621 a Fu(j)p FA(x)1920 4633 y Ft(1)1976 4621 y Fu(\000)g FA(x)2109 4592 y Fr(0)2109 4643 y Ft(1)2146 4621 y Fu(j)2169 4597 y Ft(4)2217 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1766 986 a Fv(p)1808 958 y Ft(2)1808 1009 y(1)1863 986 y Fx(+)18 b(1)g(+)2089 914 y Fs(p)p 2172 914 222 4 v 72 x Fv(p)2214 958 y Ft(2)2214 1009 y(1)2270 986 y Fx(+)g(1)2404 775 y Fs(!)2480 770 y Fk(1)p 2480 779 29 3 v 2480 812 a(2)956 1279 y Fv(A)p Fx(\()p Fv(p)1092 1291 y Ft(1)1130 1279 y Fx(\))46 b(=)1363 1223 y(1)p 1329 1260 111 4 v 1329 1277 a Fu(p)p 1398 1277 42 4 v 69 x Fx(2)1463 1138 y Fs( )1529 1279 y Fx(1)32 b(+)1841 1223 y(1)p 1710 1260 306 4 v 1710 1277 a Fs(p)p 1793 1277 222 4 v 72 x Fv(p)1835 1321 y Ft(2)1835 1371 y(1)1890 1349 y Fx(+)18 b(1)2025 1138 y Fs(!)2100 1132 y Fk(1)p 2100 1141 29 3 v 2100 1175 a(2)2189 1279 y Fx(=)2344 1223 y(1)p 2309 1260 111 4 v 2309 1277 a Fu(p)p 2379 1277 42 4 v 2379 1346 a Fx(2)2462 1279 y(+)35 b(~)-45 b Fv(g)s Fx(\()p Fv(p)2676 1291 y Ft(1)2713 1279 y Fx(\))p Fv(;)359 b Fx(\(8.12\))1048 1561 y(~)-45 b Fv(g)s Fx(\()p Fv(p)1162 1573 y Ft(1)1199 1561 y Fx(\))23 b(:=)1433 1504 y(1)p 1398 1542 111 4 v 1398 1558 a Fu(p)p 1467 1558 42 4 v 69 x 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b(the)g(cuts)f(are)g(c)n(hosen)386 2129 y(from)35 b Fv(i)h Fx(to)f Fu(1)h Fx(and)g(from)f Fu(\0001)g Fx(to)h Fu(\000)p Fv(i)p Fx(,)f(resp)r(ectiv)n(ely)-7 b(.)35 b(According)g(to)g(a)h(corollary)d(of)i(the)386 2229 y(P)n(aley-Wiener)23 b(theorem)i([17,)g(Thm)h(IX.14],)f(for)30 b(~)-47 b Fv(x)24 b Fx(:=)e Fu(j)p FA(x)2225 2199 y Fr(0)2225 2249 y Ft(1)2277 2229 y Fu(\000)14 b FA(x)2406 2241 y Ft(1)2444 2229 y Fu(j)37 b Fv(>)22 b Fx(0)p Fv(;)49 b Fx(it)25 b(follo)n(ws)g(that)3259 2207 y(\024)3257 2229 y Fv(k)3302 2241 y Ft(~)-35 b Fo(g)386 2335 y Fx(and)556 2313 y(\024)554 2335 y Fv(k)597 2347 y Fo(g)670 2335 y Fx(\(if)35 b(de\014ned)f(with)h(con)n(v)n(ergence)c(generating)i (factors\))g(are)g(b)r(ounded)i(con)n(tin)n(uous)386 2434 y(functions)24 b(deca)n(ying)e(exp)r(onen)n(tially)h(at)28 b(~)-47 b Fv(x)24 b Fu(!)f(1)p Fv(;)g Fx(i.e.)h(for)f(an)n(y)g Fv(b)f(<)h Fx(1)g(and)g(an)n(y)g Fv(\016)j(>)d Fx(0,)g(there)386 2534 y(exists)k(a)g(constan)n(t)g Fv(c)1055 2546 y Ft(1)1115 2534 y Fv(>)c 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a(p)1969 4072 y Ft(1)2039 4003 y Fx([\()p Fv(\017p)2170 4015 y Ft(1)2207 4003 y Fx(\))2239 3969 y Ft(2)2300 4003 y Fv(e)2339 3969 y Fr(\000)p Fo(\017p)2453 3977 y Fk(1)2489 4003 y Fx(])2512 3886 y Fs(\033)2574 3883 y(\014)2574 3933 y(\014)2574 3982 y(\014)2574 4032 y(\014)2648 4003 y Fu(\024)2793 3947 y Fv(c)p 2768 3984 85 4 v 2773 4060 a Fx(~)e Fv(x)2815 4036 y Ft(3)3127 4003 y Fx(\(8.15\))386 4191 y(b)r(ecause)29 b(the)h(b)r(oundary)f(terms)g(v)-5 b(anish,)29 b Fu(j)14 b Fx(sin)g Fv(p)1925 4203 y Ft(1)1967 4191 y Fx(~)-47 b Fv(x)q Fu(j)26 b(\024)g Fx(1)j(and)g(the)h(term)g(in)f(curly)g(brac)n (k)n(ets)386 4291 y(is)e(b)r(ounded)h(b)n(y)1036 4258 y Fo(c)p 934 4272 235 4 v 934 4319 a Ft(\()p Fo(p)994 4327 y Fk(1)1026 4319 y Ft(+1\))1136 4303 y Fk(2)1206 4291 y Fx(for)f(all)h Fv(\017)p Fx(,)f(whic)n(h)h(is)f(an)g(in)n (tegrable)g(function.)575 4416 y(The)g(con)n(tin)n(uit)n(y)f(of)1232 4394 y(\024)1229 4416 y Fv(k)1274 4428 y Ft(~)-35 b Fo(g)1338 4416 y Fx(at)32 b(~)-47 b Fv(x)23 b(>)g Fx(0)j(together)g(with)h (\(8.13\))f(th)n(us)h(ensures)f(the)h(estimate)386 4522 y Fu(j)411 4500 y Fx(\024)409 4522 y Fv(k)454 4534 y Ft(~)-35 b Fo(g)491 4522 y Fx(\()5 b(~)-47 b Fv(x)q Fx(\))p Fu(j)37 b(\024)794 4490 y Fo(c)p 774 4504 71 4 v 778 4551 a Ft(~)-37 b Fo(x)812 4534 y Fk(3)854 4522 y Fv(:)575 4633 y Fx(F)-7 b(or)24 b(the)g(second)g(k)n(ernel,)1392 4611 y(\024)1389 4633 y Fv(k)1432 4652 y Fg(\033)1480 4636 y Fk(\(1\))1557 4652 y Fm(p)1599 4660 y Fk(1)1631 4652 y Fo(g)1670 4633 y Fx(\()5 b(~)-47 b Fv(x)p Fx(\))p Fv(;)25 b Fx(w)n(e)f(need)g(more)g(careful)g(estimates)g(due)g(to)g (the)386 4746 y(presence)h(of)h(the)g(deriv)-5 b(ativ)n(e)25 b Fn(\033)1390 4716 y Ft(\(1\))1479 4746 y Fu(r)1551 4758 y Ft(~)1548 4759 y Fm(x)1618 4746 y Fx(in)h(\(3.7\).)g(First)g(w)n (e)f(pro)n(v)n(e)f(analyticit)n(y)i(of)f(the)i(k)n(ernel)p eop %%Page: 19 