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b(hat)f(on)h(a)f(function)h(denotes)f(its)h(F)-7 b(ourier)26 b(transform.)451 924 y(It)g(is)e(w)n(ell-kno)n(wn)g(that)h Fo(H)32 b Ft(is)25 b(not)g(b)r(ounded)h(from)e(b)r(elo)n(w.)36 b(As)25 b(long)f(as)h(pair)f(creation)g(is)451 1024 y(neglected,)31 b(the)g(con)n(v)n(en)n(tional)d(w)n(a)n(y)i(to)g(circum)n(v)n(en)n(t)f (this)i(de\014ciency)f(is)h(the)g(in)n(tro)r(duc-)451 1124 y(tion)25 b(of)g(the)g(semib)r(ounded)g(op)r(erator)e Fo(P)1734 1136 y Fp(+)1789 1124 y Fo(H)7 b(P)1918 1136 y Fp(+)1998 1124 y Ft(where)24 b Fo(P)2288 1136 y Fp(+)2369 1124 y Ft(pro)5 b(jects)23 b(on)n(to)h(the)h(p)r(ositiv)n(e)451 1223 y(sp)r(ectral)35 b(subspace)g(of)h Fo(H)7 b Ft(.)60 b(As)36 b(a)f(\014rst)g(appro)n(ximation,)h(Bro)n(wn)e(and)i(Ra)n(v)n (enhall)e([1])451 1323 y(in)n(tro)r(duced)28 b(the)g(op)r(erator)1018 1547 y Fo(B)g Ft(:=)45 b(\003)1300 1559 y Fp(+)1355 1547 y Fo(H)7 b Ft(\003)1489 1559 y Fp(+)1544 1547 y Fo(;)346 b Ft(\003)1971 1559 y Fn(\006)2049 1547 y Ft(:=)2193 1490 y(1)p 2193 1528 42 4 v 2193 1604 a(2)2281 1430 y Fh(\022)2343 1547 y Ft(1)18 b Fm(\006)2519 1490 y Fo(D)2588 1502 y Fp(0)p 2495 1528 153 4 v 2495 1604 a Fm(j)p Fo(D)2587 1616 y Fp(0)2625 1604 y Fm(j)2658 1430 y Fh(\023)3115 1547 y Ft(\(1.2\))451 1771 y(with)35 b(\003)705 1783 y Fp(+)794 1771 y Ft(pro)5 b(jecting)33 b(on)n(to)g(the)h(p)r(ositiv)n (e)g(sp)r(ectral)f(subspace)h(of)g(the)g(free)g(Dirac)g(op-)451 1870 y(erator)j Fo(D)776 1882 y Fp(0)813 1870 y Fo(;)h Ft(and)g Fm(j)p Fo(D)1138 1882 y Fp(0)1175 1870 y Fm(j)55 b Ft(=)1358 1798 y Fh(p)p 1441 1798 109 4 v 72 x Fo(D)1512 1842 y Fp(2)1510 1892 y(0)1628 1870 y Ft(is)38 b(the)h(free)f(energy)-7 b(.)67 b(In)38 b(momen)n(tum)h(space)e(one)451 1970 y(has)641 2156 y(~)621 2177 y Fo(D)690 2189 y Fp(0)727 2177 y Ft(\()p Fi(p)p Ft(\))24 b(:=)1002 2060 y Fh(\022)1096 2121 y Fo(D)1165 2133 y Fp(0)p 1073 2158 153 4 v 1073 2234 a Fm(j)p Fo(D)1165 2246 y Fp(0)1202 2234 y Fm(j)1235 2060 y Fh(\023)1310 2177 y Ft(\()p Fi(p)p Ft(\))46 b(=)1594 2121 y Fj(\013)p Fi(p)19 b Ft(+)f Fo(\014)t(m)p 1594 2158 342 4 v 1715 2234 a(E)1776 2246 y Fl(p)1946 2177 y Fo(;)346 b(E)2376 2189 y Fl(p)2438 2177 y Ft(:=)2571 2101 y Fh(p)p 2654 2101 291 4 v 76 x Fo(p)2696 2153 y Fp(2)2752 2177 y Ft(+)18 b Fo(m)2908 2153 y Fp(2)3115 2177 y Ft(\(1.3\))451 2403 y(with)29 b(the)h(electron)e(mass)g Fo(m:)53 b Ft(By)29 b(construction,)f(the)h(Bro)n(wn-Ra)n(v)n(enhall)d (op)r(erator)h Fo(B)451 2503 y Ft(is)35 b(of)f(\014rst)g(order)f(in)i (the)g(p)r(oten)n(tial)g Fo(V)53 b Ft(and)34 b(has)g(b)r(een)h(sho)n (wn)f(to)g(b)r(e)h(b)r(ounded)g(from)451 2602 y(b)r(elo)n(w)27 b(for)g(sub)r(critical)h(p)r(oten)n(tial)f(strength)g Fo(\015)33 b Ft([5)o(].)451 2702 y(An)h(alternativ)n(e)f(w)n(a)n(y)f (to)h(deriv)n(e)g(a)g(semib)r(ounded)h(op)r(erator)e(from)h Fo(H)41 b Ft(has)33 b(b)r(een)h(sug-)451 2802 y(gested)d(b)n(y)g (Douglas)f(and)g(Kroll)g([4],)i(using)f(the)g(F)-7 b(oldy-W)g(outh)n (uysen)31 b(transformation)451 2901 y(tec)n(hnique)f([6].)44 b(The)31 b(decoupling)e(of)h(the)h(p)r(ositiv)n(e)e(and)h(negativ)n(e)f (sp)r(ectral)h(subspaces)451 3001 y(of)25 b Fo(H)33 b Ft(to)25 b(order)f Fo(n)h Ft(in)h Fo(V)44 b Ft(is)25 b(ac)n(hiev)n(ed)f(b)n(y)h(means)g(of)h Fo(n)13 b Ft(+)h(1)25 b(successiv)n(e)f(unitary)h(transfor-)451 3101 y(mations)i Fo(U)831 3070 y Fn(0)822 3122 y Fl(j)857 3101 y Fo(;)60 b(j)28 b Ft(=)23 b(0)p Fo(;)14 b Ft(1)p Fo(;)g(:::;)g(n)923 3292 y Ft(\()p Fo(U)1021 3258 y Fn(0)1012 3313 y Fl(n)1071 3292 y Fm(\001)g(\001)g(\001)g Fo(U)1248 3258 y Fn(0)1239 3313 y Fp(1)1294 3292 y Fm(\001)19 b Fo(U)1402 3258 y Fn(0)1393 3313 y Fp(0)1430 3292 y Ft(\))k Fo(H)30 b Ft(\()p Fo(U)1682 3258 y Fn(0)1673 3313 y Fl(n)1732 3292 y Fm(\001)14 b(\001)g(\001)g Fo(U)1909 3258 y Fn(0)1900 3313 y Fp(1)1955 3292 y Fm(\001)19 b Fo(U)2063 3258 y Fn(0)2054 3313 y Fp(0)2091 3292 y Ft(\))2123 3258 y Fn(\000)p Fp(1)2249 3292 y Ft(=:)k Fo(H)2436 3258 y Fn(0)2429 3313 y Fl(n)2515 3292 y Ft(+)41 b Fo(R)2684 3304 y Fl(n)p Fp(+1)3115 3292 y Ft(\(1.4\))451 3471 y(whic)n(h)24 b(cast)g(the)g(tranformed)f(op)r (erator)g(in)n(to)h(a)f(blo)r(c)n(k-diagonal)f(con)n(tribution)h Fo(H)3075 3441 y Fn(0)3068 3492 y Fl(n)3138 3471 y Ft(plus)451 3571 y(an)39 b(error)f(term)i Fo(R)1068 3583 y Fl(n)p Fp(+1)1237 3571 y Ft(with)g(p)r(oten)n(tial)f(strength)g(giv)n(en)g(b)n (y)g(the)h Fo(n)26 b Ft(+)g(1st)40 b(p)r(o)n(w)n(er)e(of)451 3670 y Fo(\015)5 b(:)63 b(U)651 3640 y Fn(0)642 3691 y Fp(0)712 3670 y Ft(is)32 b(the)i(free)e(F)-7 b(oldy-W)g(outh)n(uysen) 32 b(transformation)f(whic)n(h)i(blo)r(c)n(k-diagonalises)451 3770 y Fo(D)520 3782 y Fp(0)585 3770 y Ft(exactly)27 b([14)o(],)805 4027 y Fo(U)871 3993 y Fn(0)862 4048 y Fp(0)922 4027 y Ft(:=)45 b Fo(A)1155 3910 y Fh(\022)1216 4027 y Ft(1)c(+)g Fo(\014)1559 3971 y Fj(\013)p Fi(p)p 1480 4008 275 4 v 1480 4084 a Fo(E)1541 4096 y Fl(p)1598 4084 y Ft(+)18 b Fo(m)1764 3910 y Fh(\023)1839 4027 y Fo(;)346 b(A)23 b Ft(:=)2427 3910 y Fh(\022)2498 3971 y Fo(E)2559 3983 y Fl(p)2616 3971 y Ft(+)18 b Fo(m)p 2498 4008 V 2564 4084 a Ft(2)p Fo(E)2667 4096 y Fl(p)2782 3910 y Fh(\023)2853 3905 y Fg(1)p 2853 3914 29 3 v 2853 3947 a(2)2909 4027 y Fo(;)183 b Ft(\(1.5\))451 4257 y(and)28 b(for)f Fo(U)806 4227 y Fn(0)797 4279 y Fl(j)831 4257 y Ft(,)h(Douglas)f(and)g(Kroll)f([4])i(use)1053 4461 y Fo(U)1119 4427 y Fn(0)1110 4482 y Fl(j)1182 4461 y Ft(=)45 b(\(1)18 b(+)h Fo(W)1558 4427 y Fp(2)1546 4482 y Fl(j)1595 4461 y Ft(\))1637 4404 y Fg(1)p 1637 4413 V 1637 4447 a(2)1721 4461 y Ft(+)32 b Fo(W)1896 4473 y Fl(j)1931 4461 y Fo(;)346 b(j)28 b Ft(=)23 b(1)p Fo(;)14 b(:::;)g(n)430 b Ft(\(1.6\))451 4640 y(with)35 b(an)n(tisymmetric)f(op) r(erators)e Fo(W)1651 4652 y Fl(j)1687 4640 y Fo(:)69 b Ft(It)34 b(should)h(b)r(e)f(noted)h(that)f(the)h(c)n(hoice)f(of)g Fo(U)3260 4610 y Fn(0)3251 4661 y Fl(j)451 4739 y Ft(is)g(not)f (unique,)j(and)d(neither)g(is)h(the)g(resulting)f(op)r(erator)f Fo(H)2453 4709 y Fn(0)2446 4760 y Fl(n)2524 4739 y Ft(from)i(\(1.4\))f (for)g Fo(n)g(>)f Ft(4)451 4839 y(as)g(has)h(b)r(een)g(sho)n(wn)f(b)n (y)g(W)-7 b(olf,)35 b(Reiher,)f(and)e(Hess)h([15)o(].)53 b(Applying)33 b(the)g(free)f(F)-7 b(oldy-)451 4939 y(W)g(outh)n(uysen) 28 b(transformation)e(\(1.5\))h(one)g(obtains)g([4,)h(9)o(])1301 5130 y Fo(U)1367 5096 y Fn(0)1358 5151 y Fp(0)1395 5130 y Fo(H)7 b(U)1537 5070 y Ff(0)1559 5095 y Fn(\000)p Fp(1)1528 5152 y(0)1694 5130 y Ft(=)46 b Fo(\014)t(E)1917 5142 y Fl(p)1998 5130 y Ft(+)41 b Fm(E)2148 5142 y Fp(1)2226 5130 y Ft(+)g Fm(O)2398 5142 y Fp(1)1108 5338 y Fu(Document)l(a)25 b(Ma)l(thema)l(tica)h(10)f(\(2005\))g(331{356)p eop %%Page: 333 3 333 2 bop 1291 518 a Fr(Pr)n(ojected)32 b(Dira)n(c)f(Opera)-6 b(tor)716 b Ft(333)487 762 y Fm(E)531 774 y Fp(1)591 762 y Ft(:=)23 b Fo(A)778 645 y Fh(\022)839 762 y Fo(V)37 b Ft(+)1089 706 y Fj(\013)q Fi(p)p 1017 743 261 4 v 1017 819 a Fo(E)1078 831 y Fl(p)1122 819 y Ft(+)18 b Fo(m)1288 762 y(V)1436 706 y Fj(\013)q Fi(p)p 1364 743 V 1364 819 a Fo(E)1425 831 y Fl(p)1469 819 y Ft(+)g Fo(m)1635 645 y Fh(\023)1709 762 y Fo(A;)98 b Fm(O)1958 774 y Fp(1)2018 762 y Ft(:=)23 b Fo(\014)t(A)2256 645 y Fh(\022)2399 706 y Fj(\013)p Fi(p)p 2327 743 V 2327 819 a Fo(E)2388 831 y Fl(p)2432 819 y Ft(+)18 b Fo(m)2597 762 y(V)38 b Fm(\000)18 b Fo(V)2915 706 y Fj(\013)p Fi(p)p 2843 743 V 2843 819 a Fo(E)2904 831 y Fl(p)2947 819 y Ft(+)g Fo(m)3113 645 y Fh(\023)3188 762 y Fo(A)3115 914 y Ft(\(1.7\))451 1013 y(where)34 b(the)h(transformed)f(p)r(oten)n(tial)h(has)f(b)r(een)h (split)g(in)n(to)f(an)h(ev)n(en)f(term)h Fm(E)2972 1025 y Fp(1)3078 1013 y Ft(\(com-)451 1113 y(m)n(uting)g(with)g Fo(\014)t Ft(\))h(and)e(an)h(o)r(dd)g(term)g Fm(O)1791 1125 y Fp(1)1898 1113 y Ft(\(an)n(ticomm)n(uting)f(with)h Fo(\014)t Ft(,)j(since)c Fo(\013)3080 1125 y Fl(k)3121 1113 y Fo(\014)53 b Ft(=)451 1213 y Fm(\000)p Fo(\014)t(\013)620 1225 y Fl(k)661 1213 y Fo(;)60 b(k)26 b Ft(=)d(1)p Fo(;)14 b Ft(2)p Fo(;)g Ft(3\))p Fo(:)49 b Ft(F)-7 b(or)27 b(an)g(exp)r(onen)n (tial)g(unitary)h(transformation,)1273 1401 y Fo(U)1339 1366 y Fn(0)1330 1421 y Fl(j)1388 1401 y Ft(:=)45 b Fo(e)1560 1366 y Fn(\000)p Fl(iS)1676 1374 y Fe(j)1711 1401 y Fo(;)346 b(j)28 b Ft(=)23 b(1)p Fo(;)14 b(:::;)g(n)650 b Ft(\(1.8\))451 1589 y(with)23 b(a)f(symmetric)g(op)r(erator)f Fo(S)1479 1601 y Fl(j)1514 1589 y Ft(,)i(the)g(next)g(transformation)d(giv)n(es)i (in)g(agreemen)n(t)f(with)451 1688 y([9])556 1889 y Fo(e)595 1855 y Fn(\000)p Fl(iS)711 1863 y Fg(1)761 1889 y Fo(U)827 1855 y Fn(0)818 1910 y Fp(0)855 1889 y Fo(H)7 b(U)997 1829 y Ff(0)1019 1854 y Fn(\000)p Fp(1)988 1912 y(0)1122 1889 y Fo(e)1161 1855 y Fl(iS)1225 1863 y Fg(1)1308 1889 y Ft(=)46 b Fo(\014)t(E)1531 1901 y Fl(p)1611 1889 y Ft(+)41 b Fm(E)1761 1901 y Fp(1)1840 1889 y Ft(+)g Fm(O)2012 1901 y Fp(1)2091 1889 y Ft(+)g Fo(i)p Ft([)p Fo(\014)t(E)2361 1901 y Fl(p)2399 1889 y Fo(;)14 b(S)2487 1901 y Fp(1)2524 1889 y Ft(])42 b(+)f Fo(i)p Ft([)p Fm(E)2791 1901 y Fp(1)2846 1889 y Ft(+)18 b Fm(O)2995 1901 y Fp(1)3033 1889 y Fo(;)c(S)3121 1901 y Fp(1)3158 1889 y Ft(])1398 2128 y Fm(\000)1473 2072 y Ft(1)p 1473 2109 42 4 v 1473 2185 a(2)1547 2128 y([)g([)p Fo(\014)t(E)1719 2140 y Fl(p)1758 2128 y Fo(;)g(S)1846 2140 y Fp(1)1883 2128 y Ft(])p Fo(;)28 b(S)2008 2140 y Fp(1)2045 2128 y Ft(])42 b(+)f Fo(R)2279 2140 y Fp(3)2316 2128 y Fo(:)776 b Ft(\(1.9\))451 2309 y Fo(S)502 2321 y Fp(1)567 2309 y Ft(is)27 b(de\014ned)h(from)g(the)g(requiremen)n(t)e (that)i Fm(O)1983 2321 y Fp(1)2048 2309 y Ft(is)g(eliminated,)1241 2497 y Fo(i)23 b Ft([)p Fo(\014)t(E)1428 2509 y Fl(p)1467 2497 y Fo(;)14 b(S)1555 2509 y Fl(j)1590 2497 y Ft(])46 b(=)g Fm(\000O)1901 2509 y Fl(j)1935 2497 y Fo(;)346 b(j)28 b Ft(=)23 b(1)578 b(\(1.10\))451 2685 y(hence)47 b Fo(S)752 2697 y Fp(1)836 2685 y Ft(is)g(o)r(dd)g(and)f(of)h(\014rst)f (order)g(in)h(the)g(p)r(oten)n(tial)g(lik)n(e)f Fm(O)2728 2697 y Fp(1)2766 2685 y Fo(:)102 b Ft(After)47 b(eac)n(h)451 2784 y(transformation)36 b Fo(U)1093 2754 y Fn(0)1084 2806 y Fl(j)1118 2784 y Ft(,)k(the)d Fo(j)30 b Ft(+)24 b(1st)37 b(order)f(term)h(in)g Fo(\015)42 b Ft(of)37 b Fo(R)2423 2796 y Fl(j)s Fp(+1)2579 2784 y Ft(is)g(decomp)r(osed)f(in) n(to)451 2884 y(ev)n(en)31 b(\()p Fm(E)720 2896 y Fl(j)s Fp(+1)839 2884 y Ft(\))g(and)g(o)r(dd)g(\()p Fm(O)1332 2896 y Fl(j)s Fp(+1)1452 2884 y Ft(\))g(con)n(tributions,)h(and)e(the)i (successiv)n(e)e(transformation)451 2984 y Fo(U)517 2954 y Fn(0)508 3005 y Fl(j)s Fp(+1)664 2984 y Ft(=)22 b Fo(e)790 2954 y Fn(\000)p Fl(iS)906 2962 y Fe(j)r Fg(+1)1035 2984 y Ft(is)h(c)n(hosen)f(to)h(eliminate)g Fm(O)1893 2996 y Fl(j)s Fp(+1)2012 2984 y Fo(;)h Ft(whic)n(h)e(is)h(ac)n(hiev)n(ed)f (b)n(y)h(the)g(condition)451 3083 y(\(1.10\))32 b(for)f(the)i Fo(j)i(>)30 b Ft(1)i(under)g(consideration.)49 b(With)33 b(this)f(pro)r(cedure)f(one)h(arriv)n(es)e(at)451 3183 y(the)e(ev)n(en)f(\(and)h(hence)g(blo)r(c)n(k-diagonal\))d(op)r(erator) 1328 3371 y Fo(H)1404 3337 y Fn(0)1397 3392 y Fl(n)1488 3371 y Ft(=)46 b Fo(\014)t(E)1711 3383 y Fl(p)1792 3371 y Ft(+)41 b Fm(E)1942 3383 y Fp(1)2020 3371 y Ft(+)18 b Fo(:::)h Ft(+)41 b Fm(E)2341 3383 y Fl(n)2386 3371 y Fo(:)665 b Ft(\(1.11\))451 3562 y(The)33 b(ph)n(ysical)f(quan)n(tit)n (y)g(of)h(in)n(terest)g(is)f(the)i(exp)r(ectation)e(v)-5 b(alue)33 b(of)g(the)g(transformed)451 3661 y(Dirac)g(op)r(erator.)51 b(F)-7 b(or)33 b(the)g(Bro)n(wn-Ra)n(v)n(enhall)d(op)r(erator,)j (consider)f(the)h(exp)r(ectation)451 3761 y(v)-5 b(alue)26 b(formed)f(with)i(4-spinors)d Fo(')i Ft(in)g(the)g(p)r(ositiv)n(e)g(sp) r(ectral)f(subspace)g(of)h Fo(D)2918 3773 y Fp(0)2981 3761 y Ft(whic)n(h)g(in)451 3861 y(momen)n(tum)c(space)e(can)h(b)r(e)h (expressed)e(in)i(terms)f(of)g(P)n(auli)g(spinors)f Fo(u)j Fm(2)g Fo(H)2823 3876 y Fp(1)p Fl(=)p Fp(2)2927 3861 y Ft(\()p Fk(R)3014 3831 y Fp(3)3057 3861 y Ft(\))6 b Fm(\012)g Fk(C)3220 3831 y Fp(2)3263 3861 y Fo(;)1088 4107 y Ft(^)-56 b Fo(')q Ft(\()p Fi(p)p Ft(\))37 b(=)1671 4051 y(1)p 1404 4088 577 4 v 1404 4105 a Fh(p)p 1487 4105 494 4 v 69 x Ft(2)p Fo(E)1590 4186 y Fl(p)1642 4174 y Ft(\()p Fo(E)1735 4186 y Fl(p)1792 4174 y Ft(+)18 b Fo(m)p Ft(\))2014 3990 y Fh(\022)2075 4051 y Ft(\()p Fo(E)2168 4063 y Fl(p)2225 4051 y Ft(+)g Fo(m)p Ft(\))29 b(^)-48 b Fo(u)p Ft(\()p Fi(p)p Ft(\))2187 4164 y Fi(p)p Fj(\033)32 b Ft(^)-47 b Fo(u)o Ft(\()p Fi(p)p Ft(\))2601 3990 y Fh(\023)3074 4107 y Ft(\(1.12\))451 4356 y(where)24 b Fj(\033)k Ft(is)d(the)g(v)n(ector)f(of)g(the)h(three)g(P)n(auli)f (matrices.)35 b(Then,)25 b(an)g(op)r(erator)e Fo(b)2977 4368 y Fl(m)3064 4356 y Ft(acting)451 4455 y(on)k Fo(H)635 4470 y Fp(1)p Fl(=)p Fp(2)740 4455 y Ft(\()p Fk(R)826 4425 y Fp(3)869 4455 y Ft(\))19 b Fm(\012)f Fk(C)1057 4425 y Fp(2)1142 4455 y Ft(ma)n(y)27 b(b)r(e)h(de\014ned)g(b)n(y)f([5]) 1497 4643 y(\()p Fo(';)14 b(B)t(')p Ft(\))38 b(=:)23 b(\()p Fo(u;)14 b(b)2075 4655 y Fl(m)2137 4643 y Fo(u)p Ft(\))p Fo(:)834 b Ft(\(1.13\))451 4831 y(On)29 b(the)g(other)f(hand,)h (in)g(case)f(of)g(the)h(Douglas-Kroll)d(transformed)i(op)r(erator)f