19 19 18 bop 1606 259 a Ft(HVZ)30 b(THEOREM)1153 b(19)388 437 y Fx(\024)386 459 y Fv(k)429 471 y Fo(g)468 459 y Fv(:)69 b Fx(W)-7 b(e)34 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Fx(\()p Fv(y)s Fx(\))24 b Fv(e)2538 743 y Fo(i\017y)2651 777 y Fv(dy)2738 660 y Fs(\023)1097 1102 y Fx(~)1079 1124 y Fv(f)8 b Fx(\()p Fv(y)s Fx(\))24 b(:=)46 b Fv(y)16 b(e)1490 1089 y Fr(\000)t Ft(~)-37 b Fo(x)o(y)1687 1068 y Fx(1)p 1652 1105 111 4 v 1652 1121 a Fu(p)p 1721 1121 42 4 v 69 x Fx(2)1810 982 y Fs( )2181 1068 y Fx(1)p 1885 1105 633 4 v 1885 1194 a(1)18 b Fu(\000)g Fv(y)2072 1170 y Ft(2)2128 1194 y Fx(+)2211 1121 y Fs(p)p 2294 1121 224 4 v 73 x Fx(1)g Fu(\000)g Fv(y)2481 1170 y Ft(2)2528 982 y Fs(!)2603 977 y Fk(1)p 2603 986 29 3 v 2603 1019 a(2)3127 1124 y Fx(\(8.16\))386 1335 y(Using)33 b(analyticit)n(y)g(of)h(the)f(in)n(tegrands)g(outside)g(the)h(cuts)g (as)f(w)n(ell)g(as)g(their)g(exp)r(onen)n(tial)386 1434 y(deca)n(y)e(as)g(Re\()p Fv(y)s Fx(\))g Fu(!)f(1)p Fx(,)i(w)n(e)g(can)f (use)h(Cauc)n(h)n(y's)f(in)n(tegral)g(theorem)g(to)h(deform)g(the)g(in) n(te-)386 1534 y(gration)c(paths)i(to)g(the)g(real)f(axis.)g(The)h (resulting)g(in)n(tegrals)e(are)h(\014nite)i(for)e(all)35 b(~)-47 b Fv(x)27 b(>)f Fx(0)k(and)386 1633 y Fv(\017)23 b Fx(=)f(0)27 b(since)h(the)g(singularit)n(y)e(at)i Fv(y)d Fx(=)e(1)k(is)h(in)n(tegrable.)e(Th)n(us)h(\(8.16\))g(giv)n(es)1090 1839 y(1)p 1032 1876 159 4 v 1037 1961 a(~)-47 b Fv(x)1080 1892 y Fu(p)p 1149 1892 42 4 v 69 x Fx(2)1214 1782 y Fs(Z)1297 1802 y Fr(1)1260 1970 y Ft(1)1381 1895 y Fv(y)s(dy)26 b(e)1574 1861 y Fr(\000)t Ft(~)-37 b Fo(x)o(y)1726 1895 y Fx(Re)1838 1753 y Fs( )2223 1839 y Fx(1)p 1914 1876 661 4 v 1914 1965 a Fv(y)1958 1941 y Ft(2)2013 1965 y Fu(\000)18 b Fx(1)g(+)g Fv(i)2268 1892 y Fs(p)p 2351 1892 224 4 v 73 x Fv(y)2395 1941 y Ft(2)2450 1965 y Fu(\000)g Fx(1)2584 1753 y Fs(!)2660 1748 y Fk(1)p 2660 1757 29 3 v 2660 1790 a(2)3127 1895 y Fx(\(8.17\))386 2123 y(with)24 b(p)r(ositiv)n(e)f(c)n(hoice)g(of)g(the)h(real)f(part.)g(Its)g(deriv)-5 b(ativ)n(e)23 b(with)h(resp)r(ect)f(to)29 b(~)-47 b Fv(x)24 b Fx(also)e(con)n(v)n(erges)386 2223 y(absolutely)27 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2635 y Fx(as)32 b(~)-47 b Fv(x)24 b Fu(!)f(1)273 b Fx(\(8.18\))386 2835 y(with)28 b(a)f(constan)n(t)g Fv(z)1018 2847 y Ft(0)1055 2835 y Fx(.)575 2935 y(F)-7 b(or)35 b(the)i(estimate)f(of)1330 2913 y(\024)1328 2935 y Fv(k)1371 2947 y Fo(g)1446 2935 y Fx(near)41 b(~)-47 b Fv(x)38 b Fx(=)e(0)p Fv(;)g Fx(w)n(e)g(separate)f(the)h('Coulom)n (bic')g(tail)g(from)386 3037 y Fv(g)s(;)60 b(g)s Fx(\()p Fv(p)p Fx(\))22 b(=)825 3004 y Ft(1)p 781 3018 122 4 v 781 3027 a Fr(p)p 836 3027 34 3 v 48 x Ft(2)p Fo(p)931 3037 y Fx(+)32 b(\()p Fv(g)s Fx(\()p Fv(p)p Fx(\))19 b Fu(\000)1365 3004 y Ft(1)p 1321 3018 122 4 v 1321 3027 a Fr(p)p 1375 3027 34 3 v 1375 3075 a Ft(2)q Fo(p)1453 3037 y Fx(\))51 b(and)27 b(obtain)g(with)i(\(8.11\))d(and)i(\(3.7\)) 1182 3251 y(\024)1179 3273 y Fv(k)1222 3285 y Fo(g)1261 3273 y Fx(\()5 b(~)-47 b Fv(x)q Fx(\))46 b(=)1639 3217 y(1)p 1540 3254 240 4 v 1540 3339 a(2)1582 3271 y Fu(p)p 1651 3271 42 4 v 68 x Fx(2)o Fv(\031)1742 3315 y Ft(2)1827 3156 y Fs(\022)1919 3217 y Fx(1)p 1898 3254 85 4 v 1903 3330 a(~)-47 b Fv(x)1945 3306 y Ft(2)2034 3273 y Fu(\000)47 b Fx(lim)2140 3325 y Fo(\017)p Fr(!)p Ft(0)2304 3273 y Fv(I)2340 3285 y Fo(\017)2372 3273 y Fx(\()5 b(~)-47 b Fv(x)q Fx(\))2484 3156 y Fs(\023)3127 3273 y Fx(\(8.19\))560 3578 y Fv(I)596 3590 y Fo(\017)629 3578 y Fx(\()5 b(~)-47 b Fv(x)q Fx(\))23 b(:=)911 3522 y(1)p 908 3559 48 4 v 913 3635 a(~)-47 b Fv(x)979 3465 y Fs(Z)1062 3486 y Fr(1)1025 3654 y Ft(0)1146 3578 y Fv(dp)1231 3590 y Ft(1)1305 3578 y Fx(sin)14 b Fv(p)1463 3590 y Ft(1)1505 3578 y Fx(~)-47 b Fv(x)1967 3522 y Fx(\(1)19 b(+)2143 3450 y Fs(p)p 2226 3450 222 4 v 72 x Fv(p)2268 3493 y Ft(2)2268 3544 y(1)2323 3522 y Fx(+)f(1\))2490 3469 y Fk(1)p 2490 3478 29 3 v 2490 3512 a(2)p 1580 3559 1339 4 v 1580 3648 a Fx(\()p Fv(p)1654 3619 y Ft(2)1654 3670 y(1)1710 3648 y Fx(+)g(1\))1877 3597 y Fk(1)p 1877 3606 29 3 v 1877 3639 a(4)1933 3648 y Fx([)p Fv(p)1998 3660 y Ft(1)2054 3648 y Fx(+)g(\()p Fv(p)2211 3619 y Ft(2)2211 