Fo(H)3108 4801 y Fn(0)3101 4852 y Fl(n)3146 4831 y Fo(;)i Ft(its)451 4931 y(upp)r(er)38 b(blo)r(c)n(k)g(corresp)r(onds)e(to)h (the)i(particle)e(states)g(\(ha)n(ving)h(p)r(ositiv)n(e)f(energy\))g (and)451 5031 y(therefore)25 b(the)h(exp)r(ectation)f(v)-5 b(alue)26 b(has)f(to)g(b)r(e)h(formed)g(with)g(the)g(four-spinor)e Fo( )i Ft(:=)3170 4963 y Fh(\000)3209 4994 y Fl(u)3212 5059 y Fp(0)3248 4963 y Fh(\001)451 5130 y Ft(with)i Fo(u)23 b Fm(2)g Fo(H)858 5145 y Fp(1)p Fl(=)p Fp(2)963 5130 y Ft(\()p Fk(R)1049 5100 y Fp(3)1092 5130 y Ft(\))c Fm(\012)f Fk(C)1280 5100 y Fp(2)1351 5130 y Ft(as)27 b(ab)r(o)n(v)n(e.)1108 5338 y Fu(Document)l(a)e(Ma)l(thema)l(tica)h(10) f(\(2005\))g(331{356)p eop %%Page: 334 4 334 3 bop 451 518 a Ft(334)727 b Fr(D.)30 b(H.)h(Jakubassa-Amundsen)451 725 y Ft(F)-7 b(or)32 b(the)i(\014rst-order)c(term,)k Fo(H)1451 695 y Fn(0)1444 746 y Fp(1)1482 725 y Fo(;)f Ft(it)g(is)f(easy)g(to)h(sho)n(w)f([2)o(])h(that)g(its)g(exp)r (ectation)g(v)-5 b(alue)451 825 y(agrees)26 b(with)i(the)g(exp)r (ectation)f(v)-5 b(alue)28 b(of)f(the)h(Bro)n(wn-Ra)n(v)n(enhall)d(op)r (erator,)h(i.e.)678 1060 y(\()p Fo(u;)14 b(b)831 1072 y Fl(m)893 1060 y Fo(u)p Ft(\))46 b(=)1130 943 y Fh(\022\022)1252 1004 y Fo(u)1255 1117 y Ft(0)1300 943 y Fh(\023)1361 1060 y Fo(;)14 b(H)1474 1025 y Fn(0)1467 1080 y Fp(1)1504 943 y Fh(\022)1565 1004 y Fo(u)1568 1117 y Ft(0)1613 943 y Fh(\023\023)1749 1060 y Fo(;)369 b(u)22 b Fm(2)i Fo(H)2359 1075 y Fp(1)p Fl(=)p Fp(2)2463 1060 y Ft(\()p Fk(R)2549 1025 y Fp(3)2593 1060 y Ft(\))18 b Fm(\012)g Fk(C)2780 1025 y Fp(2)2824 1060 y Fo(;)227 b Ft(\(1.14\))451 1299 y(suc)n(h)19 b(that)g Fo(H)877 1269 y Fn(0)870 1320 y Fl(n)934 1299 y Ft(ma)n(y)f(b)r(e)h(considered)f(as)g(the)h(natural)f (con)n(tin)n(uation)g(of)h Fo(B)k Ft(to)18 b(higher)g(order)451 1399 y(in)31 b Fo(V)19 b Ft(.)44 b(While)31 b Fo(H)1006 1369 y Fn(0)999 1420 y Fl(n)1075 1399 y Ft(is)f(kno)n(wn)f(explicitly)i (up)f(to)g Fo(n)e Ft(=)f(5)i([15)o(],)j(the)e(sp)r(ectral)g(prop)r (erties,)451 1499 y(in)e(particular)e(the)i(b)r(oundedness)f(from)g(b)r (elo)n(w,)g(ha)n(v)n(e)g(only)g(b)r(een)g(in)n(v)n(estigated)g(for)g (the)451 1598 y(Jansen-Hess)f(op)r(erator)g(\(i.e.)37 b Fo(n)23 b Ft(=)g(2\))k([2,)h(8)o(].)451 1701 y(The)g(aim)f(of)h(this) g(w)n(ork)e(is)h(to)h(pro)n(v)n(e)e(t)n(w)n(o)h(theorems.)451 1878 y Fr(Theorem)35 b Ft(1.1)p Fr(.)42 b Fd(L)l(et)31 b Fo(H)j Ft(=)27 b Fo(D)1444 1890 y Fp(0)1502 1878 y Ft(+)19 b Fo(V)51 b Fd(b)l(e)33 b(the)f(one-p)l(article)h(Dir)l(ac)g (op)l(er)l(ator)g(acting)g(on)451 1978 y Fm(S)6 b Ft(\()p Fk(R)594 1948 y Fp(3)637 1978 y Ft(\))22 b Fm(\012)f Fk(C)831 1948 y Fp(4)908 1978 y Fd(with)34 b Fm(S)40 b Fd(the)34 b(Schwartz)h(sp)l(ac)l(e)f(of)h(smo)l(oth)f(str)l(ongly)g (lo)l(c)l(alise)l(d)h(functions.)451 2077 y(L)l(et)i Fo(\015)42 b Fd(b)l(e)37 b(the)g(str)l(ength)g(of)h(the)f(Coulomb)h(p)l (otential)g Fo(V)56 b Fd(and)38 b Fo(p)e Ft(:=)g Fm(j)p Fi(p)p Fm(j)p Fo(:)74 b Fd(Then)38 b(ther)l(e)451 2177 y(exists)e(a)g(se)l(quenc)l(e)g(of)h(unitary)f(tr)l(ansformations)h Fo(U)2173 2189 y Fl(k)2262 2177 y Ft(=)d Fo(e)2400 2147 y Fl(iB)2473 2156 y Fe(k)2513 2177 y Fo(;)83 b(k)38 b Ft(=)c(1)p Fo(;)14 b(:::;)g(n;)70 b Fd(such)451 2277 y(that)30 b(the)g(tr)l(ansforme)l(d)g(Dir)l(ac)g(op)l(er)l(ator)h(c)l (an)f(b)l(e)g(written)f(in)h(the)g(fol)t(lowing)i(way)1065 2467 y Ft(\()p Fo(U)1154 2479 y Fp(1)1206 2467 y Fm(\001)14 b(\001)g(\001)f Fo(U)1373 2479 y Fl(n)1418 2467 y Ft(\))1450 2432 y Fn(\000)p Fp(1)1540 2467 y Fo(H)20 b(U)1686 2479 y Fp(1)1737 2467 y Fm(\001)14 b(\001)g(\001)g Fo(U)1905 2479 y Fl(n)1996 2467 y Ft(=:)22 b Fo(H)2182 2432 y Fp(\()p Fl(n)p Fp(\))2321 2467 y Ft(+)41 b Fo(R)2491 2432 y Fp(\()p Fl(n)p Fp(+1\))998 2742 y Fo(H)1074 2707 y Fp(\()p Fl(n)p Fp(\))1193 2742 y Ft(:=)46 b(\003)1385 2754 y Fp(+)1454 2600 y Fh( )1559 2638 y Fl(n)1520 2663 y Fh(X)1520 2842 y Fl(k)q Fp(=0)1654 2742 y Fo(H)1723 2754 y Fl(k)1764 2600 y Fh(!)1844 2742 y Ft(\003)1902 2754 y Fp(+)1989 2742 y Ft(+)31 b(\003)2143 2754 y Fn(\000)2213 2600 y Fh( )2319 2638 y Fl(n)2279 2663 y Fh(X)2279 2842 y Fl(k)q Fp(=0)2413 2742 y Fo(H)2482 2754 y Fl(k)2523 2600 y Fh(!)2603 2742 y Ft(\003)2661 2754 y Fn(\000)2716 2742 y Fo(:)335 b Ft(\(1.15\))451 2976 y Fd(Her)l(e,)28 b Ft(\003)731 2988 y Fp(+)812 2976 y Fd(pr)l(oje)l(cts)f(onto)g(the)f(p)l(ositive)i (sp)l(e)l(ctr)l(al)f(subsp)l(ac)l(e)g(of)g Fo(D)2515 2988 y Fp(0)2552 2976 y Fo(;)60 b Ft(\003)2693 2988 y Fn(\000)2772 2976 y Ft(=)23 b(1)11 b Fm(\000)g Ft(\003)3047 2988 y Fp(+)3102 2976 y Fd(,)27 b(and)451 3076 y Fo(H)520 3088 y Fl(k)587 3076 y Fd(is)f(a)g Fo(p)p Fd(-form)g(b)l(ounde)l(d)g (op)l(er)l(ator,)i(its)e(form)g(b)l(ound)g(b)l(eing)g(pr)l(op)l (ortional)i(to)e Fo(\015)3029 3046 y Fl(k)3069 3076 y Fo(;)60 b(k)26 b Ft(=)451 3176 y(1)p Fo(;)14 b(:::;)g(n:)84 b Fd(The)41 b(r)l(emainder)h Fo(R)1445 3145 y Fp(\()p Fl(n)p Fp(+1\))1667 3176 y Fd(which)h(stil)t(l)e(c)l(ouples)h(the)f(sp) l(e)l(ctr)l(al)g(subsp)l(ac)l(es)g(of)451 3275 y Fo(D)520 3287 y Fp(0)594 3275 y Fd(is)c Fo(p)p Fd(-form)g(b)l(ounde)l(d)h(with)f (form)h(b)l(ound)f 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b(reco)n(v)n(ered.)34 b(\(1.15\))26 b(implies)h(that)f(the)h(transformed)451 4053 y(Dirac)34 b(op)r(erator)f(can)h(b)r(e)h(expressed)e(in)i(terms)f(of)g(pro)5 b(jectors)33 b(to)h(arbitrary)f(order)g(in)451 4153 y(the)25 b(p)r(oten)n(tial)f(strength.)35 b(Similar)24 b(transformation)f(sc)n (hemes)g(are)g(kno)n(wn)h(for)f(b)r(ounded)451 4253 y(op)r(erators)j (on)h(lattices,)h(see)f(e.g.)36 b([3])28 b(and)f([13)o(].)451 4356 y(The)h(next)g(theorem)f(states)g(the)h(unitary)g(equiv)-5 b(alence)27 b(of)h(the)g(transformed)e(Dirac)i(op-)451 4455 y(erators)i(obtained)i(with)g(either)g(the)g(transformation)f(sc)n (heme)g(from)h(Theorem)f(1.1)g(or)451 4555 y(the)d(Douglas-Kroll)d (transformation)h(sc)n(heme)h(\(1.8\))h(-)f(\(1.11\).)451 4732 y Fr(Theorem)34 b Ft(1.2)p Fr(.)41 b Fd(L)l(et)31 b Fo(')c Fm(2)f Ft(\003)1398 4744 y Fp(+)1453 4732 y Ft(\()p Fm(S)6 b Ft(\()p Fk(R)1628 4702 y Fp(3)1671 4732 y Ft(\))20 b Fm(\012)g Fk(C)1862 4702 y Fp(4)1905 4732 y Ft(\))32 b Fd(b)l(e)f(a)h(4-spinor)g(in)g(the)g(p)l(ositive)h(sp)l(e) l(ctr)l(al)451 4831 y(subsp)l(ac)l(e)k(of)g Fo(D)964 4843 y Fp(0)1001 4831 y Fd(,)h(which)g(de\014nes)f(a)f(Pauli)i(spinor)f Fo(u)f Fd(by)g(me)l(ans)h(of)g(\(1.12\).)60 b(L)l(et)36 b Fo(H)3248 4801 y Fn(0)3241 4852 y Fl(n)451 4931 y Fd(b)l(e)30 b(the)h(Douglas-Kr)l(ol)t(l)g(tr)l(ansforme)l(d)g(Dir)l(ac)f(op)l(er)l (ator)i(to)e Fo(n)p Fd(-th)f(or)l(der)i(in)g(the)f(p)l(otential)451 5031 y(str)l(ength,)38 b(obtaine)l(d)f(with)g(exp)l(onential)f(unitary) g(op)l(er)l(ators)h Fo(U)2494 5000 y Fn(0)2485 5052 y Fl(j)2520 5031 y Fd(.)58 b(Then)36 b(to)h(any)f(or)l(der)451 5130 y Fo(n)p Fd(,)d(its)f(exp)l(e)l(ctation)g(value)h(agr)l(e)l(es)f (with)g(the)h(one)f(of)h(the)f(tr)l(ansforme)l(d)g(Dir)l(ac)h(op)l(er)l (ator)1108 5338 y Fu(Document)l(a)25 b(Ma)l(thema)l(tica)h(10)f (\(2005\))g(331{356)p eop %%Page: 335 5 335 4 bop 1291 518 a Fr(Pr)n(ojected)32 b(Dira)n(c)f(Opera)-6 b(tor)716 b Ft(335)451 725 y Fd(fr)l(om)30 b(\(1.15\),)575 986 y Ft(\()p Fo(';)14 b(H)774 951 y Fp(\()p Fl(n)p Fp(\))885 986 y Fo(')p Ft(\))47 b(=)1128 844 y Fh( )1194 986 y Fo(';)1325 882 y Fl(n)1286 907 y Fh(X)1285 1086 y Fl(k)q Fp(=0)1420 986 y Fo(H)1489 998 y Fl(k)1544 986 y Fo(')1598 844 y Fh(!)1701 986 y Ft(=)1811 869 y Fh(\022\022)1934 930 y Fo(u)1937 1043 y Ft(0)1981 869 y Fh(\023)2042 986 y Fo(;)14 b(H)2155 951 y Fn(0)2148 1006 y Fl(n)2193 869 y Fh(\022)2254 930 y Fo(u)2257 1043 y Ft(0)2302 869 y Fh(\023\023)2438 986 y Fo(;)268 b(n)23 b Ft(=)g(1)p Fo(;)14 b Ft(2)p Fo(;)g(:::)22 b(:)3074 1164 y Ft(\(1.16\))451 1335 y(This)40 b(extends)f(the)h(\014rst-order)d(equation)i(\(1.14\))f (to)h(arbitrary)f(order)g Fo(n)p Ft(.)72 b(Actually)451 1435 y(\(1.16\))28 b(holds)h(also)f(for)h(other)f(t)n(yp)r(es)h(of)g (unitary)f(op)r(erators,)g(pro)n(vided)g(the)h(same)g(t)n(yp)r(e)451 1535 y(is)f(used)f(in)h(all)f(transformations)f Fo(U)1592 1547 y Fl(k)1660 1535 y Ft(and)i Fo(U)1888 1505 y Fn(0)1879 1558 y Fl(k)1919 1535 y Fo(;)60 b(k)26 b Fm(\025)d Ft(1)p Fo(:)451 1768 y Fr(2)92 b(Pr)n(oof)31 b(of)h(Theorem)f(1.1)451 1953 y(2.1)91 b(Deriv)-7 b(a)h(tion)32 b(of)f(unit)-6 b(ar)g(y)32 b(transf)n(orma)-6 b(tions)451 2109 y Ft(The)30 b(sequence)g(of)g(unitary)g(op)r(erators)e Fo(U)1791 2121 y Fl(k)1873 2109 y Ft(=)f Fo(e)2004 2079 y Fl(iB)2077 2088 y Fe(k)2162 2109 y Ft(is)j(constructed)g(with)h(the)f(help)h(of) 451 2209 y(an)h(iteration)g(sc)n(heme.)50 b(F)-7 b(ollo)n(wing)32 b(Sob)r(olev)f([13)o(])i(w)n(e)f(consider)f Fo(U)2608 2221 y Fl(k)2681 2209 y Ft(as)h(an)g(elemen)n(t)g(of)451 2309 y(the)c(group)f Fo(U)887 2321 y Fl(k)927 2309 y Ft(\()p Fo(t)p Ft(\))38 b(=)22 b Fo(e)1185 2279 y Fl(iB)1258 2288 y Fe(k)1295 2279 y Fl(t)1324 2309 y Fo(;)60 b(t)23 b Fm(2)h Fk(R)p Fo(:)451 2410 y Ft(Let)37 b Fo(A)g Ft(b)r(e)g(an)f (arbitrary)f Fo(t)p Ft(-indep)r(enden)n(t)i(op)r(erator.)62 b(The)37 b(deriv)-5 b(ativ)n(e)35 b(of)i(the)g(trans-)451 2510 y(formed)27 b(op)r(erator)f(is)i(giv)n(en)f(b)n(y)878 2678 y Fo(d)p 863 2715 74 4 v 863 2791 a(dt)946 2734 y(A)p Ft(\()p Fo(t)p Ft(\))d(:=)1285 2678 y Fo(d)p 1270 2715 V 1270 2791 a(dt)1367 2667 y Fh(\000)1405 2734 y Fo(e)1444 2700 y Fn(\000)p Fl(iB)1569 2709 y Fe(k)1605 2700 y Fl(t)1649 2734 y Fo(A)14 b(e)1764 2700 y Fl(iB)1837 2709 y Fe(k)1873 2700 y Fl(t)1902 2667 y Fh(\001)1987 2734 y Ft(=)45 b Fo(i)14 b(U)2197 2746 y Fl(k)2237 2734 y Ft(\()p Fm(\000)p Fo(t)p Ft(\))24 b([)p Fo(A;)14 b(B)2605 2746 y Fl(k)2646 2734 y Ft(])23 b Fo(U)2749 2746 y Fl(k)2789 2734 y Ft(\()p Fo(t)p Ft(\))232 b(\(2.1\))451 2948 y(where)33 b(the)i(comm)n(utator)d([)p Fo(A;)14 b(B)1506 2960 y Fl(k)1547 2948 y Ft(])34 b(:=)f Fo(AB)1850 2960 y Fl(k)1913 2948 y Fm(\000)22 b Fo(B)2063 2960 y Fl(k)2104 2948 y Fo(A:)68 b Ft(This)33 b(equation)h(is)f(easily)g(in)n(te-)451 3048 y(grated,)27 b(noting)g(that)h Fo(A)p Ft(\(0\))23 b(=)g Fo(A;)630 3291 y(A)p Ft(\()p Fo(t)p Ft(\))47 b(=)e Fo(U)1000 3303 y Fl(k)1041 3291 y Ft(\()p Fm(\000)p Fo(t)p Ft(\))14 b Fo(A)23 b(U)1356 3303 y Fl(k)1397 3291 y Ft(\()p Fo(t)p Ft(\))46 b(=)g Fo(A)c Ft(+)f Fo(i)1901 3178 y Fh(Z)1983 3199 y Fl(t)1946 3367 y Fp(0)2026 3291 y Fo(d\034)33 b(U)2195 3303 y Fl(k)2236 3291 y Ft(\()p Fm(\000)p Fo(\034)9 b Ft(\))24 b([)p Fo(A;)14 b(B)2619 3303 y Fl(k)2660 3291 y Ft(])23 b Fo(U)2763 3303 y Fl(k)2803 3291 y Ft(\()p Fo(\034)9 b Ft(\))p Fo(:)180 b Ft(\(2.2\))451 3524 y(Iterating)31 b(once,)g(i.e.)47 b(replacing)30 b Fo(A)i Ft(b)n(y)e(the)i(op)r(erator) d([)p Fo(A;)14 b(B)2409 3536 y Fl(k)2450 3524 y Ft(])31 b(in)h(\(2.2\))f(and)f(inserting)451 3623 y(the)e(resulting)f(equation) g(in)n(to)h(the)g(r.h.s.)36 b(of)28 b(\(2.2\),)f(one)g(obtains)g(for)g Fo(t)c Ft(=)g(1)548 3869 y Fo(A)p Ft(\(1\))46 b(=)g Fo(A)33 b Ft(+)e Fo(i)p Ft([)p Fo(A;)14 b(B)1278 3881 y Fl(k)1319 3869 y Ft(])32 b(+)g Fo(i)1500 3834 y Fp(2)1551 3756 y Fh(Z)1634 3776 y Fp(1)1597 3944 y(0)1685 3869 y Fo(d\034)1788 3756 y Fh(Z)1871 3776 y Fl(\034)1834 3944 y Fp(0)1926 3869 y Fo(dt)1999 3834 y Fn(0)2036 3869 y Fo(U)2093 3881 y Fl(k)2134 3869 y Ft(\()p Fm(\000)p Fo(t)2261 3834 y Fn(0)2284 3869 y Ft(\))14 b([[)p Fo(A;)g(B)2538 3881 y Fl(k)2579 3869 y Ft(])p Fo(;)g(B)2702 3881 y Fl(k)2743 3869 y Ft(])g Fo(U)2837 3881 y Fl(k)2878 3869 y Ft(\()p Fo(t)2940 3834 y Fn(0)2963 3869 y Ft(\))p Fo(:)97 b Ft(\(2.3\))451 4097 y(After)28 b Fo(n)g Ft(iterations)e(the)i(follo)n(wing)f(represen) n(tation)f(of)h Fo(A)2308 4109 y Fp(1)2374 4097 y Ft(is)g(obtained,)451 4317 y Fo(A)p Ft(\(1\))37 b(=)g Fo(A)29 b Ft(+)g Fo(i)p Ft([)p Fo(A;)14 b(B)1157 4329 y Fl(k)1197 4317 y Ft(])39 b(+)1374 4261 y(1)p 1362 4298 65 4 v 1362 4374 a(2!)1437 4317 y Fo(i)1466 4283 y Fp(2)1526 4317 y Ft([[)p Fo(A;)14 b(B)1734 4329 y Fl(k)1775 4317 y Ft(])p Fo(;)g(B)1898 4329 y Fl(k)1938 4317 y Ft(])39 b(+)14 b Fo(:::)i Ft(+)2292 4261 y(1)p 2276 4298 73 4 v 2276 4374 a Fo(n)p Ft(!)2359 4317 y Fo(i)2388 4283 y Fl(n)2456 4317 y Ft([[)p Fo(:::)p Ft([)p Fo(A;)e(B)2756 4329 y Fl(k)2797 4317 y Ft(])p Fo(;)g(:::;)g(B)3026 4329 y Fl(k)3067 4317 y Ft(])39 b(+)28 b Fo(R)3115 4445 y Ft(\(2.4\))451 4544 y(where)e(the)g Fo(n)p Ft(-th)g(term)f(con)n(tains)g Fo(n)h Ft(comm)n(utators)e(with)j Fo(B)2359 4556 y Fl(k)2400 4544 y Fo(;)f Ft(and)f(the)i(remainder)d Fo(R)j Ft(is)451 4644 y(an)g(\()p Fo(n)19 b Ft(+)f(1\)-fold)27 b(in)n(tegral.)451 4745 y(Let)19 b(us)h(apply)e(this)i(sc)n(heme)f (inductiv)n(ely)g(to)g(the)g(Dirac)g(op)r(erator)e Fo(H)30 b Ft(=)23 b Fo(D)2783 4757 y Fp(0)2821 4745 y Ft(+)r Fo(V)5 b(:)42 b Ft(Assume)451 4845 y(that)f(to)f(order)f Fo(n)26 b Fm(\000)h Ft(1)40 b(the)g(transformation)f(has)g(b)r(een)i (ac)n(hiev)n(ed)e(with)i(a)e(resulting)451 4944 y(op)r(erator)26 b(of)i(the)g(form)f(giv)n(en)g(in)g(Theorem)g(1.1,)706 5130 y(\()p Fo(U)795 5142 y Fp(1)846 5130 y Fm(\001)14 b(\001)g(\001)f Fo(U)1013 5142 y Fl(n)p Fn(\000)p Fp(1)1143 5130 y Ft(\))1175 5096 y Fn(\000)p Fp(1)1288 5130 y Fo(H)30 b(U)1444 5142 y Fp(1)1494 5130 y Fm(\001)14 b(\001)g(\001)g Fo(U)1662 5142 y Fl(n)p Fn(\000)p Fp(1)1838 5130 y Ft(=)46 b Fo(H)2025 5096 y Fp(\()p Fl(n)p Fn(\000)p Fp(1\))2248 5130 y Ft(+)41 b Fo(H)2423 5142 y Fl(n)2510 5130 y Ft(+)2634 5109 y(~)2616 5130 y Fo(R)2680 5096 y Fp(\()p Fl(n)p Fp(+1\))3115 5130 y Ft(\(2.5\))1108 5338 y Fu(Document)l(a)25 b(Ma)l(thema)l(tica)h(10)f(\(2005\))g(331{356)p eop %%Page: 336 6 336 5 bop 451 518 a Ft(336)727 b Fr(D.)30 b(H.)h(Jakubassa-Amundsen)451 725 y Ft(where)g Fo(H)764 737 y Fl(n)841 725 y Ft(and)1025 704 y(~)1007 725 y Fo(R)1071 695 y Fp(\()p Fl(n)p Fp(+1\))1283 725 y Ft(are)g(resp)r(ectiv)n(ely)f(of)i(order)e Fo(\015)2250 695 y Fl(n)2327 725 y Ft(and)h Fo(\015)2540 695 y Fl(n)p Fp(+1)2669 725 y Fo(;)h Ft(and)f(still)h(couple)451 825 y(the)c(sp)r(ectral)f(subspaces.)36 b(Decomp)r(ose)27 b Fo(H)1822 837 y Fl(n)1895 825 y Ft(in)n(to)717 1008 y Fo(H)786 1020 y Fl(n)877 1008 y Ft(=)46 b Fo(V)1036 1020 y Fl(n)1123 1008 y Ft(+)41 b Fo(W)1307 1020 y Fl(n)1353 1008 y Fo(;)498 b(V)1922 1020 y Fl(n)1991 1008 y Ft(:=)45 b(\003)2182 1020 y Fp(+)2237 1008 y Fo(H)2306 1020 y Fl(n)2351 1008 y Ft(\003)2409 1020 y Fp(+)2506 1008 y Ft(+)c(\003)2670 1020 y Fn(\000)2725 1008 y Fo(H)2794 1020 y Fl(n)2840 1008 y Ft(\003)2898 1020 y Fn(\000)1874 1132 y Fo(W)1952 1144 y Fl(n)2021 1132 y Ft(:=)k(\003)2212 1144 y Fp(+)2267 1132 y Fo(H)2336 1144 y Fl(n)2381 1132 y Ft(\003)2439 1144 y Fn(\000)2536 1132 y Ft(+)d(\003)2701 1144 y Fn(\000)2756 1132 y Fo(H)2825 1144 y Fl(n)2870 1132 y Ft(\003)2928 1144 y Fp(+)2997 1132 y Fo(:)95 b Ft(\(2.6\))451 1315 y(The)29 b(next)g(transformation,)e Fo(U)1458 1327 y Fl(n)1527 1315 y Ft(=)d Fo(e)1655 1285 y Fl(iB)1728 1293 y Fe(n)1774 1315 y Fo(;)k Ft(aims)g(at)h(eliminating) f(the)h(term)g Fo(W)2979 1327 y Fl(n)3053 1315 y Ft(whic)n(h,)451 1415 y(in)j(con)n(trast)e(to)h Fo(V)1030 1427 y Fl(n)1076 1415 y Ft(,)h(couples)f(the)g(sp)r(ectral)g(subspaces.)47 b(This)32 b(condition)f(will)g(\014x)h Fo(B)3218 1427 y Fl(n)3263 1415 y Ft(.)451 1515 y(W)-7 b(e)21 b(note)f(that)g(from)g (\(2.4\),)h(the)g(transformation)d(repro)r(duces)h(the)i(op)r(erator)d (itself,)k(suc)n(h)451 1614 y(that)31 b(the)g(term)g Fo(H)1058 1584 y Fp(\()p Fl(n)p Fn(\000)p Fp(1\))1240 1614 y Ft(,)h(already)d(in)i(the)g(desired)f(form,)i(is)e(preserv)n (ed.)45 b(F)-7 b(rom)30 b(this)h(it)451 1714 y(follo)n(ws)d(that)h Fo(H)982 1684 y Fp(\()p Fl(n)p Fn(\000)p Fp(1\))1193 1714 y Ft(con)n(tains)f(the)h(zero-order)d(term)j(\003)2316 1726 y Fp(+)2371 1714 y Fo(D)2440 1726 y Fp(0)2477 1714 y Ft(\003)2535 1726 y Fp(+)2609 1714 y Ft(+)19 b(\003)2751 1726 y Fn(\000)2806 1714 y Fo(D)2875 1726 y Fp(0)2912 1714 y Ft(\003)2970 1726 y Fn(\000)3065 1714 y Ft(=)25 b Fo(D)3224 1726 y Fp(0)451 1814 y Ft(\(note)j(that)g(\003)906 1826 y Fn(\006)989 1814 y Ft(comm)n(utes)f(with)i Fo(D)1638 1826 y Fp(0)1702 1814 y Ft(and)f(\003)1922 1783 y Fp(2)1922 1834 y(+)1995 1814 y Ft(+)18 b(\003)2136 1783 y Fp(2)2136 1834 y Fn(\000)2215 1814 y Ft(=)k(1\).)451 1913 y(W)-7 b(e)28 b(obtain)674 2097 y Fo(U)740 2062 y Fn(\000)p Fp(1)731 2117 y Fl(n)843 2097 y Ft(\()p Fo(H)951 2062 y Fp(\()p Fl(n)p Fn(\000)p Fp(1\))1165 2097 y Ft(+)k Fo(H)1331 2109 y Fl(n)1376 2097 y Ft(\))24 b Fo(U)1489 2109 y Fl(n)1580 2097 y Ft(=)45 b Fo(H)1766 2062 y Fp(\()p Fl(n)p Fn(\000)p Fp(1\))1980 2097 y Ft(+)32 b Fo(V)2125 2109 y Fl(n)2203 2097 y Ft(+)g Fo(W)2378 2109 y Fl(n)2456 2097 y Ft(+)g Fo(i)p Ft([)p Fo(D)2674 2109 y Fp(0)2710 2097 y Fo(;)14 b(B)2810 2109 y Fl(n)2855 2097 y Ft(])237 b(\(2.7\))997 2359 y(+)23 b Fo(i)14 b Ft([\(\003)1241 2371 y Fp(+)1309 2256 y Fl(n)p Fn(\000)p Fp(1)1312 2280 y Fh(X)1312 2459 y Fl(k)q Fp(=1)1449 2359 y Fo(H)1518 2371 y Fl(k)1559 2359 y Ft(\003)1617 2371 y Fp(+)1704 2359 y Ft(+)32 b(\003)1859 2371 y Fn(\000)1928 2256 y Fl(n)p Fn(\000)p Fp(1)1931 2280 y Fh(X)1931 2459 y Fl(k)q Fp(=1)2068 2359 y Fo(H)2137 2371 y Fl(k)2178 2359 y Ft(\003)2236 2371 y Fn(\000)2292 2359 y Ft(\))p Fo(;)c(B)2438 2371 y Fl(n)2483 2359 y Ft(])41 b(+)2671 2338 y(~)2653 2359 y Fo(R)q(;)451 2603 y Ft(where)710 2582 y(~)692 2603 y Fo(R)30 b Ft(collects)e(the)h(terms)f(con)n(taining)g(at)h(least)f(t) n(w)n(o)g(comm)n(utators)f(with)i Fo(B)3069 2615 y Fl(n)3114 2603 y Ft(.)41 b Fo(B)3241 2615 y Fl(n)451 2702 y Ft(is)28 b(determined)f(from)h(the)g(requiremen)n(t)1452 2885 y Fo(W)1530 2897 y Fl(n)1618 2885 y Ft(+)41 b Fo(i)14 b Ft([)p Fo(D)1859 2897 y Fp(0)1895 2885 y Fo(;)g(B)1995 2897 y Fl(n)2040 2885 y Ft(])46 b(=)g(0)p Fo(:)830 b Ft(\(2.8\))451 3069 y(Since)41 b Fo(W)759 3081 y Fl(n)846 3069 y Ft(is)f(of)h(order)e Fo(\015)1328 3038 y Fl(n)1373 3069 y Fo(;)104 b(B)1563 3081 y Fl(n)1649 3069 y Ft(is)41 b(prop)r(ortional)e(to)h Fo(\015)2399 3038 y Fl(n)2530 3069 y Ft(\(the)h(b)r(oundedness)g(of)451 3168 y Fo(B)514 3180 y Fl(n)599 3168 y Ft(is)f(sho)n(wn)g(later\).)74 b(Moreo)n(v)n(er,)40 b(the)h(comm)n(utators)d(of)i(the)h(t)n(yp)r(e)f ([\(\003)2903 3180 y Fp(+)2958 3168 y Fo(H)3027 3180 y Fl(k)3068 3168 y Ft(\003)3126 3180 y Fp(+)3221 3168 y Ft(+)451 3268 y(\003)509 3280 y Fn(\000)565 3268 y Fo(H)634 3280 y Fl(k)675 3268 y Ft(\003)733 3280 y Fn(\000)788 3268 y Ft(\))p Fo(;)14 b(B)920 3280 y Fl(n)966 3268 y Ft(])31 b(are)e(of)i(order)e Fo(\015)1527 3238 y Fl(n)p Fp(+)p Fl(k)1690 3268 y Ft(with)i Fo(k)g Fm(\025)d Ft(1)p Fo(;)59 b Ft(and)2355 3247 y(~)2337 3268 y Fo(R)31 b Ft(is)g(of)g(order)e Fo(\015)2884 3238 y Fp(2)p Fl(n)2962 3268 y Fo(:)59 b Ft(Hence,)451 3367 y(these)33 b(terms)g(are)e (disregarded)g(\(together)h(with)i(the)f(remainder)2622 3346 y(~)2604 3367 y Fo(R)2668 3337 y Fp(\()p Fl(n)p Fp(+1\))2881 3367 y Ft(from)g(\(2.5\)\))451 3467 y(in)28 b(constructing)f(the)h(transformed)e(op)r(erator)g(to)i(order)e Fo(n)p Ft(,)900 3650 y Fo(H)976 3616 y Fp(\()p Fl(n)p Fp(\))1119 3650 y Ft(=)45 b Fo(H)1305 3616 y Fp(\()p Fl(n)p Fn(\000)p Fp(1\))1529 3650 y Ft(+)c Fo(V)1683 3662 y Fl(n)1775 3650 y Ft(=)k Fo(D)1954 3662 y Fp(0)2023 3650 y Ft(+)32 b Fo(V)2168 3662 y Fp(1)2238 3650 y Ft(+)g Fo(V)2383 3662 y Fp(2)2453 3650 y Ft(+)18 b Fo(:::)h Ft(+)32 b Fo(V)2769 3662 y Fl(n)2814 3650 y Fo(:)278 b Ft(\(2.9\))451 3833 y(P)n(articularly)25 b(in)n(teresting)i(are)g (the)h(cases)e Fo(n)d Ft(=)g(1)k(and)h Fo(n)22 b Ft(=)h(2)p Fo(:)50 b Ft(F)-7 b(or)27 b Fo(n)c Ft(=)g(1)p Fo(;)k Ft(w)n(e)g(ha)n(v)n(e)675 4016 y Fo(H)751 3982 y Fp(\(1\))886 4016 y Ft(=)46 b Fo(D)1066 4028 y Fp(0)1144 4016 y Ft(+)41 b Fo(V)1298 4028 y Fp(1)1382 4016 y Ft(=)46 b(\003)1551 4028 y Fp(+)1620 4016 y Ft(\()p Fo(D)1721 4028 y Fp(0)1776 4016 y Ft(+)18 b Fo(V)h Ft(\))14 b(\003)2030 4028 y Fp(+)2127 4016 y Ft(+)41 b(\003)2291 4028 y Fn(\000)2360 4016 y Ft(\()p Fo(D)2461 4028 y Fp(0)2517 4016 y Ft(+)18 b Fo(V)h Ft(\))14 b(\003)2771 4028 y Fn(\000)2827 4016 y Fo(:)224 b Ft(\(2.10\))451 4200 y(Restricting)26 b Fo(H)952 4169 y Fp(\(1\))1066 4200 y Ft(to)g(the)g(p)r(ositiv)n(e)g(sp)r(ectral)f (subspace)g(\003)2326 4212 y Fp(+)2381 4200 y Ft(\()p Fm(S)6 b Ft(\()p Fk(R)2556 4169 y Fp(3)2599 4200 y Ft(\))15 b Fm(\012)g Fk(C)2780 4169 y Fp(4)2823 4200 y Ft(\))p Fo(;)26 b Ft(the)h(second)451 4299 y(term)k(on)g(the)h(r.h.s.)48 b(of)31 b(\(2.10\))f(v)-5 b(anishes)31 b(and)g(the)h(remaining)e(term)h (agrees)f(with)i(the)451 4399 y(Bro)n(wn-Ra)n(v)n(enhall)25 b(op)r(erator.)451 4499 y(Let)31 b(us)f(no)n(w)g(consider)g Fo(n)e Ft(=)f(2)p Fo(:)58 b Ft(F)-7 b(rom)30 b(\(2.9\))h(it)g(follo)n (ws)e(that)i(the)g(transformed)e(Dirac)451 4598 y(op)r(erator)d(in)j (second)e(order)g(is)h(determined)g(b)n(y)f(the)i(\014rst)f (transformation,)e Fo(U)2965 4610 y Fp(1)3026 4598 y Ft(=)d Fo(e)3153 4568 y Fl(iB)3226 4576 y Fg(1)3263 4598 y Fo(;)451 4698 y Ft(only)-7 b(.)47 b(Ho)n(w)n(ev)n(er,)30 b(the)h(existence)f(of)h(the)h(second)e(transformation,)g Fo(U)2699 4710 y Fp(2)2765 4698 y Ft(=)e Fo(e)2897 4668 y Fl(iB)2970 4676 y Fg(2)3007 4698 y Fo(;)j Ft(has)f(to)451 4798 y(b)r(e)k(established)g(to)f(sho)n(w)g(that)h Fo(H)1583 4767 y Fp(\(2\))1706 4798 y Ft(is)f(indeed)h(the)g(transformed)f(op)r (erator,)g(with)h(a)451 4897 y(remainder)27 b(of)g(order)f Fo(\015)1203 4867 y Fp(3)1240 4897 y Fo(:)51 b Ft(W)-7 b(e)28 b(ha)n(v)n(e)492 5115 y Fo(U)558 5079 y Fn(\000)p Fp(1)549 5137 y(1)661 5115 y Fo(H)20 b(U)807 5127 y Fp(1)890 5115 y Ft(=)46 b Fo(D)1070 5127 y Fp(0)1125 5115 y Ft(+)18 b Fo(V)1256 5127 y Fp(1)1313 5115 y Ft(+)g Fo(W)1474 5127 y Fp(1)1544 5115 y Ft(+)31 b Fo(i)14 b Ft([)p Fo(D)1775 5127 y Fp(0)1812 5115 y Fo(;)g(B)1912 5127 y Fp(1)1949 5115 y Ft(])32 b(+)g Fo(i)14 b Ft([)p Fo(V)5 b(;)14 b(B)2320 5127 y Fp(1)2357 5115 y Ft(])32 b Fm(\000)2505 5059 y Ft(1)p 2505 5096 42 4 v 2505 5172 a(2)2557 5115 y([[)p Fo(D)2672 5127 y Fp(0)2709 5115 y Fo(;)14 b(B)2809 5127 y Fp(1)2846 5115 y Ft(])p Fo(;)g(B)2969 5127 y Fp(1)3006 5115 y Ft(])32 b(+)g Fo(R)q(;)1108 5338 y Fu(Document)l(a)25 b(Ma)l(thema)l(tica)h(10)f(\(2005\))g(331{356)p eop %%Page: 337 7 337 6 bop 1291 518 a Fr(Pr)n(ojected)32 b(Dira)n(c)f(Opera)-6 b(tor)716 b Ft(337)995 772 y Fo(R)47 b Ft(=)e Fm(\000)1294 659 y Fh(Z)1377 679 y Fp(1)1340 848 y(0)1428 772 y Fo(d\034)1531 659 y Fh(Z)1614 679 y Fl(\034)1577 848 y Fp(0)1669 772 y Fo(dt)1742 738 y Fn(0)1788 772 y Fo(U)1845 784 y Fp(1)1882 772 y Ft(\()p Fm(\000)p Fo(t)2009 738 y Fn(0)2033 772 y Ft(\))23 b([[)p Fo(V)5 b(;)14 b(B)2287 784 y Fp(1)2324 772 y Ft(])p Fo(;)g(B)2447 784 y Fp(1)2484 772 y Ft(])23 b Fo(U)2587 784 y Fp(1)2624 772 y Ft(\()p Fo(t)2686 738 y Fn(0)2710 772 y Ft(\))332 b(\(2.11\))797 1062 y Fm(\000)22 b Fo(i)927 949 y Fh(Z)1010 969 y Fp(1)973 1137 y(0)1061 1062 y Fo(d\034)1164 949 y Fh(Z)1247 969 y Fl(\034)1210 1137 y Fp(0)1302 1062 y Fo(dt)1375 1027 y Fn(0)1412 949 y Fh(Z)1495 969 y Fl(t)1520 944 y Ff(0)1458 1137 y Fp(0)1561 1062 y Fo(d\034)1649 1027 y Fn(0)1696 1062 y Fo(U)1753 1074 y Fp(1)1790 1062 y Ft(\()p Fm(\000)p Fo(\034)1932 1027 y Fn(0)1956 1062 y Ft(\))h([[[)p Fo(D)2149 1074 y Fp(0)2186 1062 y Fo(;)14 b(B)2286 1074 y Fp(1)2323 1062 y Ft(])p Fo(;)g(B)2446 1074 y Fp(1)2484 1062 y Ft(])p Fo(;)g(B)2607 1074 y Fp(1)2644 1062 y Ft(])23 b Fo(U)2747 1074 y Fp(1)2784 1062 y Ft(\()p Fo(\034)2861 1027 y Fn(0)2885 1062 y Ft(\))p Fo(:)451 1265 y Ft(Making)28 b(use)g(of)g(the)h (de\014ning)f(relation)g(for)g Fo(B)1946 1277 y Fp(1)1983 1265 y Fo(;)62 b(W)2146 1277 y Fp(1)2203 1265 y Ft(+)19 b Fo(i)p Ft([)p Fo(D)2408 1277 y Fp(0)2444 1265 y Fo(;)14 b(B)2544 1277 y Fp(1)2581 1265 y Ft(])25 b(=)f(0)p Fo(;)52 b Ft(the)29 b(op)r(erator)451 1365 y Fo(H)527 1335 y Fp(\(2\))644 1365 y Ft(tak)n(es)d(the)i(form)1099 1553 y Fo(H)1175 1518 y Fp(\(2\))1310 1553 y Ft(=)46 b Fo(D)1490 1565 y Fp(0)1545 1553 y Ft(+)18 b Fo(V)1676 1565 y Fp(1)1746 1553 y Ft(+)32 b(\003)1901 1565 y Fp(+)1956 1553 y Fo(H)2025 1565 y Fp(2)2062 1553 y Ft(\003)2120 1565 y Fp(+)2207 1553 y Ft(+)g(\003)2362 1565 y Fn(\000)2418 1553 y Fo(H)2487 1565 y Fp(2)2524 1553 y Ft(\003)2582 1565 y Fn(\000)3074 1553 y Ft(\(2.12\))1325 1788 y Fo(H)1394 1800 y Fp(2)1454 1788 y Ft(:=)46 b Fo(i)14 b Ft([)p Fo(V)1702 1800 y Fp(1)1739 1788 y Fo(;)g(B)1839 1800 y Fp(1)1876 1788 y Ft(])33 b(+)2045 1732 y Fo(i)p 2039 1769 42 4 v 2039 1845 a Ft(2)2090 1788 y([)p Fo(W)2191 1800 y Fp(1)2229 1788 y Fo(;)14 b(B)2329 1800 y Fp(1)2366 1788 y Ft(])p Fo(:)451 2040 y Fr(2.2)91 b(Integral)40 b(opera)-6 b(tors)41 b(in)f(F)n(ourier)g(sp) -6 b(a)n(ce)42 b(and)e(the)g(determina)-6 b(tion)661 2140 y(of)32 b Fo(B)853 2152 y Fp(1)451 2298 y Ft(Since)f Fo(D)740 2310 y Fp(0)807 2298 y Ft(is)g(a)f(m)n(ultiplication)h(op)r (erator)d(in)j(momen)n(tum)g(space,)f(it)h(is)g(con)n(v)n(enien)n(t)e (to)451 2397 y(set)23 b(up)f(the)h(calculus)f(in)h(F)-7 b(ourier)21 b(space.)35 b(Let)23 b Fo(')g Fm(2)g(S)6 b Ft(\()p Fk(R)2206 2367 y Fp(3)2249 2397 y Ft(\))i Fm(\012)g Fk(C)2416 2367 y Fp(4)2460 2397 y Fo(:)45 b Ft(W)-7 b(e)23 b(de\014ne)g(an)f(in)n(tegral)451 2497 y(op)r(erator)k Fo(W)40 b Ft(acting)27 b(on)g Fo(')h Ft(b)n(y)f(means)g(of)1044 2723 y(\()p Fo(W)f(')p Ft(\)\()p Fi(x)p Ft(\))f(:=)1658 2666 y(1)p 1548 2703 261 4 v 1548 2781 a(\(2)p Fo(\031)s Ft(\))1704 2757 y Fp(3)p Fl(=)p Fp(2)1856 2610 y Fh(Z)1953 2723 y Fo(d)p Fi(p)e Fo(e)2111 2688 y Fl(i)p Fc(p)n(x)2232 2723 y Fo(w)r Ft(\()p Fi(x)p Fo(;)14 b Fi(p)p Ft(\))38 b(^)-56 b Fo(')q Ft(\()p Fi(p)p Ft(\))381 b(\(2.13\))451 2960 y(where)32 b(here)g(and)g(in)g(the)h(follo)n(wing)e(the)i (\(three-dimensional\))f(momen)n(tum)g(in)n(tegrals)451 3060 y(extend)26 b(o)n(v)n(er)d(the)j(whole)f Fk(R)1322 3030 y Fp(3)1365 3060 y Fo(:)h Ft(This)f(agrees)e(with)j(the)g(formal)e (de\014nition)i(of)f(a)g(pseudo)r(d-)451 3160 y(i\013eren)n(tial)i(op)r (erator)e([13)o(])i(and)g(w)n(e)g(will)g(call)f Fo(w)r Ft(\()p Fi(x)p Fo(;)14 b Fi(p)p Ft(\))29 b(the)e(sym)n(b)r(ol)g(of)g Fo(W)12 b Ft(.)37 b(In)n(tro)r(ducing)451 3259 y(the)28 b(F)-7 b(ourier)27 b(transform)43 b(^)-59 b Fo(w)s Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))p Fo(;)61 b(W)12 b(')28 b Ft(tak)n(es)e(the)i(form)757 3490 y(\()p