3670 y(1)2266 3648 y 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Fs(\032)1743 4087 y Fx(\()1824 4031 y Fv(d)p 1785 4068 123 4 v 1785 4144 a(dp)1870 4156 y Ft(1)1931 4087 y Fx([)p Fu(\001)14 b(\001)g(\001)g Fx(]\))g Fv(e)2173 4052 y Fr(\000)p Fo(\017p)2287 4060 y Fk(1)2355 4087 y Fu(\000)2462 4031 y Fx([)p Fu(\001)g(\001)g(\001)g Fx(])p 2462 4068 157 4 v 2501 4144 a Fv(p)2543 4156 y Ft(1)2643 4087 y Fx(\()p Fv(\017p)2751 4099 y Ft(1)2801 4087 y Fv(e)2840 4052 y Fr(\000)p Fo(\017p)2954 4060 y Fk(1)2990 4087 y Fx(\))3022 3970 y Fs(\033)3127 4087 y Fx(\(8.20\))386 4295 y(where)34 b(the)i(\014rst)e(term)h(comes)g (from)f(the)h(b)r(oundary)g(at)f Fv(p)2306 4307 y Ft(1)2379 4295 y Fx(=)h(0)f(and)h([)p Fu(\001)14 b(\001)g(\001)g Fx(])35 b(denotes)f(the)386 4394 y(fraction)j(in)h(\(8.19\).)f(The)h (in)n(tegral)e(is)i(\014nite)g(\(and)g(indep)r(enden)n(t)h(of)k(~)-48 b Fv(x)39 b Fx(for)j(~)-47 b Fv(x)40 b Fu(!)g Fx(0\))e(b)r(e-)386 4494 y(cause)d(the)i(curly)f(brac)n(k)n(et)f(can)h(b)r(e)g(estimated)h (b)n(y)2179 4460 y Fo(c)2209 4468 y Fk(0)p 2093 4475 235 4 v 2093 4523 a Ft(\(1+)p Fo(p)2237 4531 y Fk(1)2269 4523 y Ft(\))2295 4506 y Fk(2)2412 4494 y Fx(\(indep)r(enden)n(t)g(of)f Fv(\017)p Fx(\))h(whic)n(h)386 4619 y(is)f(in)n(tegrable.)e(Hence,)1175 4598 y(\024)1172 4619 y Fv(k)1215 4631 y Fo(g)1254 4619 y Fx(\()5 b(~)-47 b Fv(x)q Fx(\))37 b Fu(\030)1537 4587 y Ft(~)-35 b Fo(c)p 1515 4601 71 4 v 1519 4648 a Ft(~)e Fo(x)1553 4632 y Fk(2)1667 4619 y Fx(for)41 b(~)-47 b Fv(x)37 b Fu(!)g Fx(0)e(with)h(some)g(constan)n(t)h(~)-44 b Fv(c:)36 b Fx(P)n(erforming)386 4745 y(the)d(deriv)-5 b(ativ)n(e)32 b(inheren)n(t)h(in)g Fn(\033)1413 4715 y Ft(\(1\))1502 4745 y FA(p)1555 4757 y Ft(1)1593 4745 y Fx(,)g(w)n(e)f(\014nd)1950 4723 y(\024)1948 4745 y Fv(k)1991 4764 y Fg(\033)2038 4748 y Fk(\(1)2093 4764 y Ft(\))p Fm(p)2161 4772 y Fk(1)2193 4764 y Fo(g)2231 4745 y Fx(\()5 b(~)-47 b Fv(x)q Fx(\))32 b Fu(\030)g Fx(2)p Fv(i)r Fx(~)-44 b Fv(c)2602 4713 y Fg(\033)2649 4688 y Fk(\(1\))2729 4712 y Ft(~)2726 4713 y Fm(x)p 2602 4727 165 4 v 2653 4774 a Ft(~)-37 b Fo(x)2687 4757 y Fk(4)2840 4745 y Fx(\()5 b(~)-47 b Fv(x)32 b Fu(!)g Fx(0\))65 b(and)p eop %%Page: 20 20 20 19 bop 386 259 a Ft(20)904 b(D.)22 b(H.)g(JAKUBASSA-AMUNDSEN)386 465 y Fu(\030)29 b Fv(iz)548 477 y Ft(0)608 432 y Fg(\033)655 407 y Fk(\(1\))736 431 y Ft(~)733 432 y Fm(x)p 608 446 165 4 v 660 493 a Ft(~)-38 b Fo(x)693 477 y Fk(4)796 465 y Fx(\()5 b(~)-47 b Fv(x)875 434 y Ft(5)p Fo(=)p Ft(4)1002 465 y Fx(+)1097 432 y Ft(7)p 1097 446 34 4 v 1097 493 a(4)1146 465 y Fx(~)f Fv(x)1187 434 y Ft(1)p Fo(=)p Ft(4)1292 465 y Fx(\))14 b Fv(e)1377 434 y Fr(\000)t Ft(~)-37 b Fo(x)1532 465 y Fx(for)36 b(~)-47 b Fv(x)30 b Fu(!)g(1)p Fv(:)i Fx(F)-7 b(rom)31 b(the)h(analyticit)n(y)f(of)2877 443 y(\024)2875 465 y Fv(k)2918 477 y Fo(g)2989 465 y Fx(w)n(e)g(\014nally)386 564 y(obtain)1224 665 y Fu(j)1249 643 y Fx(\024)1247 665 y Fv(k)1290 684 y Fg(\033)1338 668 y Fk(\(1\))1415 684 y Fm(p)1457 692 y Fk(1)1489 684 y Fo(g)1528 665 y Fx(\()5 b(~)-47 b Fv(x)q Fx(\))p Fu(j)46 b(\024)1854 609 y Fv(c)p 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Fv(p)1579 1213 y Fo(k)1619 1201 y Fx(\))p Fv(;)60 b(k)26 b Fx(=)d(1)p Fv(;)14 b Fx(2\))p Fv(:)860 1375 y(M)941 1387 y Ft(1)1000 1375 y Fx(:=)46 b([)p Fv(A)1219 1387 y Ft(1)1257 1375 y Fv(;)14 b( )1348 1387 y Fo(j)1383 1375 y Fx(])1580 1319 y(1)p 1439 1356 323 4 v 1439 1432 a Fu(j)p FA(x)1512 1444 y Ft(1)1568 1432 y Fu(\000)k FA(x)1701 1444 y Ft(2)1739 1432 y Fu(j)1790 1375 y(\001)h(j)p FA(x)1905 1387 y Ft(1)1961 1375 y Fu(\000)f FA(x)2094 1387 y Ft(2)2131 1375 y Fu(j)24 b Fv(A)2240 1387 y Ft(2)2451 1319 y Fx(1)p 2310 1356 V 2310 1432 a Fu(j)p FA(x)2383 1444 y Ft(1)2439 1432 y Fu(\000)18 b FA(x)2572 1444 y Ft(2)2610 1432 y Fu(j)2666 1375 y Fv(A)2728 1387 y Ft(1)2765 1375 y Fv(A)2827 1387 y Ft(2)1189 1607 y Fx(+)k Fv(A)1338 1619 y Ft(1)1399 1607 y Fx([)p Fv(A)1484 1619 y Ft(2)1521 1607 y Fv(;)14 b( )1612 1619 y Fo(j)1647 1607 y Fx(])1844 1551 y(1)p 1704 1588 V 1704 1664 a Fu(j)p FA(x)1777 1676 y Ft(1)1833 1664 y Fu(\000)k FA(x)1966 1676 y Ft(2)2003 1664 y Fu(j)2059 1607 y Fv(A)2121 1619 y Ft(1)2159 1607 y Fv(A)2221 