Fo(W)e(')p Ft(\)\()p Fi(x)p Ft(\))48 b(=)1338 3433 y(1)p 1262 3470 194 4 v 1262 3546 a(\(2)p Fo(\031)s Ft(\))1418 3522 y Fp(3)1502 3377 y Fh(Z)1599 3490 y Fo(d)p Fi(p)23 b Fo(e)1757 3455 y Fl(i)p Fc(p)n(x)1901 3377 y Fh(Z)1998 3490 y Fo(d)p Fi(q)g Fo(e)2153 3455 y Fl(i)p Fc(qx)2300 3490 y Ft(^)-59 b Fo(w)s Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))37 b(^)-55 b Fo(')p Ft(\()p Fi(p)p Ft(\))p Fo(:)307 b Ft(\(2.14\))451 3721 y(F)-7 b(rom)27 b(this,)h(the)g(F)-7 b(ourier)27 b(transform)f(of)i Fo(W)12 b(')28 b Ft(is)f(found)1016 3947 y(\()1060 3926 y Fh(d)1048 3947 y Fo(W)12 b(')p Ft(\)\()p Fi(p)1309 3913 y Fn(0)1333 3947 y Ft(\))47 b(=)1642 3891 y(1)p 1532 3928 261 4 v 1532 4005 a(\(2)p Fo(\031)s Ft(\))1688 3981 y Fp(3)p Fl(=)p Fp(2)1840 3834 y Fh(Z)1937 3947 y Fo(d)p Fi(p)40 b Ft(^)-59 b Fo(w)s Ft(\()p Fi(p)2203 3913 y Fn(0)2245 3947 y Fm(\000)18 b Fi(p)p Fo(;)c Fi(p)p Ft(\))37 b(^)-56 b Fo(')q Ft(\()p Fi(p)p Ft(\))p Fo(:)353 b Ft(\(2.15\))451 4185 y(With)27 b Fo(')g Ft(in)f(\(2.14\))f(replaced)g(b)n(y)h Fo(G')p Ft(,)h(the)f(sym)n(b)r(ol)g(of)g(a)g(pro)r(duct)g Fo(W)12 b(G)26 b Ft(of)g(t)n(w)n(o)f(in)n(tegral)451 4284 y(op)r(erators)h(is)h (deriv)n(ed,)658 4510 y(\()p Fo(W)g(G)14 b(')p Ft(\)\()p Fi(x)p Ft(\))38 b(=)1309 4454 y(1)p 1233 4491 194 4 v 1233 4567 a(\(2)p Fo(\031)s Ft(\))1389 4543 y Fp(3)1473 4397 y Fh(Z)1570 4510 y Fo(d)p Fi(p)1666 4476 y Fn(0)1713 4510 y Fo(e)1752 4476 y Fl(i)p Fc(p)1817 4451 y Ff(0)1839 4476 y Fc(x)1920 4397 y Fh(Z)2017 4510 y Fo(d)p Fi(p)i Ft(^)-59 b Fo(w)s Ft(\()p Fi(p)2283 4476 y Fn(0)2325 4510 y Fm(\000)18 b Fi(p)p Fo(;)c Fi(p)p Ft(\))2624 4489 y Fh(c)2606 4510 y Fo(G')q Ft(\()p Fi(p)p Ft(\))p Fo(:)208 b Ft(\(2.16\))451 4767 y(Using)34 b(\(2.15\))f(for)1089 4746 y Fh(c)1071 4767 y Fo(G')q Ft(\()p Fi(p)p Ft(\))p Fo(;)h Ft(as)g(w)n(ell)g(as)f(the)i(de\014nition)f(\(2.14\))f(of)h(the) h(F)-7 b(ourier)33 b(trans-)451 4867 y(formed)27 b(sym)n(b)r(ol)38 b Fh(c)-93 b Fo(w)r(g)31 b Ft(of)c Fo(W)12 b(G)p Ft(,)28 b(one)f(gets)901 5093 y Fh(c)-93 b Fo(w)r(g)s Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))38 b(=)1467 5036 y(1)p 1357 5073 261 4 v 1357 5151 a(\(2)p Fo(\031)s Ft(\))1513 5127 y Fp(3)p Fl(=)p Fp(2)1665 4980 y Fh(Z)1762 5093 y Fo(d)p Fi(p)1858 5058 y Fn(0)1922 5093 y Ft(^)-59 b Fo(w)r Ft(\()p Fi(q)19 b Fm(\000)f Fi(p)2203 5058 y Fn(0)2227 5093 y Fo(;)c Fi(p)k Ft(+)g Fi(p)2471 5058 y Fn(0)2494 5093 y Ft(\))27 b(^)-45 b Fo(g)r Ft(\()p Fi(p)2677 5058 y Fn(0)2701 5093 y Fo(;)14 b Fi(p)p Ft(\))p Fo(:)228 b Ft(\(2.17\))1108 5338 y Fu(Document)l(a)25 b(Ma)l(thema)l(tica)h(10)f (\(2005\))g(331{356)p eop %%Page: 338 8 338 7 bop 451 518 a Ft(338)727 b Fr(D.)30 b(H.)h(Jakubassa-Amundsen)451 725 y Ft(F)-7 b(or)18 b(the)h(goal)f(of)g(determining)h(the)g (transformation)e Fo(U)2178 737 y Fp(1)2215 725 y Ft(,)j(its)f(exp)r (onen)n(t)f Fo(B)2776 737 y Fp(1)2832 725 y Ft(is)h(considered)451 825 y(as)i(an)g(in)n(tegral)g(op)r(erator.)33 b(One)21 b(has)g(to)h(solv)n(e)e(\(2.8\))i(for)f Fo(n)i Ft(=)f(1,)h(using)e Fo(W)2773 837 y Fp(1)2834 825 y Ft(=)h(\003)2979 837 y Fp(+)3034 825 y Fo(V)d Ft(\003)3159 837 y Fn(\000)3221 825 y Ft(+)451 924 y(\003)509 936 y Fn(\000)565 924 y Fo(V)g Ft(\003)690 936 y Fp(+)772 924 y Ft(and)28 b(\(1.2\),)996 1154 y Fm(\000)p Fo(i)22 b Ft([)p Fo(D)1204 1166 y Fp(0)1241 1154 y Fo(;)14 b(B)1341 1166 y Fp(1)1378 1154 y Ft(])46 b(=)g Fo(W)1636 1166 y Fp(1)1720 1154 y Ft(=)1841 1098 y(1)p 1841 1135 42 4 v 1841 1211 a(2)1929 1037 y Fh(\022)1990 1154 y Fo(V)60 b Fm(\000)2238 1098 y Fo(D)2307 1110 y Fp(0)p 2214 1135 153 4 v 2214 1211 a Fm(j)p Fo(D)2306 1223 y Fp(0)2344 1211 y Fm(j)2390 1154 y Fo(V)2504 1098 y(D)2573 1110 y Fp(0)p 2481 1135 V 2481 1211 a Fm(j)p Fo(D)2573 1223 y Fp(0)2610 1211 y Fm(j)2643 1037 y Fh(\023)2718 1154 y Fo(:)333 b Ft(\(2.18\))451 1383 y(Let)28 b Fo(\036)649 1395 y Fp(1)714 1383 y Ft(b)r(e)g(the)g(sym)n(b)r(ol)f(of)h Fo(B)1412 1395 y Fp(1)1449 1383 y Ft(.)37 b(F)-7 b(rom)27 b(\(2.14\))g(and)h(with)g Fo(D)2387 1395 y Fp(0)2451 1383 y Ft(from)g(\(1.1\))f(one)g(has)805 1605 y(\()p Fo(D)906 1617 y Fp(0)957 1605 y Fo(B)1020 1617 y Fp(1)1071 1605 y Fo(')p Ft(\)\()p Fi(x)p Ft(\))38 b(=)1506 1549 y(1)p 1430 1586 194 4 v 1430 1662 a(\(2)p Fo(\031)s Ft(\))1586 1638 y Fp(3)1670 1492 y Fh(Z)1767 1605 y Fo(d)p Fi(p)24 b Fo(d)p Fi(q)f Fo(D)2072 1617 y Fp(0)2132 1605 y Fo(e)2171 1571 y Fl(i)p Fp(\()p Fc(p)p Fp(+)p Fc(q)p Fp(\))p Fc(x)2457 1584 y Ft(^)2446 1605 y Fo(\036)2495 1617 y Fp(1)2532 1605 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))38 b(^)-56 b Fo(')q Ft(\()p Fi(p)p Ft(\))672 1884 y(=)868 1828 y(1)p 792 1865 V 792 1941 a(\(2)p Fo(\031)s Ft(\))948 1917 y Fp(3)1033 1771 y Fh(Z)1130 1884 y Fo(d)p Fi(p)23 b Fo(d)p Fi(q)14 b Ft([)p Fj(\013)q Ft(\()p Fi(p)k Ft(+)h Fi(q)p Ft(\))42 b(+)f Fo(\014)t(m)p Ft(])23 b Fo(e)2069 1850 y Fl(i)p Fp(\()p Fc(p)p Fp(+)p Fc(q)p Fp(\))p Fc(x)2354 1863 y Ft(^)2343 1884 y Fo(\036)2392 1896 y Fp(1)2430 1884 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))37 b(^)-55 b Fo(')p Ft(\()p Fi(p)p Ft(\))p Fo(:)222 b Ft(\(2.19\))451 2076 y(The)28 b(F)-7 b(ourier)26 b(transforms)h(of)g Fo(D)1487 2088 y Fp(0)1524 2076 y Fo(')h Ft(and)g(of)f Fo(V)19 b(')28 b Ft(are,)f(resp)r(ectiv)n(ely)-7 b(,)26 b(obtained)i(from)1300 2256 y(\()1352 2235 y Fh(d)1332 2256 y Fo(D)1401 2268 y Fp(0)1438 2256 y Fo(')p Ft(\)\()p Fi(p)p Ft(\))38 b(=)46 b(\()p Fj(\013)p Fi(p)c Ft(+)f Fo(\014)t(m)p Ft(\))c(^)-56 b Fo(')q Ft(\()p Fi(p)p Ft(\))671 2485 y(\()p Fo(V)33 b(')p Ft(\)\()p Fi(x)p Ft(\))39 b(=)1253 2429 y(1)p 1143 2466 261 4 v 1143 2544 a(\(2)p Fo(\031)s Ft(\))1299 2520 y Fp(3)p Fl(=)p Fp(2)1451 2372 y Fh(Z)1548 2485 y Fo(d)p Fi(q)23 b Fo(e)1703 2451 y Fl(i)p Fc(qx)1847 2368 y Fh(\022)1908 2485 y Fm(\000)2062 2429 y Fo(\015)p 1983 2466 207 4 v 1983 2542 a Ft(2)p Fo(\031)2075 2518 y Fp(2)2112 2542 y Fo(q)2152 2518 y Fp(2)2199 2368 y Fh(\023)2297 2372 y(Z)2394 2485 y Fo(d)p Fi(p)g Fo(e)2552 2451 y Fl(i)p Fc(p)n(x)2695 2485 y Ft(^)-55 b Fo(')p Ft(\()p Fi(p)p Ft(\))221 b(\(2.20\))451 2698 y(suc)n(h)38 b(that)h(the)g(sym)n(b)r(ol)f Fo(v)j Ft(of)e Fo(V)57 b Ft(is)38 b(de\014ned)h(b)n(y)j(^)-46 b Fo(v)t Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))55 b(=)41 b Fm(\000)2572 2628 y Fh(p)p 2655 2628 134 4 v 70 x Ft(2)p Fo(=\031)16 b(\015)5 b(=q)2932 2668 y Fp(2)2968 2698 y Fo(:)53 b Ft(Acting)451 2798 y(\(2.18\))21 b(on)g Fo(')h Ft(and)f(equating)g(the) g(resp)r(ectiv)n(e)g(sym)n(b)r(ols)g(leads)f(to)i(the)f(follo)n(wing)g (algebraic)451 2907 y(equation)27 b(for)928 2885 y(^)917 2907 y Fo(\036)966 2919 y Fp(1)1027 2907 y Ft(:)593 3086 y([)p Fj(\013)p Ft(\()p Fi(p)19 b Ft(+)f Fi(q)p Ft(\))42 b(+)f Fo(\014)t(m)p Ft(])1277 3065 y(^)1266 3086 y Fo(\036)1315 3098 y Fp(1)1353 3086 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))42 b Fm(\000)1716 3065 y Ft(^)1706 3086 y Fo(\036)1755 3098 y Fp(1)1792 3086 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))24 b([)p Fj(\013)p Fi(p)33 b Ft(+)18 b Fo(\014)t(m)p Ft(])46 b(=)g Fo(i)40 b Ft(^)-59 b Fo(w)2690 3098 y Fp(1)2727 3086 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))143 b(\(2.21\))1267 3306 y(=)46 b Fm(\000)1453 3250 y Fo(i\015)1525 3262 y Fp(0)p 1452 3287 109 4 v 1468 3363 a Fo(q)1508 3339 y Fp(2)1594 3306 y Ft([1)32 b Fm(\000)1807 3285 y Ft(~)1788 3306 y Fo(D)1857 3318 y Fp(0)1894 3306 y Ft(\()p Fi(q)19 b Ft(+)f Fi(p)p Ft(\))h Fm(\001)2243 3285 y Ft(~)2223 3306 y Fo(D)2292 3318 y Fp(0)2329 3306 y Ft(\()p Fi(p)p Ft(\)])451 3512 y(with)43 b Fo(\015)698 3524 y Fp(0)783 3512 y Ft(:=)61 b Fo(\015)5 b(=)1022 3444 y Fm(p)p 1090 3444 92 4 v 1090 3512 a Ft(2)p Fo(\031)93 b Ft(and)1467 3491 y(~)1448 3512 y Fo(D)1517 3524 y Fp(0)1554 3512 y Ft(\()p Fi(p)p Ft(\))43 b(the)g(op)r(erator)d(from)i(\(1.3\))g(with)h (norm)f(unit)n(y)-7 b(.)468 3612 y(^)-59 b Fo(w)510 3624 y Fp(1)548 3612 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))p Fo(;)85 b Ft(b)r(eha)n(ving)40 b(lik)n(e)g Fo(q)1431 3582 y Fn(\000)p Fp(1)1560 3612 y Ft(for)g Fo(q)47 b Fm(!)d Ft(0)p Fo(;)c Ft(is)g(less)g(singular)f(than)44 b(^)-45 b Fo(v)s Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\),)44 b(suc)n(h)451 3712 y(that)28 b(the)g(prescription)f(\(2.6\))g(for)g Fo(W)1635 3724 y Fp(1)1700 3712 y Ft(implies)h(a)f(regularisation)e(of) j(the)g(p)r(oten)n(tial)g Fo(V)18 b Ft(.)451 3884 y Fr(Lemma)31 b Ft(2.1)p Fr(.)41 b Fd(A)29 b(solution)1351 3862 y Ft(^)1340 3884 y Fo(\036)1389 3896 y Fp(1)1427 3884 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))30 b Fd(for)h(the)f(symb)l(ol)g(of)h Fo(B)2362 3896 y Fp(1)2429 3884 y Fd(is)f(given)g(by)780 4082 y Ft(^)769 4104 y Fo(\036)818 4116 y Fp(1)855 4104 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))47 b(=)f Fm(\000)1292 4048 y Fo(i\015)1364 4060 y Fp(0)p 1292 4085 109 4 v 1307 4161 a Fo(q)1347 4137 y Fp(2)1642 4048 y Ft(1)p 1444 4085 439 4 v 1444 4161 a Fo(E)1505 4173 y Fl(p)1562 4161 y Ft(+)18 b Fo(E)1706 4176 y Fn(j)p Fc(q)p Fp(+)p Fc(p)p Fn(j)1929 4012 y Fh(\020)1998 4083 y Ft(~)1978 4104 y Fo(D)2047 4116 y Fp(0)2084 4104 y Ft(\()p Fi(q)h Ft(+)f Fi(p)p Ft(\))33 b Fm(\000)2502 4083 y Ft(~)2483 4104 y Fo(D)2552 4116 y Fp(0)2589 4104 y Ft(\()p Fi(p)p Ft(\))2706 4012 y Fh(\021)3074 4104 y Ft(\(2.22\))451 4341 y Fd(which)f(satis\014es)d(the)h(c)l(ondition)h(for)g(symmetry)f (of)g Fo(B)2171 4353 y Fp(1)2238 4341 y Fd([13)q(],)1371 4508 y Ft(^)1360 4530 y Fo(\036)1409 4542 y Fp(1)1446 4530 y Ft(\()p Fm(\000)p Fi(q)p Fo(;)14 b Fi(p)19 b Ft(+)f Fi(q)p Ft(\))1867 4496 y Fn(\003)1952 4530 y Ft(=)2074 4508 y(^)2063 4530 y Fo(\036)2112 4542 y Fp(1)2149 4530 y Ft(\()p Fi(q)p Fo(;)c Fi(p)p Ft(\))p Fo(:)698 b Ft(\(2.23\))451 4710 y Fd(It)29 b(is)i(estimate)l(d)f(by)1356 4844 y Fm(j)1390 4822 y Ft(^)1379 4844 y Fo(\036)1428 4856 y Fp(1)1465 4844 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))p Fm(j)48 b(\024)1862 4788 y Fo(c)p 1860 4825 41 4 v 1860 4901 a(q)2137 4788 y Ft(1)p 1943 4825 429 4 v 1943 4901 a(\()p Fo(q)22 b Ft(+)c Fo(p)g Ft(+)g(1\))2334 4877 y Fp(2)3074 4844 y Ft(\(2.24\))451 5045 y Fd(with)31 b(some)f(c)l (onstant)f Fo(c)23 b Fm(2)g Fk(R)1366 5057 y Fp(+)1427 5045 y Fo(:)46 b(B)1559 5057 y Fp(1)1626 5045 y Fd(is)30 b(a)g(b)l(ounde)l(d)h(op)l(er)l(ator)f(on)g Fo(L)2606 5057 y Fp(2)2643 5045 y Ft(\()p Fk(R)2729 5015 y Fp(3)2772 5045 y Ft(\))19 b Fm(\012)f Fk(C)2960 5015 y Fp(4)3003 5045 y Fo(:)1108 5338 y Fu(Document)l(a)25 b(Ma)l(thema)l(tica)h(10)f (\(2005\))g(331{356)p eop %%Page: 339 9 339 8 bop 1291 518 a Fr(Pr)n(ojected)32 b(Dira)n(c)f(Opera)-6 b(tor)716 b Ft(339)451 725 y Fd(Pr)l(o)l(of.)136 b Ft(a\))41 b(Calculation)27 b(of)1467 703 y(^)1456 725 y Fo(\036)1505 737 y Fp(1)1543 725 y Fo(:)659 858 y Ft(In)h(order)e(to)h(solv)n(e)g (\(2.21\))f(the)i(ansatz)f(is)h(made)1221 1058 y(^)1210 1079 y Fo(\036)1259 1091 y Fp(1)1296 1079 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))47 b(=)f Fm(\000)1733 1023 y Fo(i\015)1805 1035 y Fp(0)p 1733 1060 109 4 v 1748 1136 a Fo(q)1788 1112 y Fp(2)1875 1079 y Ft(\()p Fo(c)1943 1091 y Fp(1)1980 1079 y Fj(\013)p Fi(q)c Ft(+)f Fo(c)2277 1091 y Fp(2)2314 1079 y Fj(\013)p Fi(p)h Ft(+)f Fo(c)2614 1091 y Fp(3)2651 1079 y Fo(\014)t Ft(\))340 b(\(2.25\))659 1310 y(and)33 b(from)g(the)h(prop)r(erties)f(of)g(the)h(Dirac)f(matrices)g Fo(\014)2444 1280 y Fp(2)2515 1310 y Ft(=)f(1)p Fo(;)47 b(\013)2777 1280 y Fp(2)2777 1332 y Fl(i)2847 1310 y Ft(=)33 b(1)p Fo(;)46 b(\014)t(\013)3160 1322 y Fl(i)3221 1310 y Ft(=)659 1410 y Fm(\000)p Fo(\013)777 1422 y Fl(i)804 1410 y Fo(\014)t(;)g(i)31 b Ft(=)h(1)p Fo(;)14 b Ft(2)p Fo(;)g Ft(3)p Fo(;)75 b(\013)1432 1422 y Fl(i)1460 1410 y Fo(\013)1513 1422 y Fl(k)1586 1410 y Ft(=)31 b Fm(\000)p Fo(\013)1800 1422 y Fl(k)1841 1410 y Fo(\013)1894 1422 y Fl(i)1953 1410 y Ft(\()p Fo(i)h Fm(6)p Ft(=)f Fo(k)s Ft(\))p Fo(;)65 b Ft(the)33 b(follo)n(wing)f(iden)n(tities)h(are)659 1509 y(deriv)n(ed)890 1691 y Fj(\013)q Fi(p)18 b Fm(\001)h Fj(\013)p Fi(p)46 b Ft(=)g Fo(p)1382 1657 y Fp(2)1419 1691 y Fo(;)346 b Fj(\013)p Fi(q)19 b Fm(\001)f Fj(\013)p Fi(p)46 b Ft(=)g(2)p Fi(p)s(q)41 b Fm(\000)g Fj(\013)p Fi(p)19 b Fm(\001)f Fj(\013)q Fi(q)p Fo(:)232 b Ft(\(2.26\))659 1872 y(Insertion)27 b(of)g(\(2.25\))g(in)n(to)g(\(2.21\))g(then)h (leads)f(to)h(an)f(equation)g(of)h(the)g(t)n(yp)r(e)969 2054 y Fo(\025)1017 2066 y Fp(1)1077 2054 y Fj(\013)q Fi(p)18 b Fm(\001)h Fj(\013)p Fi(q)42 b Ft(+)f Fo(\025)1563 2066 y Fp(2)1623 2054 y Fj(\013)q Fi(q)18 b Fm(\001)h Fo(\014)46 b Ft(+)41 b Fo(\025)2044 2066 y Fp(3)2105 2054 y Fj(\013)p Fi(p)18 b Fm(\001)h Fo(\014)46 b Ft(+)41 b Fo(\025)2528 2066 y Fp(4)2611 2054 y Ft(=)46 b(0)310 b(\(2.27\))659 2235 y(where)33 b(the)g Fo(\025)1101 2247 y Fl(k)1143 2235 y Fo(;)79 b(k)35 b Ft(=)e(1)p Fo(;)14 b(:::;)g Ft(4)p Fo(;)65 b Ft(are)32 b(scalars)g(dep)r(ending)h(on)g Fi(p)h Ft(and)f Fi(q)p Fo(:)67 b Ft(\(2.27\))659 2335 y(m)n(ust)36 b(hold)g(for)f Fi(p)p Fo(;)14 b Fi(q)38 b Fm(2)g Fk(R)1526 2305 y Fp(3)1605 2335 y Ft(whence)e Fo(\025)1952 2347 y Fl(k)2030 2335 y Ft(=)h(0)p Fo(;)88 b(k)40 b Ft(=)d(1)p Fo(;)14 b(:::;)g Ft(4)p Fo(:)72 b Ft(The)36 b(resulting)659 2434 y(system)27 b(of)h(4)f(equations)g(for)g (the)h Fo(c)1776 2446 y Fl(i)1803 2434 y Fo(;)60 b(i)23 b Ft(=)g(1)p Fo(;)14 b Ft(2)p Fo(;)g Ft(3)49 b(has)27 b(a)g(unique)h(solution,)659 2654 y Fo(c)695 2666 y Fp(1)755 2654 y Ft(\()p Fo(q)827 2619 y Fp(2)865 2654 y Ft(+2)p Fi(p)s(q)p Ft(\))46 b(=)g(1)23 b Fm(\000)1499 2597 y Fo(E)1560 2609 y Fl(p)p 1430 2634 238 4 v 1430 2711 a Fo(E)1491 2726 y Fn(j)p Fc(q)p Fp(+)p Fc(p)p Fn(j)1677 2654 y Fo(;)180 b(c)1916 2666 y Fp(2)1999 2654 y Ft(=)46 b(2)p Fo(c)2188 2666 y Fp(1)2248 2654 y Fm(\000)2501 2597 y Ft(1)p 2346 2634 351 4 v 2346 2711 a Fo(E)2407 2723 y Fl(p)2460 2711 y Fo(E)2521 2726 y Fn(j)p Fc(q)p Fp(+)p Fc(p)p Fn(j)2707 2654 y Fo(;)180 b(c)2946 2666 y Fp(3)3029 2654 y Ft(=)46 b Fo(c)3176 2666 y Fp(2)3213 2654 y Fo(m)3074 2811 y Ft(\(2.28\))659 2910 y(suc)n(h)27 b(that)710 3099 y(^)699 3121 y Fo(\036)748 3133 y Fp(1)786 3121 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))9 b(=)g Fm(\000)1148 3065 y Fo(i\015)1220 3077 y Fp(0)p 1149 3102 109 4 v 1164 3178 a Fo(q)1204 3154 y Fp(2)1267 3004 y Fh(\024)1311 3121 y Ft([\()p Fi(q)c Ft(+)g(2)p Fi(p)p Ft(\))p Fj(\013)t Ft(+)g(2)p Fo(\014)t(m)p Ft(])2068 3065 y(1)p 1954 3102 272 4 v 1954 3178 a Fo(q)1994 3154 y Fp(2)2021 3178 y Ft(+)-9 b(2)p Fi(p)s(q)2234 3004 y Fh(\022)2281 3121 y Ft(1)5 b Fm(\000)2476 3065 y Fo(E)2537 3077 y Fl(p)p 2407 3102 238 4 v 2407 3178 a Fo(E)2468 3193 y Fn(j)p Fc(q)p Fp(+)p Fc(p)p Fn(j)2654 3004 y Fh(\023)2720 3121 y Fm(\000)2813 3065 y Fi(p)p Fj(\013)18 b Ft(+)g Fo(\014)t(m)p 2813 3102 342 4 v 2815 3178 a(E)2876 3190 y Fl(p)2915 3178 y Fo(E)2976 3193 y Fn(j)p Fc(q)p Fp(+)p Fc(p)p Fn(j)3165 3004 y Fh(\025)3222 3121 y Fo(:)3074 3278 y Ft(\(2.29\))659 3377 y(It)27 b(is)f(readily)g(v)n(eri\014ed)g (that)h(\(2.29\))f(can)h(b)r(e)g(cast)f(in)n(to)h(the)g(form)f (\(2.22\),)g(pro)n(ving)659 3486 y(that)850 3464 y(^)839 3486 y Fo(\036)888 3498 y Fp(1)925 3486 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))29 b(is)e(con)n(tin)n(uous)g(in)h(b)r(oth)g(v)-5 b(ariables)26 b(except)i(for)f Fo(q)f Ft(=)d(0)p Fo(:)539 3652 y Ft(b\))42 b(The)22 b(symmetry)f(condition)h(\(2.23\))g(follo)n (ws)f(immediately)h(from)g(\(2.22\))f(using)h(the)659 3751 y(self-adjoin)n(tness)k(of)i Fj(\013)g Ft(and)f Fo(\014)t Ft(.)548 3917 y(c\))42 b(W)-7 b(e)32 b(de\014ne)f(the)h (class)f(of)g(our)g(in)n(tegral)f(op)r(erators)f(\(2.14\))i(b)n(y)g (means)g(of)h(the)g(es-)659 4016 y(timate)h(of)f(their)h(sym)n(b)r(ols) e(in)i(the)g(six-dimensional)e(space)h(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))32 b Fm(2)g Fk(R)2994 3986 y Fp(6)3037 4016 y Fo(:)64 b Ft(This)659 4116 y(estimate)31 b(determines)g(the)g (con)n(v)n(ergence)e(prop)r(erties)h(of)h(in)n(tegrals)e(without)j(the) 659 4216 y(precise)20 b(kno)n(wledge)g(of)h(the)g(sym)n(b)r(ols)f (themselv)n(es,)i(and)f(it)h(is)f(an)f(easy)h(w)n(a)n(y)e(to)i(deal)659 4315 y(with)34 b(pro)r(ducts)f(of)h(in)n(tegral)e(op)r(erators)g(in)i (pro)r(ofs)f(of)g(b)r(oundedness)h(or)f Fo(p)p Ft(-form)659 4415 y(b)r(oundedness.)659 4514 y(In)f(order)e(to)i(estimate)f(a)g(sym) n(b)r(ol)h(b)n(y)f(its)h(asymptotic)f(b)r(eha)n(viour)g(for)g Fo(q)s(;)14 b(p)29 b Fm(!)h Ft(0)659 4614 y(and)36 b Fo(q)s(;)14 b(p)37 b Fm(!)g(1)p Fo(;)74 b Ft(it)37 b(m)n(ust)f(b)r(e)h (a)f(con)n(tin)n(uous)f(function)i(of)f(the)h(t)n(w)n(o)e(v)-5 b(ariables)659 4714 y(in)37 b Fk(R)819 4726 y Fp(+)906 4714 y Fm(\002)24 b Fk(R)1049 4726 y Fp(+)1110 4714 y Fo(:)38 b Ft(This)f(condition)h(is)f(ful\014lled)h(for)2293 4692 y(^)2282 4714 y Fo(\036)2331 4726 y Fp(1)2369 4714 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))p Fo(:)78 b Ft(By)37 b(insp)r(ection)h(of)659 4822 y(\(2.22\))e(one)g(\014nds)h(that)1477 4800 y(^)1466 4822 y Fo(\036)1515 4834 y Fp(1)1552 4822 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))38 b(is)f(\014nite)g Fm(6)p Ft(=)h(0)e(for)g Fo(p)i Ft(=)g(0)e(while)h(it)g(b)r(eha)n(v)n (es)659 4922 y Fm(\030)42 b Ft(1)p Fo(=q)s(;)99 b(q)45 b Fm(!)e Ft(0)p Fo(;)97 b Fm(\030)42 b Ft(1)p Fo(=q)1613 4892 y Fp(3)1649 4922 y Fo(;)100 b(q)46 b Fm(!)c(1)e Ft(and)f Fm(\030)k Ft(1)p Fo(=p)2510 4892 y Fp(2)2546 4922 y Fo(;)99 b(p)43 b Fm(!)f(1)p Fo(:)83 b Ft(T)-7 b(ak)n(en)659 5022 y(in)n(to)27 b(consideration)g(that)h Fo(\036)1565 5034 y Fp(1)1603 5022 y Ft(\()p Fi(x)p Fo(;)14 b Fi(p)p Ft(\))29 b(is)e(dimensionless)h(as)f(is)h Fo(B)2687 5034 y Fp(1)2752 5022 y Ft(\(cf.)39 b Fo(U)2965 5034 y Fp(1)3025 5022 y Ft(=)23 b Fo(e)3152 4991 y Fl(iB)3225 4999 y Fg(1)659 5130 y Ft(and)f(\(2.13\)\))g(whence)1378 5108 y(^)1367 5130 y Fo(\036)1416 5142 y Fp(1)1453 5130 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))24 b(is)e(of)g(dimension)g (\(momen)n(tum\))2707 5100 y Fn(\000)p Fp(3)2797 5130 y Ft(,)h(the)g(estimate)1108 5338 y Fu(Document)l(a)i(Ma)l(thema)l (tica)h(10)f(\(2005\))g(331{356)p eop %%Page: 340 10 340 9 bop 451 518 a Ft(340)727 b Fr(D.)30 b(H.)h(Jakubassa-Amundsen)659 725 y Fm(j)693 703 y Ft(^)682 725 y Fo(\036)731 737 y Fp(1)769 725 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))p Fm(j)37 b(\024)23 b Fo(c=q)14 b Fm(\001)e Ft(\()p Fo(q)k Ft(+)c Fo(p)g Ft(+)g(1\))1652 695 y Fn(\000)p Fp(2)1787 725 y Ft(is)24 b(obtained.)36 b(The)24 b(constan)n(t)g Fo(c)g Ft(is)h(determined)659 834 y(b)n(y)k(the)g(maxim)n(um)g(of)h Fo(q)22 b Fm(\001)e(j)1542 812 y Ft(^)1531 834 y Fo(\036)1580 846 y Fp(1)1618 834 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))p Fm(j)30 b Ft(in)g Fo(q)s(;)14 b(p)25 b Fm(2)h Fk(R)2253 846 y Fp(+)2370 834 y Ft(\(whic)n(h)j(exists)g(due)g(to)h (con-)659 942 y(tin)n(uit)n(y\))h(and)f(b)n(y)g(the)g(deca)n(y)g (constan)n(ts)f(of)2103 920 y(^)2092 942 y Fo(\036)2141 954 y Fp(1)2179 942 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))31 b(for)f Fo(q)h Fm(!)c(1)k Ft(and)f Fo(p)d Fm(!)h(1)p Fo(;)659 1042 y Ft(resp)r(ectiv)n(ely)e(\(maximised)i(in)g(the)g (second)f(v)-5 b(ariable\).)539 1206 y(d\))42 b(W)-7 b(e)26 b(presen)n(t)f(here)g(only)g(the)h(pro)r(of)g(of)f(the)h(form)g (b)r(oundedness)f(of)h Fo(B)2871 1218 y Fp(1)2908 1206 y Ft(;)h(the)f(op)r(er-)659 1306 y(ator)g(b)r(oundedness)i(can)f(b)r(e) h(sho)n(wn)f(along)f(the)i(same)f(lines.)659 1438 y(The)e(basic)h (ingredien)n(t)f(is)g(the)h Fd(Lieb)j(and)f(Y)-6 b(au)28 b(formula)e Ft(whic)n(h)g(is)g(a)f(consequence)659 1538 y(of)31 b(the)h(Sc)n(h)n(ur)e(test)i([12)o(])g(and)f(whic)n(h)g(also)f (can)h(b)r(e)h(deriv)n(ed)f(from)g(Sc)n(h)n(w)n(arz's)e(in-)659 1637 y(equalit)n(y)e([11)o(].)37 b(W)-7 b(e)28 b(giv)n(e)e(it)i(in)g(a) f(sligh)n(tly)h(generalised)e(form,)1407 1745 y Fh(\014)1407 1795 y(\014)1407 1845 y(\014)1407 1894 y(\014)1435 1752 y(Z)1531 1865 y Fo(d)p Fi(q)e Fo(d)p Fi(p)p 1767 1793 170 4 v 36 w Ft(^)-55 b Fo(')q Ft(\()p Fi(q)p Ft(\))24 b Fm(j)p Fo(K)6 b Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))g Fm(j)36 b Ft(^)-55 b Fo(')p Ft(\()p Fi(p)p Ft(\))2509 1745 y Fh(\014)2509 1795 y(\014)2509 1845 y(\014)2509 1894 y(\014)3074 1865 y Ft(\(2.30\))659 2205 y Fm(\024)719 2063 y Fh( )771 2092 y(Z)840 2205 y Fo(d)p Fi(q)p Fo(d)p Fi(p)p Fm(j)13 b Ft(^)-55 b Fo(')q Ft(\()p Fi(p)p Ft(\))p Fm(j)1247 2171 y Fp(2)1285 2205 y Fm(j)p Fo(K)6 b Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))p Fm(j)1626 2085 y Fh(\014)1626 2135 y(\014)1626 2184 y(\014)1626 2234 y(\014)1664 2149 y Fo(f)9 b Ft(\()p Fo(p)p Ft(\))p 1664 2186 156 4 v 1665 2262 a Fo(f)g Ft(\()p Fo(q)s Ft(\))1830 2085 y Fh(\014)1830 2135 y(\014)1830 2184 y(\014)1830 2234 y(\014)1857 2105 y Fp(2)1895 2063 y Fh(!)1948 2058 y Fg(1)p 1948 2067 29 3 v 1948 2100 a(2)1977 2205 y Fm(\001)2014 2063 y Fh( )2066 2092 y(Z)2136 2205 y Fo(d)p Fi(q)p Fo(d)p Fi(p)p Fm(j)k Ft(^)-55 b Fo(')q Ft(\()p Fi(p)p Ft(\))p Fm(j)2543 2171 y Fp(2)2581 2205 y Fm(j)p Fo(K)6 b Ft(\()p Fi(p)p Fo(;)14 b Fi(q)p Ft(\))p Fm(j)2922 2085 y Fh(\014)2922 2135 y(\014)2922 2184 y(\014)2922 2234 y(\014)2960 2149 y Fo(f)9 b Ft(\()p Fo(p)p Ft(\))p 2960 2186 156 4 v 2961 2262 a Fo(f)g Ft(\()p Fo(q)s Ft(\))3126 2085 y Fh(\014)3126 2135 y(\014)3126 2184 y(\014)3126 2234 y(\014)3153 2105 y Fp(2)3190 2063 y Fh(!)3243 2058 y Fg(1)p 3243 2067 29 3 v 3243 2100 a(2)659 2427 y Ft(with)30 b Fo(f)9 b Ft(\()p Fo(p)p Ft(\))28 b Fo(>)f Ft(0)j(for)g Fo(p)d(>)g Ft(0)j(a)g(smo)r(oth)g(con)n(v)n(ergence)d(generating)i(function.)46 b(F)-7 b(or)659 2526 y(a)27 b(symmetric)h(k)n(ernel,)f Fo(K)6 b Ft(\()p Fi(p)p Fo(;)14 b Fi(q)p Ft(\))38 b(=)24 b Fo(K)6 b Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))p Fo(;)52 b Ft(\(2.30\))27 b(simpli\014es)h(to)g(the)g(con)n(v)n(en-)659 2626 y(tional)f(form)g([11)o(,)h(5].)659 2758 y(F)-7 b(rom)29 b(the)h(condition)f(\(2.23\))g(w)n(e)h(ha)n(v)n(e)e(the)i (follo)n(wing)f(symmetry)g(with)h(resp)r(ect)659 2858 y(to)d(in)n(terc)n(hange)f(of)i Fi(q)g Ft(and)f Fi(p)p Ft(,)919 3046 y Fm(j)953 3024 y Ft(^)942 3046 y Fo(\036)991 3012 y Fn(\003)991 3067 y Fp(1)1030 3046 y Ft(\()p Fi(q)19 b Fm(\000)f Fi(p)p Fo(;)c Fi(p)p Ft(\))p Fm(j)46 b Ft(=)g Fm(j)1603 3024 y Ft(^)1592 3046 y Fo(\036)1641 3058 y Fp(1)1679 3046 y Ft(\()p Fi(p)18 b Fm(\000)g Fi(q)p Fo(;)c Fi(q)p Ft(\))p Fm(j)2105 2995 y Fc(q)p Fn($)p Fc(p)2137 3046 y Fm(7!)78 b(j)2332 3024 y Ft(^)2321 3046 y Fo(\036)2370 3058 y Fp(1)2408 3046 y Ft(\()p Fi(q)19 b Fm(\000)f Fi(p)p Fo(;)c Fi(p)p Ft(\))p Fm(j)p Fo(:)261 b Ft(\(2.31\))659 3225 y(One)27 b(then)h(obtains)g(for)f Fo(')d Fm(2)f Fo(L)1655 3237 y Fp(2)1692 3225 y Ft(\()p Fk(R)1778 3194 y Fp(3)1822 3225 y Ft(\))18 b Fm(\012)h Fk(C)2009 3194 y Fp(4)2081 3225 y Ft(applying)27 b(\(2.15\))g(and)g(subsequen)n(tly) 659 3324 y(\(2.30\))g(and)g(\(2.31\))681 3545 y Fm(j)p Ft(\()p Fo(';)14 b(B)890 3557 y Fp(1)928 3545 y Fo(')p Ft(\))p Fm(j)23 b Ft(=)37 b Fm(j)p Ft(\()13 b(^)-55 b Fo(';)1326 3524 y Fh(d)1308 3545 y Fo(B)1371 3557 y Fp(1)1409 3545 y Fo(')p Ft(\))p Fm(j)37 b(\024)1750 3489 y Ft(1)p 1667 3526 209 4 v 1667 3614 a(\(2)p Fo(\031)s Ft(\))1833 3563 y Fg(3)p 1833 3572 29 3 v 1833 3605 a(2)1913 3432 y Fh(Z)1996 3545 y Fo(d)p Fi(q)14 b Fm(j)p 2126 3473 170 4 v 13 w Ft(^)-55 b Fo(')q Ft(\()p Fi(q)p Ft(\))q Fm(j)2346 3432 y Fh(Z)2429 3545 y Fo(d)p Fi(p)14 b Fm(j)2573 3523 y Ft(^)2562 3545 y Fo(\036)2611 3557 y Fp(1)2649 3545 y Ft(\()p Fi(q)19 b Fm(\000)f Fi(p)p Fo(;)c Fi(p)p Ft(\))p Fm(j)g(j)f Ft(^)-55 b Fo(')q Ft(\()p Fi(p)p Ft(\))p Fm(j)3074 3705 y Ft(\(2.32\))1036 3908 y Fm(\024)1218 3852 y Ft(1)p 1134 3889 209 4 v 1134 3977 a(\(2)p Fo(\031)s Ft(\))1300 3926 y Fg(3)p 1300 3935 29 3 v 1300 3968 a(2)1366 3766 y Fh( )1432 3795 y(Z)1529 3908 y Fo(d)p Fi(q)14 b Fo(d)p Fi(p)g Fm(j)f Ft(^)-55 b Fo(')q Ft(\()p Fi(p)p Ft(\))p Fm(j)1964 3874 y Fp(2)2016 3908 y Fm(j)2050 3886 y Ft(^)2039 3908 y Fo(\036)2088 3920 y Fp(1)2125 3908 y Ft(\()p Fi(q)20 b Fm(\000)e Fi(p)p Fo(;)c Fi(p)p Ft(\))p Fm(j)2522 3788 y Fh(\014)2522 3838 y(\014)2522 3887 y(\014)2522 3937 y(\014)2559 3852 y Fo(f)9 b Ft(\()p Fo(p)p Ft(\))p 2559 3889 156 4 v 2560 3965 a Fo(f)g Ft(\()p Fo(q)s Ft(\))2725 3788 y Fh(\014)2725 3838 y(\014)2725 3887 y(\014)2725 3937 y(\014)2753 3808 y Fp(2)2790 3766 y Fh(!)2866 3761 y Fg(1)p 2866 3770 29 3 v 2866 3803 a(2)1164 4241 y Fm(\001)1201 4099 y Fh( )1267 4128 y(Z)1363 4241 y Fo(d)p Fi(q)14 b Fo(d)p Fi(p)g Fm(j)f Ft(^)-55 b Fo(')q Ft(\()p Fi(p)p Ft(\))p Fm(j)1798 4207 y Fp(2)1850 4241 y Fm(j)1884 4219 y Ft(^)1873 4241 y Fo(\036)1922 4207 y Fn(\003)1922 4262 y Fp(1)1961 4241 y Ft(\()p Fi(q)19 b Fm(\000)f Fi(p)p Fo(;)c Fi(p)p Ft(\))p Fm(j)2357 4121 y Fh(\014)2357 4171 y(\014)2357 4221 y(\014)2357 4270 y(\014)2395 4185 y Fo(f)9 b Ft(\()p Fo(p)p Ft(\))p 2395 4222 156 4 v 2396 4298 a Fo(f)g Ft(\()p Fo(q)s Ft(\))2561 4121 y Fh(\014)2561 4171 y(\014)2561 4221 y(\014)2561 4270 y(\014)2588 4141 y Fp(2)2626 4099 y Fh(!)2701 4094 y Fg(1)p 2701 4103 29 3 v 2701 4137 a(2)2757 4241 y Fo(:)659 4458 y Ft(Both)36 b(sym)n(b)r(ol)g(and)g(its)g(adjoin)n(t)g(can)g(b)r(e)h(estimated)f(b)n (y)g(the)h(same)e(expression)659 4558 y(\(2.24\))27 b(\(from)g (\(2.22\))g(one)g(ev)n(en)g(has)1867 4536 y(^)1856 4558 y Fo(\036)1905 4528 y Fn(\003)1905 4579 y Fp(1)1944 4558 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))38 b(=)22 b Fm(\000)2349 4536 y Ft(^)2338 4558 y Fo(\036)2387 4570 y Fp(1)2425 4558 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\)\),)29 b(yielding)1227 4779 y Fm(j)p Ft(\()p Fo(';)14 b(B)1436 4791 y Fp(1)1474 4779 y Fo(')p Ft(\))p Fm(j)47 b(\024)1837 4723 y Fo(c)p 1750 4760 209 4 v 1750 4848 a Ft(\(2)p Fo(\031)s Ft(\))1916 4796 y Fg(3)p 1917 4805 29 3 v 1917 4839 a(2)1983 4666 y Fh(Z)2080 4779 y Fo(d)p Fi(p)23 b Fm(j)13 b Ft(^)-55 b Fo(')p Ft(\()p Fi(p)p Ft(\))p Fm(j)2416 4745 y Fp(2)2496 4779 y Fm(\001)18 b Fo(I)2573 4791 y Fp(1)2611 4779 y Ft(\()p Fo(p)p Ft(\))936 5093 y Fo(I)972 5105 y Fp(1)1010 5093 y Ft(\()p Fo(p)p Ft(\))24 b(:=)1273 4980 y Fh(Z)1370 5093 y Fo(d)p Fi(q)1602 5037 y Ft(1)p 1497 5074 252 4 v 1497 5150 a Fm(j)p Fi(q)19 b Fm(\000)f Fi(p)p Fm(j)2097 5037 y Ft(1)p 1791 5074 653 4 v 