1619 y Ft(2)2300 1607 y Fu(\000)41 b Fv(h:c:)654 1822 y(M)735 1834 y Ft(2)795 1822 y Fx(:=)23 b([)p Fv(A)991 1834 y Ft(1)1028 1822 y Fv(;)14 b( )1119 1834 y Fo(j)1154 1822 y Fx(])1351 1766 y(1)p 1211 1803 V 1211 1879 a Fu(j)p FA(x)1284 1891 y Ft(1)1340 1879 y Fu(\000)k FA(x)1473 1891 y Ft(2)1510 1879 y Fu(j)1562 1822 y(\001)g(j)p FA(x)1676 1834 y Ft(1)1732 1822 y Fu(\000)g FA(x)1865 1834 y Ft(2)1903 1822 y Fu(j)23 b Fv(g)1989 1834 y Ft(2)2026 1822 y Fn(\033)2086 1788 y Ft(\(2\))2175 1822 y FA(p)2228 1834 y Ft(2)2439 1766 y Fx(1)p 2299 1803 V 2299 1879 a Fu(j)p FA(x)2372 1891 y Ft(1)2428 1879 y Fu(\000)18 b FA(x)2561 1891 y Ft(2)2598 1879 y Fu(j)2654 1822 y Fv(A)2716 1834 y Ft(1)2754 1822 y Fv(g)2794 1834 y Ft(2)2831 1822 y Fn(\033)2891 1788 y Ft(\(2\))2980 1822 y FA(p)3033 1834 y Ft(2)972 2037 y Fx(+)23 b Fv(A)1122 2049 y Ft(1)1182 2037 y Fx([)p Fv(g)1245 2049 y Ft(2)1282 2037 y Fn(\033)1342 2002 y Ft(\(2\))1431 2037 y FA(p)1484 2049 y Ft(2)1521 2037 y Fv(;)14 b( )1612 2049 y Fo(j)1647 2037 y Fx(])1844 1980 y(1)p 1704 2017 V 1704 2094 a Fu(j)p FA(x)1777 2106 y Ft(1)1833 2094 y Fu(\000)k FA(x)1966 2106 y Ft(2)2003 2094 y Fu(j)2059 2037 y Fv(A)2121 2049 y Ft(1)2159 2037 y Fv(g)2199 2049 y Ft(2)2236 2037 y Fn(\033)2296 2002 y Ft(\(2\))2385 2037 y FA(p)2438 2049 y Ft(2)2517 2037 y Fu(\000)41 b Fv(h:c:)374 b Fx(\(8.22\))438 2251 y Fv(M)519 2263 y Ft(4)578 2251 y Fx(:=)46 b([)p Fv(g)775 2263 y Ft(1)812 2251 y Fn(\033)872 2217 y Ft(\(1\))961 2251 y FA(p)1014 2263 y Ft(1)1052 2251 y Fv(;)14 b( )1143 2263 y Fo(j)1178 2251 y Fx(])1374 2195 y(1)p 1234 2232 V 1234 2308 a Fu(j)p FA(x)1307 2320 y Ft(1)1363 2308 y Fu(\000)k FA(x)1496 2320 y Ft(2)1533 2308 y Fu(j)1585 2251 y(\001)g(j)p FA(x)1699 2263 y Ft(1)1756 2251 y Fu(\000)g FA(x)1889 2263 y Ft(2)1926 2251 y Fu(j)23 b Fv(g)2012 2263 y Ft(2)2049 2251 y Fn(\033)2109 2217 y Ft(\(2\))2198 2251 y FA(p)2251 2263 y Ft(2)2462 2195 y Fx(1)p 2322 2232 V 2322 2308 a Fu(j)p FA(x)2395 2320 y Ft(1)2451 2308 y Fu(\000)18 b FA(x)2584 2320 y Ft(2)2621 2308 y Fu(j)2654 2251 y Fv(g)2694 2263 y Ft(1)2731 2251 y Fn(\033)2791 2217 y Ft(\(1\))2880 2251 y FA(p)2933 2263 y Ft(1)2971 2251 y Fv(g)3011 2263 y Ft(2)3048 2251 y Fn(\033)3108 2217 y Ft(\(2\))3197 2251 y FA(p)3250 2263 y Ft(2)764 2466 y Fx(+)23 b Fv(g)892 2478 y Ft(1)929 2466 y Fn(\033)989 2432 y Ft(\(1\))1078 2466 y FA(p)1131 2478 y Ft(1)1191 2466 y Fx([)p Fv(g)1254 2478 y Ft(2)1291 2466 y Fn(\033)1351 2432 y Ft(\(2\))1440 2466 y FA(p)1493 2478 y Ft(2)1531 2466 y Fv(;)14 b( )1622 2478 y Fo(j)1657 2466 y Fx(])1853 2410 y(1)p 1713 2447 V 1713 2523 a Fu(j)p FA(x)1786 2535 y Ft(1)1842 2523 y Fu(\000)k FA(x)1975 2535 y Ft(2)2013 2523 y Fu(j)2069 2466 y Fv(g)2109 2478 y Ft(1)2145 2466 y Fn(\033)2205 2432 y Ft(\(1\))2295 2466 y FA(p)2348 2478 y Ft(1)2385 2466 y Fv(g)2425 2478 y Ft(2)2462 2466 y Fn(\033)2522 2432 y Ft(\(2\))2611 2466 y FA(p)2664 2478 y Ft(2)2733 2466 y Fu(\000)32 b Fv(h:c:)386 2628 y Fx(where)27 b Fv(h:c:)h Fx(denotes)f(the)h(hermitean)g(conjugate,)f (and)h Fv(M)2262 2640 y Ft(3)2326 2628 y Fx(results)f(from)h Fv(M)2868 2640 y Ft(2)2932 2628 y Fx(up)r(on)g(in)n(ter-)386 2727 y(c)n(hanging)e FA(p)787 2739 y Ft(1)824 2727 y Fv(;)14 b Fn(\033)921 2697 y Ft(\(1\))1038 2727 y Fx(with)28 b FA(p)1280 2739 y Ft(2)1318 2727 y Fv(;)14 b Fn(\033)1414 2697 y Ft(\(2\))1531 2727 y Fx(and)28 b(therefore)e(need)i(not)g(b)r(e) g(considered)e(separately)-7 b(.)575 2827 y(In)39 b(addition)g(to)g (estimate)g(the)g(comm)n(utators)e([)p Fu(O)r Fv(;)14 b( )2334 2839 y Fo(j)2370 2827 y Fx(])2546 2794 y Ft(1)p 2445 2808 236 4 v 2445 2856 a Fr(j)p Fm(x)2505 2864 y Fk(1)2537 2856 y Fr(\000)p Fm(x)2629 2864 y Fk(2)2661 2856 y Fr(j)2771 2827 y Fx(\(with)40 b(the)f(same)386 2941 y(op)r(erators)32 b Fu(O)37 b Fx(as)d(in)h Fv(a)p Fx(\))f(except)h(for)f(a)g(p)r(ossible)g(particle)g(exc)n(hange\),)f(w) n(e)h(ha)n(v)n(e)f(to)h(pro)n(v)n(e)386 3040 y(b)r(oundedness)d(of)f Fu(j)p FA(x)1047 3052 y Ft(1)1105 3040 y Fu(\000)20 b FA(x)1240 3052 y Ft(2)1278 3040 y Fu(j)14 b(O)1508 3008 y Ft(1)p 1407 3022 V 1407 3069 a Fr(j)p Fm(x)1467 3077 y Fk(1)1499 3069 y Fr(\000)p Fm(x)1591 3077 y Fk(2)1623 3069 y Fr(j)1711 3040 y Fx(whic)n(h)31 