1791 5150 a(\()p Fm(j)p Fi(q)h Fm(\000)f Fi(p)p Fm(j)32 b Ft(+)18 b Fo(p)h Ft(+)f(1\))2407 5126 y Fp(2)2491 4972 y Fh(\014)2491 5022 y(\014)2491 5072 y(\014)2491 5122 y(\014)2528 5037 y Fo(f)9 b Ft(\()p Fo(p)p Ft(\))p 2528 5074 156 4 v 2529 5150 a Fo(f)g Ft(\()p Fo(q)s Ft(\))2694 4972 y Fh(\014)2694 5022 y(\014)2694 5072 y(\014)2694 5122 y(\014)2722 4993 y Fp(2)2773 5093 y Fo(:)278 b Ft(\(2.33\))1108 5338 y Fu(Document)l(a)25 b(Ma)l(thema)l(tica)h(10)f(\(2005\))g(331{356)p eop %%Page: 341 11 341 10 bop 1291 518 a Fr(Pr)n(ojected)32 b(Dira)n(c)f(Opera)-6 b(tor)716 b Ft(341)659 725 y(F)-7 b(or)21 b(the)i(form)e(b)r (oundedness)h(of)g Fo(B)1764 737 y Fp(1)1824 725 y Ft(it)g(remains)g (to)g(pro)n(v)n(e)e(that)i Fo(I)2730 737 y Fp(1)2768 725 y Ft(\()p Fo(p)p Ft(\))h(is)e(b)r(ounded)659 825 y(for)31 b Fo(p)g Fm(2)g Fk(R)1003 837 y Fp(+)1064 825 y Fo(:)63 b Ft(The)32 b(angular)e(in)n(tegration)h(is)h(p)r(erformed)g (with)h(the)f(help)g(of)h(the)659 924 y(form)n(ula)842 1054 y Fh(Z)925 1074 y Fp(1)888 1242 y Fn(\000)p Fp(1)991 1167 y Fo(dx)1233 1111 y Ft(1)p 1105 1148 298 4 v 1105 1164 a Fm(p)p 1175 1164 229 4 v 1175 1232 a Fo(b)18 b Ft(+)g Fo(ax)1766 1111 y Ft(1)p 1437 1148 700 4 v 1437 1232 a(\()1469 1164 y Fm(p)p 1538 1164 229 4 v 68 x Fo(b)g Ft(+)g Fo(ax)33 b Ft(+)18 b Fo(p)g Ft(+)g(1\))2099 1208 y Fp(2)2192 1167 y Ft(=)2314 1111 y(2)p 2313 1148 44 4 v 2313 1224 a Fo(a)2381 1054 y Fh(Z)2464 1028 y Fn(p)p 2518 1028 117 3 v 2518 1074 a Fl(b)p Fp(+)p Fl(a)2427 1196 y Fn(p)p 2481 1196 118 3 v 2481 1242 a Fl(b)p Fn(\000)p Fl(a)2835 1111 y Fo(dz)p 2662 1148 431 4 v 2662 1224 a Ft(\()p Fo(z)k Ft(+)c Fo(p)h Ft(+)f(1\))3056 1200 y Fp(2)1175 1481 y Ft(=)1297 1425 y(2)p 1296 1462 44 4 v 1296 1538 a Fo(a)1378 1364 y Fh(\022)1703 1425 y Ft(1)p 1449 1462 551 4 v 1449 1479 a Fm(p)p 1518 1479 181 4 v 67 x Fo(b)g Fm(\000)g Fo(a)32 b Ft(+)18 b Fo(p)g 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Fo(\036)645 3013 y Fl(n)714 3001 y Ft(denote)f(the)h(sym)n(b)r(ol)f(of)g Fo(B)1549 3013 y Fl(n)1594 3001 y Fo(:)47 b Ft(It)24 b(is)f(brie\015y)g(indicated)g(ho)n(w)g Fo(\036)2654 3013 y Fp(2)2715 3001 y Ft(can)g(b)r(e)h(obtained)451 3100 y(explicitly)-7 b(,)39 b(but)d(for)g Fo(B)1197 3112 y Fl(n)1242 3100 y Fo(;)89 b(n)37 b Fm(\025)f Ft(2)p Fo(;)g Ft(the)h(calculus)e(with)i(op)r(erator)d(classes)h(is)h(applied) 451 3200 y(instead.)h Fo(B)833 3212 y Fp(2)898 3200 y Ft(is)27 b(de\014ned)h(b)n(y)456 3435 y Fm(\000)p Fo(i)p Ft([)p Fo(D)642 3447 y Fp(0)678 3435 y Fo(;)14 b(B)778 3447 y Fp(2)815 3435 y Ft(])9 b(=)g Fo(W)999 3447 y Fp(2)1046 3435 y Ft(=)g Fo(i)p Ft(\003)1207 3447 y Fp(+)1248 3318 y Fh(\024)1292 3435 y Ft([)p Fo(V)1363 3447 y Fp(1)1400 3435 y Fo(;)14 b(B)1500 3447 y Fp(1)1537 3435 y Ft(])5 b(+)1644 3378 y(1)p 1644 3415 42 4 v 1644 3492 a(2)1696 3435 y([)p Fo(W)1797 3447 y Fp(1)1834 3435 y Fo(;)14 b(B)1934 3447 y Fp(1)1972 3435 y Ft(])1995 3318 y Fh(\025)2025 3435 y Ft(\003)2083 3447 y Fn(\000)2157 3435 y Ft(+)k Fo(i)p Ft(\003)2327 3447 y Fn(\000)2368 3318 y Fh(\024)2412 3435 y Ft([)p Fo(V)2483 3447 y Fp(1)2521 3435 y Fo(;)c(B)2621 3447 y Fp(1)2658 3435 y Ft(])5 b(+)2765 3378 y(1)p 2765 3415 V 2765 3492 a(2)2816 3435 y([)p Fo(W)2917 3447 y Fp(1)2955 3435 y Fo(;)14 b(B)3055 3447 y Fp(1)3092 3435 y Ft(])3115 3318 y Fh(\025)3145 3435 y Ft(\003)3203 3447 y Fp(+)3258 3435 y Fo(:)3074 3584 y Ft(\(2.36\))451 3684 y(With)29 b Fo(W)744 3696 y Fp(1)809 3684 y Ft(from)e(\(2.18\))g(and)h Fo(V)1455 3696 y Fp(1)1515 3684 y Ft(=)23 b Fo(V)37 b Fm(\000)18 b Fo(W)1849 3696 y Fp(1)1915 3684 y Ft(one)27 b(obtains)516 3914 y Fo(W)594 3926 y Fp(2)668 3914 y Ft(=)795 3857 y Fo(i)p 789 3894 V 789 3970 a Ft(8)877 3821 y Fh(\020)927 3914 y Ft(3)14 b([)p Fo(V)5 b(;)14 b(B)1159 3926 y Fp(1)1195 3914 y Ft(])33 b(+)41 b([)1399 3893 y(~)1380 3914 y Fo(D)1449 3926 y Fp(0)1486 3914 y Fo(;)14 b(V)1609 3893 y Ft(~)1590 3914 y Fo(D)1659 3926 y Fp(0)1709 3914 y Fo(B)1772 3926 y Fp(1)1810 3914 y Ft(])41 b(+)g([)2023 3893 y(~)2003 3914 y Fo(D)2072 3926 y Fp(0)2109 3914 y Fo(;)14 b(B)2209 3926 y Fp(1)2266 3893 y Ft(~)2246 3914 y Fo(D)2315 3926 y Fp(0)2352 3914 y Fo(V)19 b Ft(])42 b(+)f(3)2665 3893 y(~)2646 3914 y Fo(D)2715 3926 y Fp(0)2765 3914 y Ft([)p Fo(B)2851 3926 y Fp(1)2888 3914 y Fo(;)14 b(V)19 b Ft(])3048 3893 y(~)3029 3914 y Fo(D)3098 3926 y Fp(0)3135 3821 y Fh(\021)3198 3914 y Fo(:)3074 4041 y Ft(\(2.37\))451 4141 y(\(2.36\))k(is,)h(lik)n (e)f(the)h(corresp)r(onding)d(equation)i(for)g Fo(B)2127 4153 y Fp(1)2164 4141 y Ft(,)h(solv)n(ed)e(in)i(momen)n(tum)g(space)e (b)n(y)451 4240 y(in)n(tro)r(ducing)28 b(the)i(resp)r(ectiv)n(e)e(sym)n (b)r(ols)g(of)h(the)g(op)r(erators.)38 b(The)29 b(F)-7 b(ourier)28 b(transformed)451 4340 y(sym)n(b)r(ol)44 b(^)-59 b Fo(w)794 4352 y Fp(2)860 4340 y Ft(of)27 b Fo(W)1032 4352 y Fp(2)1098 4340 y Ft(is)g(comp)r(osed)g(of)h (expressions)d(of)j(the)g(t)n(yp)r(e)g(\(cf)g(\(2.17\)\))1058 4550 y Fh(d)1053 4572 y Fo(v)s(\036)1145 4584 y Fp(1)1183 4572 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))47 b(=)f Fm(\000)1660 4516 y Fo(\015)p 1620 4553 130 4 v 1620 4629 a Ft(2)p Fo(\031)1712 4605 y Fp(2)1772 4459 y Fh(Z)1869 4572 y Fo(d)p Fi(p)1965 4538 y Fn(0)2157 4516 y Ft(1)p 2022 4553 312 4 v 2022 4629 a Fm(j)p Fi(q)19 b Fm(\000)f Fi(p)2250 4605 y Fn(0)2273 4629 y Fm(j)2296 4605 y Fp(2)2377 4550 y Ft(^)2366 4572 y Fo(\036)2415 4584 y Fp(1)2453 4572 y Ft(\()p Fi(p)2538 4538 y Fn(0)2562 4572 y Fo(;)c Fi(p)p Ft(\))390 b(\(2.38\))451 4832 y(where)691 4810 y Fh(d)686 4832 y Fo(v)s(\036)778 4844 y Fp(1)839 4832 y Ft(is)23 b(the)g(F)-7 b(ourier)21 b(transformed)h(sym)n(b)r(ol)g(of)g (the)i(pro)r(duct)e Fo(V)d(B)2741 4844 y Fp(1)2778 4832 y Fo(:)46 b Ft(This)23 b(implies)451 4932 y(that)28 b(|)g(in)g(con)n (trast)e(to)h(\(2.21\))g(|)h(the)g(equation)f(for)2232 4910 y(^)2221 4932 y Fo(\036)2270 4944 y Fp(2)2308 4932 y Ft(\()p Fi(p)p Fo(;)14 b Fi(q)p Ft(\))p Fo(;)593 5130 y Ft([)p Fj(\013)g Ft(\()p Fi(p)19 b Ft(+)f Fi(q)p Ft(\))33 b(+)e Fo(\014)t(m)p Ft(])1273 5108 y(^)1262 5130 y Fo(\036)1311 5142 y Fp(2)1348 5130 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))43 b Fm(\000)1712 5108 y Ft(^)1701 5130 y Fo(\036)1750 5142 y Fp(2)1788 5130 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))24 b([)p Fj(\013)p Fi(p)32 b Ft(+)18 b Fo(\014)t(m)p Ft(])37 b(=)46 b Fo(i)30 b Ft(^)-58 b Fo(w)2667 5142 y Fp(2)2704 5130 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))p Fo(;)143 b Ft(\(2.39\))1108 5338 y Fu(Document)l(a)25 b(Ma)l(thema)l(tica)h(10)f(\(2005\))g(331{356)p eop %%Page: 342 12 342 11 bop 451 518 a Ft(342)727 b Fr(D.)30 b(H.)h(Jakubassa-Amundsen) 451 725 y Ft(in)n(v)n(olv)n(es)i(an)i(extra)f(in)n(tegral)f(on)i(the)g (r.h.s.)58 b(and)34 b(is)h(solv)n(ed)f(with)h(the)g(ansatz)f(\(using) 451 834 y(\(2.22\))27 b(for)829 812 y(^)818 834 y Fo(\036)867 846 y Fp(1)905 834 y Ft(\()p Fi(p)990 804 y Fn(0)1013 834 y Fo(;)14 b Fi(p)p Ft(\)\))810 1027 y(^)799 1049 y Fo(\036)848 1061 y Fp(2)885 1049 y Ft(\()p Fi(q)p Fo(;)g Fi(p)p Ft(\))47 b(=)f Fm(\000)1329 992 y Fo(i\015)5 b(\015)1449 1004 y Fp(0)p 1322 1030 171 4 v 1322 1106 a Ft(16)p Fo(\031)1456 1082 y Fp(2)1539 936 y Fh(Z)1636 1049 y Fo(d)p Fi(p)1732 1014 y Fn(0)1817 992 y Ft(1)p 1788 1030 99 4 v 1788 1106 a Fo(p)1830 1082 y Fn(0)p Fp(2)2065 992 y Ft(1)p 1930 1030 312 4 v 1930 1106 a Fm(j)p Fi(q)18 b Fm(\000)g Fi(p)2157 1082 y Fn(0)2181 1106 y Fm(j)2204 1082 y Fp(2)2288 1049 y Ft(\()p Fo(c)2356 1061 y Fp(1)2435 1049 y Ft(+)41 b Fo(c)2577 1061 y Fp(2)2628 1049 y Fj(\013)p Fi(p)19 b Fm(\001)f Fo(\014)t(m)513 1274 y Ft(+)k Fo(c)636 1286 y Fp(3)687 1274 y Fj(\013)p Fi(q)d Fm(\001)g Fo(\014)t(m)41 b Ft(+)g Fo(c)1168 1286 y Fp(4)1219 1274 y Fj(\013)q Fi(p)1336 1240 y Fn(0)1377 1274 y Fm(\001)19 b Fo(\014)t(m)42 b Ft(+)f Fo(c)1727 1286 y Fp(5)1778 1274 y Fj(\013)p Fi(p)1894 1240 y Fn(0)1936 1274 y Fm(\001)18 b Fj(\013)p Fi(p)42 b Ft(+)f Fo(c)2277 1286 y Fp(6)2328 1274 y Fj(\013)p Fi(q)19 b Fm(\001)g Fj(\013)p Fi(p)41 b Ft(+)g Fo(c)2801 1286 y Fp(7)2852 1274 y Fj(\013)p Fi(q)19 b Fm(\001)g Fj(\013)p Fi(p)3142 1240 y Fn(0)3165 1274 y Ft(\))14 b Fo(:)3074 1374 y Ft(\(2.40\))451 1473 y(The)22 b(scalar)f(co)r (e\016cien)n(ts)g Fo(c)1297 1485 y Fl(j)1332 1473 y Fo(;)60 b(j)28 b Ft(=)23 b(1)p Fo(;)14 b(:::;)g Ft(7)44 b(\(dep)r(ending)23 b(on)f Fi(q)p Fo(;)14 b Fi(p)22 b Ft(and)g Fi(p)2743 1443 y Fn(0)2766 1473 y Ft(\))46 b(are)21 b(uniquely)451 1573 y(determined.)451 1673 y(The)29 b(matter)f(of)g(in)n(terest)g(is,) g(ho)n(w)n(ev)n(er,)f(not)h(the)h(explicit)f(form)g(of)h Fo(B)2681 1685 y Fp(2)2746 1673 y Ft(or)f(generally)-7 b(,)27 b(of)451 1772 y Fo(B)514 1784 y Fl(n)559 1772 y Fo(;)60 b(n)23 b Fm(\025)g Ft(2)p Fo(;)46 b Ft(but)24 b(the)f(existence)g(of)g(the)h(p)r(oten)n(tials)e Fo(W)2234 1784 y Fl(n)2303 1772 y Ft(and)h Fo(V)2508 1784 y Fl(n)2577 1772 y Ft(on)g(the)g(form)g(domain)451 1872 y Fo(H)520 1887 y Fp(1)p Fl(=)p Fp(2)625 1872 y Ft(\()p Fk(R)711 1842 y Fp(3)754 1872 y Ft(\))d Fm(\012)g Fk(C)945 1842 y Fp(4)1045 1872 y Ft(of)29 b(the)h(free)g(Dirac)f(op)r(erator,)g(suc)n (h)g(that)i(the)f(exp)r(ectation)f(v)-5 b(alue)30 b(of)451 1983 y Fo(H)527 1953 y Fp(\()p Fl(n)p Fp(\))652 1983 y Ft(from)d(\(2.9\))g(is)h(w)n(ell)f(de\014ned.)451 2141 y Fr(Lemma)38 b Ft(2.2)p Fr(.)45 b Fd(L)l(et)35 b Fo(U)1152 2153 y Fl(n)1231 2141 y Ft(=)f Fo(e)1369 2111 y Fl(iB)1442 2119 y Fe(n)1487 2141 y Fo(;)82 b(n)34 b Fm(\025)g Ft(1)p Fo(;)70 b Fd(b)l(e)36 b(the)g(tr)l(ansformations)g(fr)l(om)h(The)l(or)l (em)451 2241 y(1.1.)j(L)l(et)30 b Fo(\036)818 2253 y Fl(n)893 2241 y Fd(b)l(e)g(the)g(symb)l(ol)h(of)f Fo(B)1563 2253 y Fl(n)1638 2241 y Fd(and)h Fo(W)1878 2253 y Fl(n)1953 2241 y Fd(the)f(p)l(otential)h(in)e(the)h(de\014ning)h(e)l(quation)451 2341 y(for)g Fo(\036)633 2353 y Fl(n)678 2341 y Fo(:)53 b Fd(Then)31 b Fo(W)1049 2353 y Fl(n)1124 2341 y Fd(is)f Fo(p)p Fd(-form)g(b)l(ounde)l(d)g(on)g Fo(H)1989 2356 y Fp(1)p Fl(=)p Fp(2)2093 2341 y Ft(\()p Fk(R)2180 2310 y Fp(3)2223 2341 y Ft(\))19 b Fm(\012)f Fk(C)2411 2310 y Fp(4)2484 2341 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y Ft(a\))28 b Fo(p)p Ft(-form)e(b)r(oundedness)i(of)f Fo(W)1478 3366 y Fl(n)451 3501 y Ft(The)i(pro)r(of)e(is)i(b)n(y)f (induction.)40 b(The)28 b Fo(p)p Ft(-form)g(b)r(oundedness)g(of)g Fo(W)2544 3513 y Fp(1)2582 3501 y Ft(,)h(eq.)39 b(\(2.18\),)28 b(follo)n(ws)451 3600 y(from)k(Kato's)e(inequalit)n(y)h([10)o(])h(and)g (from)f(the)h(self-adjoin)n(tness)f(and)g(b)r(oundedness)h(of)471 3679 y(~)451 3700 y Fo(D)520 3712 y Fp(0)557 3700 y Fo(;)486 3907 y Fm(j)p Ft(\()p Fo(';)14 b(W)710 3919 y Fp(1)748 3907 y Fo(')p Ft(\))p Fm(j)47 b(\024)1015 3851 y Ft(1)p 1015 3888 42 4 v 1015 3964 a(2)1090 3907 y Fm(j)p Ft(\()p Fo(';)14 b(V)19 b(')p Ft(\))p Fm(j)42 b Ft(+)1570 3851 y(1)p 1570 3888 V 1570 3964 a(2)1645 3907 y Fm(j)p Ft(\()1720 3886 y(~)1700 3907 y Fo(D)1769 3919 y Fp(0)1806 3907 y Fo(';)14 b(V)1984 3886 y Ft(~)1964 3907 y Fo(D)2033 3919 y Fp(0)2070 3907 y Fo(')p Ft(\))p Fm(j)47 b(\024)2337 3851 y Fo(\015)5 b(\031)p 2337 3888 98 4 v 2366 3964 a Ft(4)2459 3907 y(\()p Fo(';)14 b(p')p 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Ft(])451 4532 y(and)31 b([)p Fo(V)5 b(;)14 b(B)792 4544 y Fp(2)829 4532 y Ft(])p Fo(:)60 b Ft(The)31 b(additionally)f(o)r (ccurring)f(factors)2234 4511 y(~)2215 4532 y Fo(D)2284 4544 y Fp(0)2352 4532 y Ft(can)h(b)r(e)h(disregarded)e(in)j(the)451 4632 y(con)n(text)21 b(of)g Fo(p)p Ft(-form)e(b)r(oundedness)i(since) 1783 4611 y(~)1764 4632 y Fo(D)1833 4644 y Fp(0)1891 4632 y Ft(is)g(a)f(b)r(ounded)h(m)n(ultiplication)g(op)r(erator)f(in) 451 4732 y(momen)n(tum)i(space.)35 b(In)22 b(the)g(general)e(case,)i (the)h(pro)r(duct)e(of)h(all)g(factors)f Fo(B)2813 4744 y Fl(k)2853 4732 y Fo(;)60 b(k)26 b Fm(\024)d Fo(n)7 b Fm(\000)g Ft(1)p Fo(;)451 4831 y Ft(whic)n(h)25 b(en)n(ter)f(in)n(to) h(a)f(giv)n(en)g(comm)n(utator)g(con)n(tributing)g(to)h Fo(W)2452 4843 y Fl(n)2522 4831 y Ft(m)n(ust)g(b)r(e)h(prop)r(ortional) 451 4931 y(to)i Fo(\015)601 4901 y Fl(n)p Fn(\000)p Fp(1)758 4931 y Ft(since)g Fo(W)1040 4943 y Fl(n)1113 4931 y Ft(is)f(of)h(the)g (order)e Fo(\015)1699 4901 y Fl(n)1744 4931 y Fo(:)451 5031 y Ft(By)36 b(induction)h(h)n(yp)r(othesis,)h Fo(W)1489 5043 y Fl(k)1567 5031 y Ft(is)e Fo(p)p Ft(-form)g(b)r(ounded)h(on)f Fo(H)2475 5046 y Fp(1)p Fl(=)p Fp(2)2579 5031 y Ft(\()p Fk(R)2665 5000 y Fp(3)2709 5031 y Ft(\))24 b Fm(\012)g Fk(C)2908 5000 y Fp(4)3002 5031 y Ft(for)36 b Fo(k)k Fm(\024)451 5130 y Fo(n)26 b Fm(\000)f Ft(1)p Fo(:)80 b Ft(This)38 b(means)g(that)h(all)f(comm)n(utators)f(of)i(smaller)f (order)f(than)h Fo(n)h Ft(in)g Fo(\015)k Ft(are)1108 5338 y Fu(Document)l(a)25 b(Ma)l(thema)l(tica)h(10)f(\(2005\))g (331{356)p eop %%Page: 343 13 343 12 bop 1291 518 a Fr(Pr)n(ojected)32 b(Dira)n(c)f(Opera)-6 b(tor)716 b Ft(343)451 725 y Fo(p)p Ft(-form)36 b(b)r(ounded.)65 b(In)37 b(the)g(induction)g(step)g(one)f(has)g(to)h(sho)n(w)f(that)h([) p Fo(V)5 b(;)14 b(B)2962 737 y Fl(n)p Fn(\000)p Fp(1)3092 725 y Ft(])37 b(and)451 825 y([[)p Fm(\001)p Ft(])p Fo(;)14 b(B)643 837 y Fl(k)684 825 y Ft(])p Fo(;)68 b(k)30 b(<)d(n)20 b Fm(\000)g Ft(1)p Fo(;)57 b Ft(are)29 b Fo(p)p 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y(and)k(vice)f(v)n (ersa,)f Fo(p)p Ft(-form)h(b)r(oundedness)g(can)g(b)r(e)h(expressed)f (in)h(the)g(follo)n(wing)e(w)n(a)n(y)555 1704 y Fm(j)p Ft(\()p Fo(';)14 b Ft([)p Fm(\001)p Ft(])p Fo(')p Ft(\))p Fm(j)9 b(\024)1057 1648 y Ft(1)p 974 1685 209 4 v 974 1773 a(\(2)p Fo(\031)s Ft(\))1140 1722 y Fg(3)p 1140 1731 29 3 v 1140 1764 a(2)1192 1591 y Fh(Z)1261 1704 y Fo(d)p Fi(p)14 b Fm(j)g Ft(^)-55 b Fo(')p Ft(\()p Fi(p)p Ft(\))p Fm(j)1590 1662 y Fp(2)1613 1591 y Fh(Z)1682 1704 y Fo(d)p Fi(q)1775 1612 y Fh(\020)1839 1609 y(\014)1839 1658 y(\014)1839 1708 y(\014)1878 1678 y(b)1867 1704 y Ft([)p Fm(\001)p Ft(]\()p Fi(q)5 b Fm(\000)g Fi(p)p Fo(;)14 b Fi(p)p Ft(\))2268 1609 y Fh(\014)2268 1658 y(\014)2268 1708 y(\014)2314 1704 y Ft(+)2384 1609 y Fh(\014)2384 1658 y(\014)2384 1708 y(\014)2422 1678 y(b)2411 1704 y Ft([)p Fm(\001)p Ft(])2480 1638 y Fn(\003)2504 1704 y Ft(\()p Fi(q)5 b Fm(\000)g Fi(p)p Fo(;)14 b Fi(p)p Ft(\))2836 1609 y Fh(\014)2836 1658 y(\014)2836 1708 y(\014)2864 1612 y(\021)2913 1584 y(\014)2913 1634 y(\014)2913 1683 y(\014)2913 1733 y(\014)2951 1648 y Fo(f)9 b Ft(\()p Fo(p)p Ft(\))p 2951 1685 156 4 v 2952 1761 a Fo(f)g Ft(\()p Fo(q)s Ft(\))3117 1584 y Fh(\014)3117 1634 y(\014)3117 1683 y(\014)3117 1733 y(\014)3145 1604 y Fp(2)1102 1997 y Fm(\024)1332 1940 y Ft(1)p 1223 1977 261 4 v 1223 2055 a(\(2)p Fo(\031)s Ft(\))1379 2031 y Fp(3)p Fl(=)p Fp(2)1507 1884 y Fh(Z)1604 1997 y Fo(d)p Fi(p)37 b Fm(j)14 b Ft(^)-56 b Fo(')q Ft(\()p Fi(p)p Ft(\))p Fm(j)1955 1955 y Fp(2)2029 1997 y Fo(c)23 b(p)46 b Ft(=)g Fo(c)2323 1962 y Fn(0)2369 1997 y Ft(\()p Fo(';)14 b(p)g(')p Ft(\))440 b(\(2.43\))451 2195 y(with)29 b(some)g(constan)n(t)e Fo(c)e(>)g Ft(0)p Fo(:)j Ft(The)h(inequalit)n(y)f(in)h(the)g(second)f(line)h(of)g (\(2.43\))f(restricts)451 2295 y(the)38 b(con)n(v)n(ergence)d (generating)h(function)j(to)e Fo(f)9 b Ft(\()p Fo(p)p Ft(\))40 b(:=)f Fo(p)2301 2265 y Fl(\025)2382 2295 y Ft(with)2591 2262 y Fp(1)p 2591 2276 34 4 v 2591 2324 a(2)2674 2295 y 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y(^)3105 3247 y Fo(\036)3154 3259 y Fl(n)3230 3247 y Ft(is)451 3347 y(related)f(to)g(the)h(estimate) f(of)47 b(^)-59 b Fo(w)1472 3359 y Fl(n)1548 3347 y Ft(b)n(y)30 b(an)g(equation)f(of)i(the)f(t)n(yp)r(e)h(\(2.39\),)f(deriv)n(ed)f (from)451 3455 y(the)d(de\014ning)f(equation)g(\(2.8\).)36 b(This)25 b(equation)g(implies)g(that)h(the)f(b)r(eha)n(viour)f(of)3077 3433 y(^)3066 3455 y Fo(\036)3115 3467 y Fl(n)3186 3455 y Ft(for)451 3555 y Fo(p)i Fm(!)g Ft(0)j(and)g Fo(q)g Fm(!)d Ft(0)j(is)g(that)h(of)46 b(^)-59 b Fo(w)1530 3567 y Fl(n)1576 3555 y Ft(,)30 b(while)f(there)g(o)r(ccurs)g(an)g(extra)f (p)r(o)n(w)n(er)g(of)h Fo(q)3033 3525 y Fn(\000)p Fp(1)3152 3555 y Ft(and)451 3655 y Fo(p)493 3625 y Fn(\000)p Fp(1)610 3655 y Ft(for)e Fo(q)f Fm(!)d(1)28 b Ft(and)f Fo(p)c Fm(!)g(1)p Fo(;)28 b Ft(resp)r(ectiv)n(ely)-7 b(.)36 b(Therefore)1243 3853 y Fm(j)1277 3831 y Ft(^)1266 3853 y Fo(\036)1315 3865 y Fl(n)1361 3853 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))p Fm(j)47 b(\024)1901 3796 y Fo(c)p 1756 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4688 a(\(2)p Fo(\031)s Ft(\))1127 4664 y Fp(3)1189 4518 y Fh(Z)1286 4631 y Fo(d)p Fi(p)1382 4597 y Fn(0)1405 4631 y Fo(d)p Fi(p)14 b Fo(d)p Fi(q)24 b Fm(j)13 b Ft(^)-55 b Fo(')p Ft(\()p Fi(p)p Ft(\))p Fm(j)1863 4539 y Fh(n)1919 4631 y Fm(j)s Ft(^)-45 b Fo(v)1985 4597 y Fn(\003)2024 4631 y Ft(\()p Fi(p)2109 4597 y Fn(0)2151 4631 y Fm(\000)18 b Fi(p)p Fo(;)c Fi(p)p Ft(\))p Fm(j)23 b(j)2489 4609 y Ft(^)2478 4631 y Fo(\036)2527 4643 y Fl(n)p Fn(\000)p Fp(1)2658 4631 y Ft(\()p Fi(p)2743 4597 y Fn(0)2785 4631 y Fm(\000)18 b Fi(q)p Fo(;)c Fi(q)p Ft(\))p Fm(j)1222 4862 y Ft(+)24 b Fm(j)1345 4840 y Ft(^)1334 4862 y Fo(\036)1383 4827 y Fn(\003)1383 4882 y Fl(n)p Fn(\000)p Fp(1)1514 4862 y Ft(\()p Fi(p)1599 4827 y Fn(0)1641 4862 y Fm(\000)18 b Fi(p)p Fo(;)c Fi(p)p Ft(\))p Fm(j)23 b(j)s Ft(^)-45 b Fo(v)s Ft(\()p Fi(p)2096 4827 y Fn(0)2138 4862 y Fm(\000)18 b Fi(q)p Fo(;)c Fi(q)p Ft(\))p Fm(j)2413 4770 y Fh(o)2484 4862 y Fm(j)f Ft(^)-55 b Fo(')p Ft(\()p Fi(q)p Ft(\))p Fm(j)p Fo(:)353 b Ft(\(2.45\))659 5031 y(Due)40 b(to)g(the)h(symmetry)e(in)i Fi(p)f Ft(and)g Fi(q)p Ft(,)j(the)e(Lieb)f(and)g(Y)-7 b(au)40 b(form)n(ula)f(can)g(b)r (e)659 5130 y(applied)31 b(in)g(the)g(same)f(w)n(a)n(y)g(as)g(in)h (\(2.32\))f(and)h(in)g(\(2.43\).)46 b(Using)30 b Fo(f)9 b Ft(\()p Fo(p)p Ft(\))29 b(:=)f Fo(p)i Ft(for)1108 5338 y Fu(Document)l(a)25 b(Ma)l(thema)l(tica)h(10)f(\(2005\))g(331{356)p eop %%Page: 344 14 344 13 bop 451 518 a Ft(344)727 b Fr(D.)30 b(H.)h(Jakubassa-Amundsen) 659 725 y Ft(the)d(con)n(v)n(ergence)d(generating)h(function)i(and)g (estimating)i(^)-45 b Fo(v)31 b Ft(one)c(gets)793 958 y Fm(j)p Ft(\()p Fo(';)14 b Ft([)p Fo(V)5 b(;)14 b(B)1115 970 y Fl(n)p Fn(\000)p Fp(1)1245 958 y Ft(])g Fo(')p Ft(\))p Fm(j)47 b(\024)36 b Fo(c)1589 845 y Fh(Z)1686 958 y Fo(d)p Fi(p)h Fm(j)14 b Ft(^)-56 b Fo(')q Ft(\()p Fi(p)p Ft(\))p Fm(j)2037 916 y Fp(2)2111 841 y Fh(\032)2174 845 y(Z)2270 958 y Fo(d)p Fi(p)2366 924 y Fn(0)2559 902 y Ft(1)p 2423 939 315 4 v 2423 1015 a Fm(j)p Fi(p)2499 991 y Fn(0)2541 1015 y Fm(\000)18 b Fi(p)p Fm(j)2700 991 y Fp(2)2791 902 y Fo(p)2833 872 y Fp(2)p 2780 939 102 4 v 2780 1015 a Fo(p)2822 974 y Ff(0)2844 991 y Fp(2)2910 958 y Fm(\001)2952 845 y Fh(Z)3048 958 y Fo(d)p Fi(q)693 1182 y Fh(\014)693 1232 y(\014)693 1282 y(\014)732 1256 y Ft(^)721 1277 y Fo(\036)770 1289 y Fl(n)p Fn(\000)p Fp(1)901 1277 y Ft(\()p Fi(p)986 1243 y Fn(0)1014 1277 y Fm(\000)5 b Fi(q)p Fo(;)14 b Fi(q)p Ft(\))1253 1182 y Fh(\014)1253 1232 y(\014)1253 1282 y(\014)1304 1221 y Fo(p)1346 1166 y Ff(0)1369 1191 y Fp(2)p 1304 1258 V 1317 1334 a Fo(q)1357 1310 y Fp(2)1450 1277 y Ft(+)1515 1164 y Fh(Z)1583 1277 y Fo(d)p Fi(p)1679 1243 y Fn(0)1717 1182 y Fh(\014)1717 1232 y(\014)1717 1282 y(\014)1755 1256 y Ft(^)1744 1277 y Fo(\036)1793 1289 y Fl(n)p Fn(\000)p Fp(1)1924 1277 y Ft(\()p Fi(p)19 b Fm(\000)f Fi(p)2164 1243 y Fn(0)2187 1277 y Fo(;)c Fi(p)2277 1243 y Fn(0)2300 1277 y Ft(\))2332 1182 y Fh(\014)2332 1232 y(\014)2332 1282 y(\014)2395 1221 y Fo(p)2437 1191 y Fp(2)p 2384 1258 V 2384 1334 a Fo(p)2426 1294 y Ff(0)2448 1310 y Fp(2)2514 1277 y Fm(\001)2542 1164 y Fh(Z)2611 1277 y Fo(d)p Fi(q)2850 1221 y Ft(1)p 2714 1258 312 4 v 2714 1334 a Fm(j)p Fi(p)2790 1310 y Fn(0)2832 1334 y Fm(\000)k Fi(q)p Fm(j)2988 1310 y Fp(2)3046 1221 y Fo(p)3088 1166 y Ff(0)3110 1191 y Fp(2)p 3046 1258 102 4 v 3058 1334 a Fo(q)3098 1310 y Fp(2)3157 1136 y Fh(\))3238 1277 y Fo(:)3074 1452 y Ft(\(2.46\))659 1551 y(The)23 b(last)g Fi(q)p Ft(-in)n(tegral)f(is)g(ev)-5 b(aluated)23 b(with)h(the)f (substitution)h Fi(q)2630 1521 y Fn(0)2677 1551 y Ft(:=)e Fi(q)p Fo(=p)2921 1521 y Fn(0)2967 1551 y Ft(and)h(with)659 1651 y Fi(e)703 1663 y Fl(p)737 1647 y Ff(0)786 1651 y Ft(:=)g Fi(p)950 1621 y Fn(0)973 1651 y Fo(=p)1057 1621 y Fn(0)1080 1651 y Fo(;)696 1789 y Fh(Z)765 1902 y Fo(d)p Fi(q)1004 1846 y Ft(1)p 868 1883 312 4 v 868 1959 a Fm(j)p Fi(q)d Fm(\000)e Fi(p)1097 1935 y Fn(0)1120 1959 y Fm(j)1143 1935 y Fp(2)1200 1846 y Fo(p)1242 1791 y Ff(0)1265 1816 y Fp(2)p 1200 1883 102 4 v 1212 1959 a Fo(q)1252 1935 y Fp(2)1335 1902 y Ft(=)23 b Fo(p)1465 1868 y Fn(0)1474 1789 y Fh(Z)1557 1810 y Fn(1)1520 1978 y Fp(0)1586 1902 y Fo(dq)1669 1868 y Fn(0)1692 1789 y Fh(Z)1738 1978 y Fl(S)1782 1961 y Fg(2)1805 1902 y Fo(d)p Ft(\012)1908 1914 y Fl(q)1940 1898 y Ff(0)2124 1846 y Ft(1)p 1977 1883 336 4 v 1977 1959 a Fm(j)p Fi(q)2050 1935 y Fn(0)2079 1959 y Fm(\000)5 b Fi(e)2193 1971 y Fl(p)2227 1955 y Ff(0)2252 1959 y Fm(j)2275 1935 y Fp(2)2346 1902 y Ft(=)22 b(2)p Fo(\031)s(p)2567 1868 y Fn(0)2590 1789 y Fh(Z)2673 1810 y Fn(1)2636 1978 y Fp(0)2753 1846 y Fo(dq)2836 1816 y Fn(0)p 2753 1883 107 4 v 2775 1959 a Fo(q)2815 1935 y Fn(0)2884 1902 y Ft(ln)2967 1782 y Fh(\014)2967 1832 y(\014)2967 1881 y(\014)2967 1931 y(\014)3004 1846 y Ft(1)c(+)g Fo(q)3187 1816 y Fn(0)p 3004 1883 207 4 v 3004 1959 a Ft(1)g Fm(\000)g Fo(q)3187 1935 y Fn(0)3221 1782 y Fh(\014)3221 1832 y(\014)3221 1881 y(\014)3221 1931 y(\014)1829 2150 y Ft(=)46 b Fo(\031)1990 2116 y Fp(3)2027 2150 y Fo(p)2069 2116 y Fn(0)2092 2150 y Fo(:)959 b Ft(\(2.47\))659 2311 y(Also)861 2289 y(^)850 2311 y Fo(\036)899 2323 y Fl(n)p Fn(\000)p Fp(1)1061 2311 y Ft(is)31 b(estimated)h(b)n(y)48 b(^)-59 b Fo(w)1709 2323 y Fl(n)p Fn(\000)p Fp(1)1871 2311 y Ft(via)31 b(\(2.44\).)47 b(One)31 b(obtains)g(for)g(the)h(second)659 2411 y(term)26 b(in)h(curly)f(brac)n(k)n(ets)f(of)h(\(2.46\))g(with)h(the)f(help)h(of) g(the)f Fo(p)p Ft(-form)g(b)r(oundedness)659 2510 y(of)h Fo(W)831 2522 y Fl(n)p Fn(\000)p Fp(1)990 2510 y Ft(b)n(y)g(means)g(of) h(\(2.43\))659 2635 y Fh(Z)714 2748 y Fo(d)p Fi(p)810 2713 y Fn(0)834 2748 y Fm(j)868 2726 y Ft(^)857 2748 y Fo(\036)906 2760 y Fl(n)p Fn(\000)p Fp(1)1036 2748 y Ft(\()p Fi(p)6 b Fm(\000)g Fi(p)1251 2713 y Fn(0)1260 2748 y Fo(;)14 b Fi(p)1350 2713 y Fn(0)1373 2748 y Ft(\))p Fm(j)1450 2692 y Fo(p)1492 2661 y Fp(2)p 1439 2729 102 4 v 1439 2805 a Fo(p)1481 2764 y Ff(0)1503 2781 y Fp(2)1556 2748 y Fm(\001)6 b Fo(\031)1635 2713 y Fp(3)1672 2748 y Fo(p)1714 2713 y Fn(0)1746 2748 y Fm(\024)25 b Ft(~)-44 b Fo(c)1856 2635 y Fh(Z)1925 2748 y Fo(d)p Fi(p)2021 2713 y Fn(0)2286 2692 y Ft(1)p 2055 2729 504 4 v 2055 2805 a Fm(j)p Fi(p)5 b Fm(\000)g Fi(p)2259 2781 y Fn(0)2281 2805 y Fm(j)g Ft(+)g Fo(p)2421 2781 y Fn(0)2447 2805 y Ft(+)g(1)2582 2748 y Fm(j)17 b Ft(^)-59 b Fo(w)2664 2760 y Fl(n)p Fn(\000)p Fp(1)2795 2748 y Ft(\()p Fi(p)6 b Fm(\000)g Fi(p)3010 2713 y Fn(0)3019 2748 y Fo(;)14 b Fi(p)3109 2713 y Fn(0)3132 2748 y Ft(\))p Fm(j)3197 2692 y Fo(p)3239 2661 y Fp(2)p 3197 2729 80 4 v 3204 2805 a Fo(p)3246 2781 y Fn(0)1861 2987 y Fm(\024)37 b Fo(c)1999 2953 y Fn(00)2041 2987 y Fo(p)991 b Ft(\(2.48\))659 3139 y(where)31 b Fm(j)p Fi(p)22 b Fm(\000)f Fi(p)1140 3109 y Fn(0)1163 3139 y Fm(j)h Ft(+)f Fo(p)1336 3109 y Fn(0)1380 3139 y Ft(+)h(1)44 b Fm(\025)30 b Fo(p)1690 3109 y Fn(0)1776 3139 y Ft(has)h(b)r(een)i(used.)50 b(F)-7 b(urther)32 b(w)n(e)g(note)g(that)h(the)659 3239 y(factor)h(\()p Fm(j)p Fi(p)1012 3209 y Fn(0)1060 3239 y Fm(\000)23 b Fi(q)p Fm(j)h Ft(+)f Fo(q)k 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Ft(suc)n(h)h(that)g(the)g(second)f(term)659 1161 y(in)28 b(the)g(curly)f(brac)n(k)n(ets)f(can)h(b)r(e)h(estimated)f(b)n(y)666 1281 y Fh(Z)735 1394 y Fo(d)p Fi(p)831 1360 y Fn(0)855 1394 y Fm(j)889 1372 y Ft(^)878 1394 y Fo(\036)927 1406 y Fl(k)968 1394 y Ft(\()p Fi(p)19 b Fm(\000)f Fi(p)1208 1360 y Fn(0)1231 1394 y Fo(;)c Fi(p)1321 1360 y Fn(0)1344 1394 y Ft(\))p Fm(j)1409 1338 y Fo(p)1451 1308 y Fp(2)p 1410 1375 80 4 v 1417 1451 a Fo(p)1459 1427 y Fn(0)1522 1394 y Fm(\024)22 b Fo(c)1659 1281 y Fh(Z)1728 1394 y Fo(d)p Fi(p)1824 1360 y Fn(0)2110 1338 y Ft(1)p 1858 1375 545 4 v 1858 1451 a Fm(j)p Fi(p)c Fm(\000)5 b Fi(p)2075 1427 y Fn(0)2098 1451 y Fm(j)19 b Ft(+)5 b Fo(p)2252 1427 y Fn(0)2292 1451 y Ft(+)g(1)2413 1394 y Fm(j)17 b Ft(^)-59 b Fo(w)2495 1406 y Fl(k)2536 1394 y Ft(\()p Fi(p)19 b Fm(\000)f Fi(p)2776 1360 y Fn(0)2799 1394 y Fo(;)c Fi(p)2889 1360 y Fn(0)2913 1394 y Ft(\))p Fm(j)2978 1338 y Fo(p)3020 1308 y Fp(2)p 2978 1375 80 4 v 2985 1451 a Fo(p)3027 1427 y Fn(0)3090 1394 y 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Ft(cannot)f(b)r(e)i(based)f(on)g(the)451 4555 y(algebra)27 b(of)h(sym)n(b)r(ol)f(estimates,)h(a)g(p)r(o)n(w)n(erful)g(metho)r(d)g (in)h(the)f(case)g(of)g(p)r(erio)r(dic)g(p)r(oten-)451 4654 y(tials)g([13)o(].)451 4875 y Fr(2.4)91 b(The)32 b(Remainder)g Fo(R)1400 4845 y Fp(\()p Fl(n)p Fp(+1\))1612 4875 y Fr(and)g(its)f Fo(p)p Fr(-f)n(orm)h(boundedness)451 5031 y Ft(F)-7 b(rom)20 b(its)g(de\014nition)g(as)f(remainder)g(after)g (m)n(ultiple)i(iterations)e(of)g(\(2.3\)-t)n(yp)r(e)h(equations)451 5130 y(\(see)32 b(e.g.)50 b(\(2.11\)\),)32 b Fo(R)1161 5100 y Fp(\()p Fl(n)p Fp(+1\))1374 5130 y Ft(is)g(comp)r(osed)f(of)h(a) g(\014nite)g(n)n(um)n(b)r(er)g(of)g(compact)f(in)n(tegrals)1108 5338 y Fu(Document)l(a)25 b(Ma)l(thema)l(tica)h(10)f(\(2005\))g (331{356)p eop %%Page: 346 16 346 15 bop 451 518 a Ft(346)727 b Fr(D.)30 b(H.)h(Jakubassa-Amundsen) 451 725 y Ft(o)n(v)n(er)j(a)i(unitary)f(transform)g(of)h(the)h(same)e (m)n(ultiple)i(comm)n(utators)d([)p Fm(\001)p Ft(])i(whic)n(h)g(w)n (ould)451 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b(F)-6 b(or)33 b(every)g Fo(k)f Fm(2)c Fk(N)t Fo(;)77 b(\017)28 b(<)1574 4324 y Fp(1)p 1530 4338 121 4 v 1530 4385 a Fl(k)q Fp(+1)1661 4357 y Fo(;)61 b Fd(the)33 b Fo(k)s Fd(-th)f(or)l(der)i(p)l(otential)f(term)g Fo(V)2878 4369 y Fl(k)2951 4357 y Fd(is)h Fo(p)p Fd(-form)659 4456 y(b)l(ounde)l(d)c (with)g(form)h(b)l(ound)e(less)h(than)g(1.)498 4613 y(\(ii\))43 b(L)l(et)25 b Fo(')e Fm(2)g(S)6 b Ft(\()p Fk(R)1096 