b(app)r(ears)e(as)h(a)h(factor)f (in)g(the)i(\014rst)e(con)n(tri-)386 3150 y(bution)e(to)f Fv(M)831 3162 y Fo(k)872 3150 y Fv(;)60 b(k)26 b Fx(=)c(1)p Fv(;)14 b(:::;)g Fx(4)p Fv(:)575 3249 y Fx(F)-7 b(or)21 b(the)h(estimate)g(of)g(the)g(comm)n(utators)e(w)n(e)i(pro)r(ceed)f(as) g(in)h Fv(a)p Fx(\))g(except)g(for)f(the)i(c)n(hoice)386 3352 y Fv(f)9 b Fx(\()p FA(x)518 3364 y Ft(1)555 3352 y Fv(;)14 b FA(x)642 3364 y Ft(2)680 3352 y Fx(\))32 b(=)g Fu(j)p FA(x)914 3364 y Ft(1)973 3352 y Fu(\000)22 b FA(x)1110 3364 y Ft(2)1147 3352 y Fu(j)1170 3322 y Ft(3)p Fo(=)p Ft(2)1275 3352 y Fv(:)65 b Fx(With)34 b(the)f(inequalit)n (y)f(\(8.6\))h(for)f(the)h(k)n(ernel)f(of)h Fu(O)j Fx(and)c(the)386 3452 y(mean)f(v)-5 b(alue)31 b(theorem)g(\(8.2\))g(and)g(\(8.3\))g(w)n (e)g(obtain)g(for)f Fv(I)7 b(;)32 b Fx(de\014ned)f(as)g(the)h(in)n (tegral)e(o)n(v)n(er)386 3551 y(the)e(k)n(ernel)f(of)g([)p Fu(O)r Fv(;)14 b( )1050 3563 y Fo(j)1086 3551 y Fx(])1234 3519 y Ft(1)p 1132 3533 V 1132 3580 a Fr(j)p Fm(x)1192 3588 y Fk(1)1224 3580 y Fr(\000)p Fm(x)1316 3588 y Fk(2)1348 3580 y Fr(j)1429 3551 y Fx(where)27 b Fu(O)j Fx(dep)r(ends)e(e.g.)f(on) g FA(p)2406 3563 y Ft(2)2444 3551 y Fv(;)447 3757 y(I)7 b Fx(\()p FA(x)572 3769 y Ft(1)610 3757 y Fv(;)14 b FA(x)697 3769 y Ft(2)734 3757 y Fx(\))24 b(:=)923 3644 y Fs(Z)969 3832 y Fh(R)1016 3816 y Fk(3)1062 3757 y Fv(d)p FA(x)1155 3722 y Fr(0)1155 3777 y Ft(2)1216 3757 y Fu(j)1241 3735 y Fx(\024)1239 3757 y Fv(k)1282 3769 y Fr(O)1341 3757 y Fx(\()p FA(x)1423 3769 y Ft(2)1461 3757 y Fv(;)14 b FA(x)1548 3722 y Fr(0)1548 3777 y Ft(2)1585 3757 y Fx(\))p Fu(j)23 b(j)p Fv( )1740 3769 y Fo(j)1776 3757 y Fx(\()p FA(x)1858 3769 y Ft(1)1896 3757 y Fv(;)14 b FA(x)1983 3769 y Ft(2)2020 3757 y Fx(\))19 b Fu(\000)f Fv( )2208 3769 y Fo(j)2243 3757 y Fx(\()p FA(x)2325 3769 y Ft(1)2363 3757 y Fv(;)c FA(x)2450 3722 y Fr(0)2450 3777 y Ft(2)2487 3757 y Fx(\))p Fu(j)2716 3701 y Fx(1)p 2576 3738 323 4 v 2576 3814 a Fu(j)p FA(x)2649 3826 y Ft(1)2705 3814 y Fu(\000)k FA(x)2838 3785 y Fr(0)2838 3836 y Ft(2)2875 3814 y Fu(j)2942 3701 y Fv(f)9 b Fx(\()p FA(x)3074 3713 y Ft(1)3111 3701 y Fv(;)14 b FA(x)3198 3713 y Ft(2)3236 3701 y Fx(\))p 2942 3738 327 4 v 2942 3814 a Fv(f)9 b Fx(\()p FA(x)3074 3826 y Ft(1)3111 3814 y Fv(;)14 b FA(x)3198 3785 y Fr(0)3198 3836 y Ft(2)3236 3814 y Fx(\))691 4010 y Fu(\024)811 3954 y Fv(c)847 3966 y Ft(0)885 3954 y Fv(c)p 811 3991 110 4 v 834 4067 a(R)944 3897 y Fs(Z)990 4086 y Fh(R)1037 4069 y Fk(3)1083 4010 y Fv(d)p FA(x)1176 3976 y Fr(0)1176 4031 y Ft(2)1406 3954 y Fx(1)p 1247 3991 361 4 v 1247 4067 a Fu(j)p FA(x)1320 4079 y Ft(2)1376 4067 y Fu(\000)k FA(x)1509 4039 y Fr(0)1509 4090 y Ft(2)1546 4067 y Fu(j)1569 4043 y Ft(3)1640 4010 y Fu(j)p FA(x)1713 4022 y Ft(2)1769 4010 y Fu(\000)g FA(x)1902 3976 y Fr(0)1902 4031 y Ft(2)1940 4010 y Fu(j)2136 3954 y Fx(1)p 1996 3991 323 4 v 1996 4067 a Fu(j)p FA(x)2069 4079 y Ft(1)2125 4067 y Fu(\000)g FA(x)2258 4039 y Fr(0)2258 4090 y Ft(2)2295 4067 y Fu(j)2361 3954 y(j)p FA(x)2434 3966 y Ft(1)2491 3954 y Fu(\000)g FA(x)2624 3966 y Ft(2)2661 3954 y Fu(j)2684 3924 y Ft(3)p Fo(=)p Ft(2)p 2361 3991 428 4 v 2361 4069 a Fu(j)p FA(x)2434 4081 y Ft(1)2491 4069 y Fu(\000)g FA(x)2624 4040 y Fr(0)2624 4091 y Ft(2)2661 4069 y Fu(j)2684 4045 y Ft(3)p Fo(=)p Ft(2)2799 4010 y Fv(:)305 b Fx(\(8.23\))386 4180 y(W)-7 b(e)28 b(abbreviate)e FA(x)987 4192 y Ft(0)1047 4180 y Fx(:=)d FA(x)1208 4192 y Ft(2)1263 4180 y Fu(\000)18 b FA(x)1396 4192 y Ft(1)1461 4180 y Fx(and)27 b(substitute)h FA(x)2060 4150 y Fr(0)2060 4201 y Ft(2)2115 4180 y Fu(\000)18 b FA(x)2248 4192 y Ft(1)2308 4180 y Fx(=)23 b Fv(x)2443 4192 y Ft(0)2481 4180 y FA(y)q Fv(:)28 b Fx(Then)f(the)h(second)f(line) 386 4280 y(in)h(\(8.23\))f(is)g(written)h(as)f(\()p FA(e)1275 4292 y Fo(x)1313 4300 y Fk(0)1372 4280 y Fx(:=)c FA(x)1533 4292 y Ft(0)1570 4280 y Fv(=x)1659 4292 y Ft(0)1697 4280 y Fx(\))1142 4457 y Fv(I)7 b Fx(\()p FA(x)1267 4469 y Ft(1)1305 4457 y Fv(;)14 b FA(x)1392 4469 y Ft(2)1429 4457 y Fx(\))46 b Fu(\024)1628 4401 y Fv(c)1664 4413 y Ft(0)1701 4401 y Fv(c)p 1628 4438 110 4 v 1651 4514 a(R)1761 4344 y Fs(Z)1807 4533 y Fh(R)1854 4516 y Fk(3)1900 4457 y Fv(d)p