4583 y Fp(3)1139 4613 y Ft(\))j Fm(\012)g Fk(C)1307 4583 y Fp(4)1351 4613 y Fo(:)48 b Fd(L)l(et)25 b Fo(\026)1611 4625 y Fl(k)1688 4613 y Fo(>)e Ft(0)i Fd(for)h Fo(k)g Fm(2)d Fk(N)35 b Fd(b)l(e)25 b(the)h(in\014mum)e(of)i(the)g(c)l (onstant)659 4713 y Fo(c)g Fd(in)h(the)g(estimate)g Fm(j)p Ft(\()p Fo(';)14 b(V)1477 4725 y Fl(k)1519 4713 y Fo(')p Ft(\))p Fm(j)24 b(\024)36 b Fo(c)14 b Ft(\()p Fo(';)g(p)1968 4682 y Fp(1)p Fn(\000)p Fl(k)q(\017)2122 4713 y Fo(')p Ft(\))p Fo(:)50 b Fd(Then)28 b Fo(V)2543 4725 y Fl(k)q Fp(+1)2695 4713 y Fd(is)f(sub)l(or)l(dinate)g(to)659 4812 y Fo(V)707 4824 y Fl(k)778 4812 y Fd(in)j(the)f(sense)1245 4971 y Fm(j)p Ft(\()p Fo(';)14 b(V)1439 4983 y Fl(k)q Fp(+1)1579 4971 y Fo(')p Ft(\))p Fm(j)46 b(\024)g Fo(\016)26 b Fm(j)p Ft(\()p Fo(';)14 b(V)2102 4983 y Fl(k)2144 4971 y Fo(')p Ft(\))p Fm(j)42 b Ft(+)f Fo(C)30 b Ft(\()p Fo(';)14 b(')p Ft(\))416 b(\(3.2\))659 5130 y Fd(with)30 b Ft(0)23 b Fo(<)f(\016)k(<)d Ft(1)52 b Fd(and)31 b Fo(C)e Fm(2)23 b Fk(R)1618 5142 y Fp(+)1709 5130 y Fd(a)30 b(c)l(onstant)f(dep)l (ending)i(on)f Fo(\016)o(:)1108 5338 y Fu(Document)l(a)25 b(Ma)l(thema)l(tica)h(10)f(\(2005\))g(331{356)p eop %%Page: 348 18 348 17 bop 451 518 a Ft(348)727 b Fr(D.)30 b(H.)h(Jakubassa-Amundsen) 473 725 y Fd(\(iii\))43 b(L)l(et)38 b Fo(R)875 695 y Fp(\()p Fl(n)p Fp(+1\))1094 725 y Ft(:=)52 b Fo(U)1300 695 y Fn(\003)1291 746 y Fl(n)1352 725 y Fm(\001)14 b(\001)g(\001)f Fo(U)1528 695 y Fn(\003)1519 746 y Fp(1)1566 725 y Fo(H)7 b(U)1699 737 y Fp(1)1750 725 y Fm(\001)14 b(\001)g(\001)f Fo(U)1917 737 y Fl(n)2001 725 y Fm(\000)38 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Ft(is)g(giv)n(en)e(b)n (y)454 4648 y(^)-45 b Fo(v)491 4660 y Fp(1)529 4648 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))46 b(=)i(^)-46 b Fo(v)t Ft(\()p Fi(q)p Ft(\))36 b Fm(\000)52 b Ft(^)-59 b Fo(w)1241 4660 y Fp(1)1279 4648 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))46 b(=)f Fm(\000)1746 4611 y Fl(\015)1781 4619 y Fg(0)p 1713 4629 134 4 v 1713 4677 a Fl(q)1745 4660 y Fg(2+)p Fe(\017)1870 4648 y Fo(f)1911 4660 y Fl(\017)1956 4648 y Ft(\(1)22 b(+)2158 4627 y(~)2138 4648 y Fo(D)2207 4660 y Fp(0)2244 4648 y Ft(\()p Fi(q)h Ft(+)e Fi(p)p Ft(\))h Fm(\001)2606 4627 y Ft(~)2587 4648 y Fo(D)2656 4660 y Fp(0)2693 4648 y Ft(\()p Fi(p)p Ft(\)\))32 b(.)52 b(Since)33 b(the)451 4770 y(m)n(ultiplier)e(of)f Fo(q)972 4740 y Fn(\000)p Fp(\(2+)p Fl(\017)p Fp(\))1222 4770 y Ft(is)g(a)f(b)r(ounded)i(op)r(erator)d(whic)n(h)j(is)f(estimated)g(b) n(y)g(a)g(constan)n(t,)451 4869 y(one)d(\014nds)h(according)e(to)h (\(2.32\))g(and)h(\(2.33\))f(with)h Fo(f)9 b Ft(\()p Fo(p)p Ft(\))23 b(:=)f Fo(p)1105 5093 y Fm(j)p 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2551 y Fn(0)2631 2581 y Fm(\000)r Fi(q)p Fm(j)2771 2551 y Fp(2+)p Fl(\017)2887 2581 y Fo(:)43 b Ft(Recalling)451 2690 y(that)33 b Fm(j)670 2668 y Ft(^)659 2690 y Fo(\036)708 2702 y Fp(1)746 2690 y Fm(j)g Ft(can)f(b)r(e)h(replaced)f(b)n(y)h(its)g (adjoin)n(t)f Fm(j)1972 2668 y Ft(^)1961 2690 y Fo(\036)2010 2660 y Fn(\003)2010 2710 y Fp(1)2049 2690 y Fm(j)65 b Ft(and)32 b(substituting)h Fi(q)2820 2660 y Fn(0)2876 2690 y Ft(:=)e Fi(q)p Fo(=p)3129 2660 y Fn(0)3217 2690 y Ft(in)451 2789 y(the)d(\014rst)g(in)n(tegral)e(one)h(obtains)936 3000 y Fm(j)p Ft(\()p Fo(';)14 b Ft([)p Fo(V)5 b(;)14 b(B)1258 3012 y Fp(1)1296 3000 y Ft(])p Fo(')p Ft(\))p Fm(j)47 b(\024)f Fo(c)1649 2887 y Fh(Z)1746 3000 y Fo(d)p Fi(p)23 b Fm(j)13 b Ft(^)-55 b Fo(')q Ft(\()p Fi(p)p Ft(\))p Fm(j)2083 2966 y Fp(2)2144 3000 y Fm(f)p Fo(I)2222 3012 y Fp(11)2292 3000 y Ft(\()p Fo(p)p Ft(\))42 b(+)f Fo(I)2582 3012 y Fp(12)2653 3000 y Ft(\()p Fo(p)p Ft(\))p Fm(g)273 b Ft(\(3.10\))537 3277 y Fo(I)573 3289 y Fp(11)644 3277 y 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%%Page: 350 20 350 19 bop 451 518 a Ft(350)727 b Fr(D.)30 b(H.)h(Jakubassa-Amundsen) 451 725 y Fo(I)487 737 y Fp(12)590 725 y Ft(is)i(treated)f(in)g(a)g (similar)g(w)n(a)n(y)-7 b(.)49 b(The)33 b Fi(q)p Ft(-in)n(tegral)e(is)h (the)h(one)f(from)g(\(3.8\))g(whic)n(h)g(is)451 825 y(estimated)h(b)n (y)f Fo(c)14 b(p)1047 770 y Ff(0)1069 795 y Fp(1)p Fn(\000)p Fl(\017)1186 825 y Fo(:)64 b Ft(The)32 b(remaining)g(in)n(tegral)f(is)h (estimated)h(b)n(y)f(replacing)f Fm(j)p Fi(p)3176 795 y Fn(0)3221 825 y Fm(\000)451 924 y Fi(p)p Fm(j)j Ft(+)19 b Fo(p)34 b Ft(+)19 b(1)56 b(with)30 b Fm(j)p Fi(p)1170 894 y Fn(0)1213 924 y Fm(\000)20 b Fi(p)p Fm(j)33 b Ft(+)20 b Fo(p:)56 b Ft(With)30 b(the)g(substitution)g Fi(p)2491 894 y Fn(0)2541 924 y Ft(=:)c Fo(p)p Fi(k)20 b Ft(+)g Fi(p)56 b Ft(and)29 b(with)451 1024 y(the)f(tec)n(hniques)g(from)f (\(2.34\))g(one)g(obtains)1103 1260 y Fo(p)1145 1226 y Fp(2)1219 1147 y Fh(Z)1316 1260 y Fo(d)p Fi(p)1412 1226 y Fn(0)2025 1204 y Ft(1)p 1469 1241 1155 4 v 1469 1317 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y(with)28 b(some)f(constan)n(t)g Fo(c:)h Ft(Using)f(\(3.7\))h(and) f(the)h(de\014nition)g(of)f Fo(\026)2519 1807 y Fl(k)2588 1795 y Ft(one)g(th)n(us)h(obtains)699 1988 y Fm(j)p Ft(\()p Fo(';)14 b(V)893 2000 y Fp(2)931 1988 y Fo(')p Ft(\))p Fm(j)47 b(\024)e Fo(c)1233 2000 y Fp(0)1294 1988 y Ft(\()p Fo(';)14 b(p)1459 1954 y Fp(1)p Fn(\000)p Fp(2)p Fl(\017)1609 1988 y Fo(')p Ft(\))46 b Fm(\024)g Fo(c)1888 2000 y Fp(0)1948 1988 y Fo(\016)27 b Ft(\()p Fo(';)14 b(p)2177 1954 y Fp(1)p Fn(\000)p Fl(\017)2294 1988 y Fo(')p Ft(\))42 b(+)f Fo(C)29 b Ft(\()p Fo(';)14 b(')p Ft(\))249 b(\(3.14\))1325 2241 y Fo(<)1446 2185 y Ft(2)p Fo(c)1524 2197 y Fp(0)1574 2185 y Fo(\016)p 1446 2222 169 4 v 1486 2298 a(\026)1536 2310 y Fp(1)1647 2241 y Fm(j)p Ft(\()p Fo(';)14 b(V)1841 2253 y Fp(1)1879 2241 y Fo(')p Ft(\))p Fm(j)43 b Ft(+)e Fo(C)6 b Ft(\()p Fo(';)14 b(')p Ft(\))451 2451 y(with)30 b(2)p Fo(c)720 2463 y Fp(0)756 2451 y Fo(\016)s(=\026)888 2463 y Fp(1)964 2451 y Fo(<)25 b Ft(1)54 b(for)29 b(a)f(suitably)h(c)n (hosen)f Fo(\016)o(:)54 b Ft(This)29 b(pro)n(v)n(es)e(\(ii\))j(of)e (Prop)r(osition)g(3.1)451 2551 y(for)f Fo(k)f Ft(=)d(1)p Fo(:)451 2656 y Ft(The)28 b(pro)r(of)f(of)h(the)g(induction)g(step)g (from)g Fo(k)i Ft(to)e Fo(k)22 b Ft(+)c(1)27 b(pro)r(ceeds)g(along)g (the)h(same)f(lines)451 2756 y(as)k(applied)h(in)g(section)g(2.3)f(to)g (sho)n(w)g(the)h Fo(p)p Ft(-form)f(b)r(oundedness)h(of)g Fo(V)2744 2768 y Fl(n)2821 2756 y Ft(and)g Fo(W)3065 2768 y Fl(n)3111 2756 y Ft(.)49 b(By)451 2855 y(induction)34 b(h)n(yp)r(othesis)f(comm)n(utators)f(of)i(order)e Fo(m)g Fm(\024)h Fo(k)j Ft(in)e(the)g(p)r(oten)n(tial)f(strength,)451 2955 y(denoted)28 b(b)n(y)f([)p Fm(\001)p Ft(])949 2967 y Fl(m)1012 2955 y Fo(;)h Ft(ha)n(v)n(e)f(the)h(follo)n(wing)e(sym)n(b) r(ol)h(estimates)g(\(compare)g(\(2.43\)\))590 3105 y Fh(Z)687 3218 y Fo(d)p Fi(q)817 3126 y Fh(\020)890 3123 y(\014)890 3172 y(\014)890 3222 y(\014)929 3191 y(b)918 3218 y Ft([)p Fm(\001)p Ft(])987 3243 y Fl(m)1050 3218 y Ft(\()p Fi(q)19 b Fm(\000)f Fi(p)p Fo(;)c Fi(p)p Ft(\))1409 3123 y Fh(\014)1409 3172 y(\014)1409 3222 y(\014)1478 3218 y Ft(+)1585 3123 y Fh(\014)1585 3172 y(\014)1585 3222 y(\014)1624 3191 y(b)1612 3218 y Ft([)p Fm(\001)p Ft(])1681 3152 y Fn(\003)1681 3243 y Fl(m)1744 3218 y Ft(\()p Fi(q)20 b Fm(\000)e Fi(p)p Fo(;)c Fi(p)p Ft(\))2104 3123 y Fh(\014)2104 3172 y(\014)2104 3222 y(\014)2155 3126 y(\021)2241 3101 y(\022)2312 3162 y Fo(p)p 2312 3199 42 4 v 2313 3275 a(q)2364 3101 y Fh(\023)2425 3118 y Fp(2)p Fl(\025)2548 3218 y Fm(\024)45 b Fo(c)23 b(p)2759 3184 y Fp(1)p Fn(\000)p Fl(m\017)3074 3218 y Ft(\(3.15\))451 3469 y(where)g Fo(\025)h Ft(can)f(b)r(e)h(c)n(hosen)e(in)i(the)f(in)n (terv)-5 b(al)46 b(])1860 3437 y Fp(1)p 1860 3451 34 4 v 1860 3498 a(2)1903 3469 y Fo(;)1950 3437 y Fp(3)p 1950 3451 V 1950 3498 a(2)1993 3469 y Ft([)23 b Fo(:)h Ft(W)-7 b(e)24 b(demonstrate)e(the)i(pro)r(of)f(for)g(the)451 3569 y(comm)n(utator)31 b([[)p Fm(\001)p Ft(])1016 3581 y Fl(m)1079 3569 y Fo(;)14 b(B)1179 3581 y Fl(k)q Fn(\000)p Fl(m)p Fp(+1)1415 3569 y Ft(])63 b(whic)n(h)32 b(con)n(tributes)f(to)h Fo(V)2333 3581 y Fl(k)q Fp(+1)2458 3569 y Fo(:)63 b Ft(F)-7 b(or)32 b(the)g(comm)n(utator)451 3669 y([)p Fo(V)5 b(;)14 b(B)627 3681 y Fl(k)668 3669 y Ft(])32 b(whic)n(h)g(also)f(con)n (tributes)h(to)g Fo(V)1727 3681 y Fl(k)q Fp(+1)1885 3669 y Ft(the)g(pro)r(of)g(is)g(similar.)50 b(Since)32 b(the)g(sym)n(b)r(ol) 451 3768 y(classes)26 b(of)i Fo(W)888 3780 y Fl(m)979 3768 y Ft(and)g([)p Fm(\001)p Ft(])1210 3780 y Fl(m)1301 3768 y Ft(are)e(equal,)h(it)h(follo)n(ws)f(from)g(\(2.44\))701 3987 y Fm(j)735 3965 y Ft(^)724 3987 y Fo(\036)773 3999 y Fl(m)837 3987 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))p Fm(j)47 b(\024)1377 3930 y Fo(c)p 1232 3968 327 4 v 1232 4044 a(q)21 b Ft(+)d Fo(p)g Ft(+)g(1)1605 3891 y Fh(\014)1605 3941 y(\014)1605 3991 y(\014)1644 3960 y(b)1633 3987 y Ft([)p Fm(\001)p Ft(])1702 4012 y Fl(m)1765 3987 y Ft(\()p Fi(q)p Fo(;)c Fi(p)p Ft(\))1969 3891 y Fh(\014)1969 3941 y(\014)1969 3991 y(\014)2043 3987 y Fm(\024)2238 3930 y Fo(c)p 2164 3968 184 4 v 2164 4044 a(q)22 b Ft(+)c(1)2394 3891 y Fh(\014)2394 3941 y(\014)2394 3991 y(\014)2433 3960 y(b)2422 3987 y Ft([)p Fm(\001)p Ft(])2491 4012 y Fl(m)2554 3987 y Ft(\()p Fi(q)p Fo(;)c Fi(p)p Ft(\))2758 3891 y Fh(\014)2758 3941 y(\014)2758 3991 y(\014)2800 3987 y Fo(:)251 b Ft(\(3.16\))451 4224 y(Then)28 b(from)f(\(2.50\),)g (one)g(has)g(with)h(some)g Fo(c)1861 4236 y Fp(0)1921 4224 y Fm(2)23 b Fk(R)2053 4236 y Fp(+)2114 4224 y Fo(;)707 4460 y Fm(j)p Ft(\()p Fo(';)14 b Ft([[)p Fm(\001)p Ft(])945 4472 y Fl(m)1008 4460 y Fo(;)g(B)1108 4472 y Fl(k)q Fn(\000)p Fl(m)p Fp(+1)1344 4460 y Ft(])g Fo(')p Ft(\))p Fm(j)47 b(\024)1718 4404 y Fo(c)1754 4416 y Fp(0)p 1657 4441 194 4 v 1657 4517 a Ft(\(2)p Fo(\031)s Ft(\))1813 4493 y Fp(3)1898 4347 y Fh(Z)1995 4460 y Fo(d)p Fi(p)23 b Fm(j)13 b Ft(^)-55 b Fo(')p Ft(\()p Fi(p)p Ft(\))p Fm(j)2331 4426 y Fp(2)2392 4460 y Ft(\()p Fo(I)2460 4472 y Fp(00)2573 4460 y Ft(+)41 b Fo(I)2715 4472 y Fp(01)2786 4460 y Ft(\))256 b(\(3.17\))475 4791 y Fo(I)511 4803 y Fp(00)605 4791 y Ft(:=)702 4678 y Fh(Z)771 4791 y Fo(d)p Fi(p)867 4757 y Fn(0)890 4791 y Fm(j)925 4764 y Fh(b)913 4791 y Ft([)p Fm(\001)p Ft(])983 4816 y Fl(m)1046 4791 y Ft(\()p Fi(p)18 b Fm(\000)g Fi(p)1285 4757 y Fn(0)1309 4791 y Fo(;)c Fi(p)1399 4757 y Fn(0)1422 4791 y Ft(\))p Fm(j)1499 4735 y Fo(p)1541 4705 y Fp(2)p 1487 4772 102 4 v 1487 4848 a Fo(p)1529 4807 y Ff(0)1552 4824 y Fp(2)1617 4791 y Fm(\001)1645 4678 y Fh(Z)1714 4791 y Fo(d)p Fi(q)2075 4735 y Ft(1)p 1817 4772 557 4 v 1817 4848 a Fm(j)p Fi(q)5 b Fm(\000)g Fi(p)2018 4824 y Fn(0)2041 4848 y Fm(j)19 b Ft(+)f Fo(p)2208 4824 y Fn(0)2249 4848 y Ft(+)g(1)2384 4791 y Fm(j)2419 4764 y Fh(b)2407 4791 y Ft([)p Fm(\001)p Ft(])2476 4816 y Fl(k)q Fn(\000)p Fl(m)p Fp(+1)2712 4791 y Ft(\()p Fi(q)h Fm(\000)f Fi(p)2949 4757 y Fn(0)2972 4791 y Fo(;)c Fi(p)3062 4757 y Fn(0)3085 4791 y Ft(\))p Fm(j)3150 4735 y Fo(p)3192 4680 y Ff(0)3215 4705 y Fp(2)p 3151 4772 102 4 v 3163 4848 a Fo(q)3203 4824 y Fp(2)466 5094 y Fo(I)502 5106 y Fp(01)596 5094 y Ft(:=)693 4981 y Fh(Z)762 5094 y Fo(d)p Fi(p)858 5060 y Fn(0)1164 5038 y Ft(1)p 891 5075 587 4 v 891 5151 a Fm(j)p Fi(p)19 b Fm(\000)f Fi(p)1122 5127 y Fn(0)1145 5151 y Fm(j)h Ft(+)f Fo(p)1312 5127 y Fn(0)1353 5151 y Ft(+)g(1)1488 5094 y Fm(j)1523 5068 y Fh(b)1511 5094 y Ft([)p Fm(\001)p Ft(])1580 5119 y Fl(k)q Fn(\000)p Fl(m)p Fp(+1)1816 5094 y Ft(\()p Fi(p)g Fm(\000)g Fi(p)2055 5060 y Fn(0)2079 5094 y Fo(;)c Fi(p)2169 5060 y Fn(0)2192 5094 y Ft(\))p Fm(j)2268 5038 y Fo(p)2310 5008 y Fp(2)p 2257 5075 102 4 v 2257 5151 a Fo(p)2299 5110 y Ff(0)2321 5127 y Fp(2)2387 5094 y Fm(\001)2415 4981 y Fh(Z)2484 5094 y Fo(d)p Fi(q)g Fm(j)2626 5068 y Fh(b)2614 5094 y Ft([)p Fm(\001)p Ft(])2684 5119 y Fl(m)2747 5094 y Ft(\()p Fi(p)2832 5060 y Fn(0)2874 5094 y Fm(\000)k Fi(q)p Fo(;)c Fi(q)p Ft(\))p Fm(j)3159 5038 y Fo(p)3201 4983 y Ff(0)3224 5008 y Fp(2)p 3160 5075 V 3172 5151 a Fo(q)3212 5127 y Fp(2)1108 5338 y Fu(Document)l(a)25 b(Ma)l(thema)l(tica)h(10)f(\(2005\))g(331{356)p eop %%Page: 351 21 351 20 bop 1291 518 a Fr(Pr)n(ojected)32 b(Dira)n(c)f(Opera)-6 b(tor)716 b Ft(351)451 725 y(where)31 b(in)h(the)g(term)f(denoted)h(b)n (y)f Fo(I)1618 737 y Fp(00)1689 725 y Fo(;)73 b Fm(j)1819 703 y Ft(^)1808 725 y Fo(\036)1857 737 y Fl(k)q Fn(\000)p Fl(m)p Fp(+1)2093 725 y Ft(\()p Fi(p)2178 695 y Fn(0)2222 725 y Fm(\000)21 b Fi(q)p Fo(;)14 b Fi(q)p Ft(\))p Fm(j)33 b Ft(w)n(as)d(estimated)i(b)n(y)f(its)451 825 y(adjoin)n(t)d(b)r(efore) f(applying)h(\(3.16\).)36 b(In)28 b Fo(I)1734 837 y Fp(01)1805 825 y Ft(,)g(the)g Fi(q)p Ft(-in)n(tegral)f(is)h(b)n(y)f(\(3.15\))g (estimated)h(b)n(y)451 924 y Fo(c)14 b(p)543 869 y Ff(0)565 894 y Fp(1)p Fn(\000)p Fl(m\017)741 924 y Fo(:)51 b Ft(F)-7 b(urther)27 b(one)g(has)g(with)h Fo(a)23 b Fm(\025)g Ft(0)k(and)h Fo(\016)e(>)c Ft(0)527 1048 y Fh(Z)624 1161 y Fo(d)p Fi(q)894 1105 y Ft(1)p 751 1142 329 4 v 751 1218 a Fo(a)c Ft(+)g Fo(q)k Ft(+)c(1)1112 1161 y Fm(j)1147 1134 y Fh(b)1135 1161 y Ft([)p Fm(\001)p Ft(])1204 1186 y Fl(n