FA(y)2184 4401 y Fx(1)p 2027 4438 355 4 v 2027 4514 a Fu(j)p FA(e)2094 4526 y Fo(x)2132 4534 y Fk(0)2187 4514 y Fu(\000)18 b FA(y)q Fu(j)2344 4490 y Ft(2)2478 4401 y Fx(1)p 2425 4438 149 4 v 2425 4516 a Fv(y)2469 4492 y Ft(5)p Fo(=)p Ft(2)1210 4700 y Fx(=)1331 4644 y(2)p Fv(\031)s(c)1459 4656 y Ft(0)1496 4644 y Fv(c)p 1331 4681 201 4 v 1399 4757 a(R)1555 4587 y Fs(Z)1638 4608 y Fr(1)1602 4776 y Ft(0)1763 4644 y Fv(dy)p 1733 4681 149 4 v 1733 4759 a(y)1777 4735 y Ft(3)p Fo(=)p Ft(2)1928 4700 y Fx(ln)2011 4580 y Fs(\014)2011 4630 y(\014)2011 4679 y(\014)2011 4729 y(\014)2048 4644 y Fx(1)g(+)g Fv(y)p 2048 4681 187 4 v 2048 4757 a Fx(1)g Fu(\000)g Fv(y)2245 4580 y Fs(\014)2245 4630 y(\014)2245 4679 y(\014)2245 4729 y(\014)2319 4700 y Fu(\024)2439 4644 y Fv(C)p 2439 4681 66 4 v 2440 4757 a(R)3127 4700 y Fx(\(8.24\))p eop %%Page: 21 21 21 20 bop 1606 259 a Ft(HVZ)30 b(THEOREM)1153 b(21)386 459 y Fx(with)31 b(the)g(same)e(estimate)i(as)e(in)i(\(8.9\))f(ab)r(o)n (v)n(e.)f(Lik)n(ewise,)h(the)g(estimate)h(\(8.10\))e(holds)h(for)386 558 y(the)e(second)f(in)n(tegral)f Fv(J)o(:)575 658 y Fx(Finally)h(w)n(e)g(ha)n(v)n(e)f(to)h(sho)n(w)g(the)g(b)r(oundedness)g (of)h Fu(j)p FA(x)2274 670 y Ft(1)2329 658 y Fu(\000)17 b FA(x)2461 670 y Ft(2)2499 658 y Fu(j)d(O)2729 625 y Ft(1)p 2628 639 236 4 v 2628 687 a Fr(j)p Fm(x)2688 695 y Fk(1)2720 687 y Fr(\000)p Fm(x)2812 695 y Fk(2)2844 687 y Fr(j)2874 658 y Fv(:)50 b Fx(Since)27 b(com-)386 767 y(m)n(utators)f(help)i(to)g(regularize)d(in)n(tegrals,)i(w)n(e)g (decomp)r(ose)847 940 y Fu(j)p FA(x)920 952 y Ft(1)976 940 y Fu(\000)18 b FA(x)1109 952 y Ft(2)1147 940 y Fu(j)23 b(O)1435 883 y Fx(1)p 1295 921 323 4 v 1295 997 a Fu(j)p FA(x)1368 1009 y Ft(1)1424 997 y Fu(\000)18 b FA(x)1557 1009 y Ft(2)1594 997 y Fu(j)1674 940 y Fx(=)45 b Fu(O)f Fx(+)d([)14 b Fu(j)p FA(x)2110 952 y Ft(1)2166 940 y Fu(\000)k FA(x)2299 952 y Ft(2)2337 940 y Fu(j)p Fv(;)c Fu(O)r Fx(])2662 883 y(1)p 2521 921 V 2521 997 a Fu(j)p FA(x)2594 1009 y Ft(1)2651 997 y Fu(\000)k FA(x)2784 1009 y Ft(2)2821 997 y Fu(j)2854 940 y Fv(:)250 b Fx(\(8.25\))386 1122 y Fu(O)38 b Fx(is)d(b)r(ounded)g(so)g(w)n(e)f(can)h(concen)n (trate)f(on)h(the)g(second)g(term.)g(The)g(di\013erence)g(to)g(the)386 1221 y(comm)n(utators)26 b(o)r(ccurring)g(in)i(\(8.22\))f(is)h(the)g (replacemen)n(t)f(of)g Fv( )2430 1233 y Fo(j)2493 1221 y Fx(b)n(y)g Fu(j)p FA(x)2681 1233 y Ft(1)2737 1221 y Fu(\000)19 b FA(x)2871 1233 y Ft(2)2908 1221 y Fu(j)p Fv(:)51 b Fx(F)-7 b(rom)27 b(the)386 1321 y(mean)g(v)-5 b(alue)28 b(theorem)f(w)n(e)g(get)g(\(for)h Fu(O)i Fx(dep)r(ending)e (on)f FA(p)2228 1333 y Ft(2)2265 1321 y Fx(\))654 1509 y Fu(j)14 b(j)p FA(x)764 1521 y Ft(1)820 1509 y Fu(\000)k FA(x)953 1521 y Ft(2)991 1509 y Fu(j)32 b(\000)g(j)p FA(x)1216 1521 y Ft(1)1272 1509 y Fu(\000)18 b FA(x)1405 1475 y Fr(0)1405 1529 y Ft(2)1443 1509 y Fu(j)c(j)46 b Fx(=)1659 1388 y Fs(\014)1659 1438 y(\014)1659 1488 y(\014)1659 1538 y(\014)1687 1509 y Fx(\()p FA(x)1769 1521 y Ft(2)1825 1509 y Fu(\000)18 b FA(x)1958 1475 y Fr(0)1958 1529 y Ft(2)1996 1509 y Fx(\))2084 1453 y Fn(\030)k Fu(\000)c FA(x)2281 1465 y Ft(1)p 2061 1490 280 4 v 2061 1566 a Fu(j)p Fn(\030)k Fu(\000)c FA(x)2281 1578 y Ft(1)2318 1566 y Fu(j)2351 1388 y Fs(\014)2351 1438 y(\014)2351 1488 y(\014)2351 1538 y(\014)2425 1509 y Fu(\024)46 b(j)p FA(x)2609 1521 y Ft(2)2665 1509 y Fu(\000)18 b FA(x)2798 1475 y Fr(0)2798 1529 y Ft(2)2836 1509 y Fu(j)268 b Fx(\(8.26\))386 1697 y(with)23 b Fn(\030)j Fx(some)c(p)r(oin)n(t)h(b)r(et)n(w)n(een)g FA(x)1419 1709 y Ft(2)1479 1697 y Fx(and)g FA(x)1686 1667 y Fr(0)1686 1717 y Ft(2)1723 1697 y Fv(:)g Fx(This)g(means)f(that) i(the)f(earlier)e(estimates)i(\(8.24\))386 1796 y(and)32 b(\(8.10\))g(hold)h(except)g(for)f(the)h(factor)f Fv(c)1812 1808 y Ft(0)1849 1796 y Fv(=R)h Fx(from)g(\(8.3\))f(whic)n(h)h(is)f (not)h(presen)n(t)f(no)n(w.)386 1896 y(This)27 b(pro)n(v)n(es)f(the)i (desired)f(b)r(oundedness.)575 1996 y(The)36 b(op)r(erators)f(in)h Fv(M)1317 2008 y Fo(k)1358 1996 y Fv(;)89 b(k)41 b Fx(=)c(1)p Fv(;)14 b(::;)g Fx(4)73 b(from)36 b(\(8.22\))g(whic)n(h)h(are)e(not)h (y)n(et)h(consid-)386 2095 y(ered,)26 b(are)f(all)h(b)r(ounded.)g (Applying)h(the)f(Lieb)g(and)h(Y)-7 b(au)26 b(form)n(ula,)f(this)h (\014nally)g(sho)n(ws)f(that)386 2195 y Fu(j)p Fx(\()p Fv(\036)490 2207 y Fo(j)526 2195 y Fv(';)14 b Fx([)p Fv(v)683 2165 y Ft(\(12\))805 2195 y Fv(;)g( )896 2207 y Fo(j)931 2195 y Fx(])g Fv(')p Fx(\))p Fu(j)38 b(\024)1223 2162 y Fo(c)p 1212 2176 51 4 v 1212 2224 a(R)1287 2195 y Fu(k)p Fv(')p Fu(k)1425 2165 y Ft(2)1512 2195 y Fx(and)27 b(completes)h(the)g(pro)r(of)f(of)g(the)h(HVZ)g(theorem.)1492 2374 y Fw(A)n(ckno)n(wledgment)575 2524 y Fx(I)21 b(should)g(lik)n(e)f (to)h(thank)g(H.Sieden)n(top,)g(H.Kalf,)g(R.Lasser,)f(L.Erd\177)-42 b(os,)19 b(S.V)-7 b(ugalter)21 b(and)386 2624 y(S.Morozo)n(v)31 b(for)j(stim)n(ulating)f(discussions)g(and)h(critical)f(commen)n(ts.)g (It)h(is)g(a)f(pleasure)g(to)386 2723 y(thank)25 b(A.Sob)r(olev)g(for)f (his)h(hin)n(t)h(to)f(w)n(ork)e(in)i(momen)n(tum)h(space)e(and)h(to)g (use)g(the)g(auxiliary)386 2823 y(function)20 b(in)g(the)g(pro)r(ofs,)f (as)g(w)n(ell)h(as)f(for)g(his)g(hospitalit)n(y)h(during)f(m)n(y)g(sta) n(y)g(at)h(the)g(Univ)n(ersit)n(y)386 2922 y(of)30 b(Sussex)f(in)h (2005.)e(Supp)r(ort)h(b)n(y)h(the)g(EU)f(Net)n(w)n(ork)g(Analysis)g (and)g(Quan)n(tum)h(\(con)n(tract)386 3022 y(HPRN-CT-2002-00277\))23 b(is)k(gratefully)g(ac)n(kno)n(wledged.)1620 3202 y Fw(References)421 3334 y Fz([1])35 b(Balinsky)25 b(A.A.)f(and)i(Ev)l(ans)f(W.D.,)f(On)i (the)g(Bro)n(wn-Ra)n(v)n(enhall)f(relativistic)g(Hamiltonian)f(and)i (the)531 3417 y(stabilit)n(y)e(of)f(matter,)g Fq(Stud.)j(A)l(dv.)g (Math.)e Ff(16)p Fz(,)e(1-9)i(\(2000\).)421 3500 y([2])35 b(Bro)n(wn)16 b(G.E.)g(and)h(Ra)n(v)n(enhall)g(D.G.,)e(On)i(the)g(in)n (teraction)h(of)e(t)n(w)n(o)h(electrons,)g Fq(Pr)l(o)l(c.)j(R)l(oy.)f (So)l(c.)g(L)l(ondon)531 3583 y Ff(A208)p Fz(,)j(552-559)j(\(1951\).) 421 3666 y([3])35 b(Burenk)n(o)n(v)f(V.I.)f(and)h(Ev)l(ans)g(W.D.,)e (On)i(the)g(ev)l(aluation)h(of)e(the)h(norm)e(of)h(an)h(in)n(tegral)g (op)r(erator)531 3750 y(asso)r(ciated)h(with)e(the)h(stabilit)n(y)g(of) f(one-electron)i(atoms,)d Fq(Pr)l(o)l(c.)k(R)l(oy.)f(So)l(c.)g (\(Edinbur)l(gh\))g Ff(128A)p Fz(,)531 3833 y(993-1005)25 b(\(1998\).)421 3916 y([4])35 b(Cycon)25 b(H.L.,)d(F)-6 b(ro)r(ese)24 b(R.G.,)e(Kirsc)n(h)h(W.,)g(Simon)g(B.,)g Fq(Schr\177)-36 b(odinger)26 b(Op)l(er)l(ators)h(with)f(Applic)l(ation) h(to)531 3999 y(Quantum)h(Me)l(chanics)g(and)h(Glob)l(al)g(Ge)l(ometry) p Fz(,)c(T)-6 b(ext)27 b(and)f(Monographs)h(in)e(Ph)n(ysics,)h(1st)g (Edition,)531 4082 y(Springer-V)-6 b(erlag,)23 b(Berlin,)f(1987.)421 4165 y([5])35 b(Douglas)23 b(M.)g(and)g(Kroll)f(N.M.,)g(Quan)n(tum)h (electro)r(dynamical)g(corrections)h(to)g(the)g(\014ne)g(structure)g (of)531 4248 y(helium,)e Fq(A)n(nn.)k(Phys.)f(\(N.Y.\))f Ff(82)p Fz(,)e(89-155)j(\(1974\).)421 4331 y([6])35 b(Ev)l(ans)c(W.D.,) e(P)n(erry)h(P)-6 b(.,)29 b(Sieden)n(top)j(H.,)e(The)g(sp)r(ectrum)g (of)g(relativistic)g(one-electron)i(atoms)e(ac-)531 4414 y(cording)24 b(to)g(Bethe)h(and)f(Salp)r(eter,)g Fq(Commun.)j(Math.)f (Phys.)e Ff(178)p Fz(,)e(733-746)j(\(1996\).)421 4497 y([7])35 b(Georgescu)30 b(V.)e(and)h(Iftimo)n(vici)e(A.,)g(Crossed)h (pro)r(ducts)i(of)e(C)2261 4473 y Fe(\003)2297 4497 y Fz(-algebras)g(and)h(sp)r(ectral)g(analysis)f(of)531 4580 y(quan)n(tum)c(Hamiltonians,)e Fq(Commun.)27 b(Math.)f(Phys.)e Ff(228)p Fz(,)e(519-560)i(\(2002\).)421 4663 y([8])35 b(Gradsh)n(teyn)25 b(I.S.)e(and)h(Ryzhik)g(I.M.,)e Fq(T)-5 b(able)25 b(of)g(Inte)l(gr)l(als,)i(Series)e(and)h(Pr)l(o)l(ducts)p Fz(,)e(Academic)g(Press,)531 4746 y(New)g(Y)-6 b(ork,)23 b(1965.)p eop %%Page: 22 22 22 21 bop 386 259 a Ft(22)904 b(D.)22 b(H.)g(JAKUBASSA-AMUNDSEN)421 459 y Fz([9])35 b(Herbst)26 b(I.W.,)f(Sp)r(ectral)i(theory)f(of)g(the)h (op)r(erator)f(\()p Fd(p)2003 435 y Fc(2)2055 459 y Fz(+)17 b Fd(m)2189 435 y Fc(2)2224 459 y Fz(\))2261 411 y Fk(1)p 2261 419 29 3 v 2261 452 a(2)2321 459 y Fb(\000)g Fd(Z)5 b(e)2479 435 y Fc(2)2512 459 y Fd(=r)r Fz(.)26 b Fq(Commun.)j(Math.)e (Phys.)531 542 y Ff(53)p Fz(,)22 b(285-294)j(\(1977\).)386 625 y([10])35 b(Hunzik)n(er)27 b(W.,)e(On)i(the)g(sp)r(ectra)g(of)f(Sc) n(hr\177)-35 b(odinger)27 b(m)n(ultiparticle)e(Hamiltonians,)g Fq(Helv.)i(Phys.)h(A)l(cta)531 708 y Ff(39)p Fz(,)22 b(451-462)j(\(1966\).)386 791 y([11])35 b(Jakuba\031a-Am)n(undsen)d (D.H.,)d(Pseudorelativistic)j(op)r(erator)f(for)g(a)g(t)n(w)n (o-electron)h(ion,)e Fq(Phys.)j(R)l(ev.)531 874 y Ff(A71)p Fz(,)23 b(032105,)i(1-8)e(\(2005\).)386 957 y([12])35 b(Kato)24 b(T.,)f Fq(Perturb)l(ation)j(The)l(ory)h(for)e(Line)l(ar)h (Op)l(er)l(ators)p Fz(,)f(Springer-V)-6 b(erlag,)22 b(Berlin,)h(1980.) 386 1040 y([13])35 b(Lewis)25 b(R.T.,)f(Sieden)n(top)j(H.,)d(V)-6 b(ugalter)26 b(S.,)f(The)g(essen)n(tial)h(sp)r(ectrum)f(of)g (relativistic)g(m)n(ulti-particle)531 1123 y(op)r(erators,)f Fq(A)n(nn.)h(Inst.)h(Henri)f(Poinc)l(ar)n(\023)-35 b(e)26 b Ff(67)p Fz(,)c(No.1,)i(1-28)f(\(1997\).)386 1206 y([14])35 b(Lieb)f(E.H.)e(and)j(Y)-6 b(au)34 b(H.-T.,)e(The)i(stabilit)n(y)g(and) g(instabilit)n(y)f(of)h(relativistic)f(matter,)g Fq(Commun.)531 1289 y(Math.)26 b(Phys.)e Ff(118)p Fz(,)e(177-213)j(\(1988\).)386 1372 y([15])35 b(Murra)n(y)23 b(J.D.,)f Fq(Asymptotic)k(A)n(nalysis)p Fz(,)d(Clarendon)i(Press,)d(Oxford,)h(1974,)h Fb(x)p Fz(2,)g(5.)386 1455 y([16])35 b(Reed)29 b(M.)f(and)i(Simon)d(B.,)h Fq(F)-5 b(unctional)31 b(A)n(nalysis)p Fz(,)e(V)-6 b(ol.)28 b(I)h(of)g(Metho)r(ds)g(of)g(Mo)r(dern)f(Mathematical)531 1538 y(Ph)n(ysics,)23 b(Academic)h(Press,)e(New)i(Y)-6 b(ork,)23 b(1980.)386 1621 y([17])35 b(Reed)27 b(M.)f(and)h(Simon)f (B.,)g Fq(F)-5 b(ourier)29 b(A)n(nalysis,)f(Self-A)l(djointness)p Fz(,)f(V)-6 b(ol.)26 b(I)r(I)h(of)g(Metho)r(ds)g(of)g(Mo)r(dern)531 1704 y(Mathematical)d(Ph)n(ysics,)f(Academic)g(Press,)g(New)h(Y)-6 b(ork,)23 b(1975.)386 1787 y([18])35 b(Reed)21 b(M.)f(and)h(Simon)e (B.,)h Fq(A)n(nalysis)j(of)g(Op)l(er)l(ators)p Fz(,)e(V)-6 b(ol.)20 b(IV)g(of)g(Metho)r(ds)i(of)e(Mo)r(dern)g(Mathematical)531 1870 y(Ph)n(ysics,)j(Academic)h(Press,)e(New)i(Y)-6 b(ork,)23 b(1978.)386 1953 y([19])35 b(Sieden)n(top)18 b(H.)e(and)h(Sto)r(c)n (kmey)n(er)h(E.,)d(An)i(analytic)g(Douglas-Kroll-Hess)e(metho)r(d,)h Fq(Phys.)j(L)l(ett.)d Ff(A341)p Fz(,)531 2036 y(473-478)25 b(\(2005\).)386 2119 y([20])35 b(Simon)29 b(B.,)h(Geometric)g(metho)r (ds)g(in)g(m)n(ultiparticle)g(quan)n(tum)g(systems,)f Fq(Commun.)k(Math.)f(Phys.)531 2202 y Ff(55)p Fz(,)22 b(259-274)j(\(1977\).)386 2285 y([21])35 b(T)-6 b(esc)n(hl)24 b(G.,)f(Lecture)i(Notes)f(on)g Fq(Schr\177)-36 b(odinger)27 b(Op)l(er)l(ators)p Fz(,)d(Section)h(9.4,)e(2004.)531 2368 y(h)n(ttp://www.mat.univie.ac.at/~gerald/ftp/index.h)n(tml)386 2451 y([22])35 b(Thaller)23 b(B.,)g Fq(The)i(Dir)l(ac)h(e)l(quation)p Fz(,)e(Springer-V)-6 b(erlag,)23 b(Berlin,)f(1992.)386 2534 y([23])35 b(Tix)28 b(C.,)f(Lo)n(w)n(er)h(b)r(ound)h(for)e(the)i (ground)f(state)h(energy)g(of)f(the)h(no-pair)e(Hamiltonian,)g Fq(Phys.)j(L)l(ett.)531 2617 y Ff(B405)p Fz(,)22 b(293-296)j(\(1997\).) 386 2700 y([24])35 b(Tix)30 b(C.,)e(Strict)j(p)r(ositivit)n(y)f(of)g(a) g(relativistic)f(Hamiltonian)g(due)i(to)f(Bro)n(wn)g(and)g(Ra)n(v)n (enhall,)g Fq(Bul)t(l.)531 2783 y(L)l(ondon)d(Math.)f(So)l(c.)e Ff(30)p Fz(,)e(283-290)j(\(1998\).)386 2866 y([25])35 b(W)-6 b(eder)30 b(R.A.,)e(Sp)r(ectral)i(analysis)f(of)g(pseudo)r (di\013eren)n(tial)i(op)r(erators,)e Fq(J.)i(F)-5 b(unctional)32 b(A)n(nalysis)e Ff(20)p Fz(,)531 2949 y(319-337)25 b(\(1975\).)386 3032 y([26])35 b(v)l(an)24 b(Win)n(ter)g(C.,)e(Theory)i(of)e(\014nite)i (systems)f(of)g(particles)g(I.)g(The)h(Green)g(function,)f Fq(Mat.)i(Fys.)h(Dan.)531 3115 y(Vid.)g(Selsk.)d Ff(2)g Fz(No.8,)g(1-60)h(\(1964\).)386 3198 y([27])35 b(W)-6 b(eidmann)24 b(J.,)f Fq(Line)l(ar)i(Op)l(er)l(ators)i(in)f(Hilb)l(ert)f (Sp)l(ac)l(es)p Fz(,)g(Springer-V)-6 b(erlag,)22 b(Berlin,)g(1980.)386 3281 y([28])35 b(Zhislin)25 b(G.M.,)f(A)i(study)g(of)f(the)i(sp)r (ectrum)e(of)h(the)g(Sc)n(hr\177)-35 b(odinger)27 b(op)r(erator)f(for)f (a)h(system)f(of)g(sev)n(eral)531 3364 y(particles,)e Fq(T)-5 b(rudy)26 b(Moskov.)g(Mat.)f(Obsc.)e Ff(9)p Fz(,)f(81-120)j (\(1960\).)575 3756 y Fy(D.H.)h(Jakuba\031a-Amundsen,)f(Ma)l(thema)l (tics)f(Institute,)j(University)e(of)h(Munich,)f(There-)386 3839 y(sienstr.)h(39,)f(80333)f(Munich,)g(Germany)575 3922 y Fq(E-mail)i(addr)l(ess)5 b Fz(:)25 b Fa(dj@mathematik.uni-muen)q (chen)q(.de)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF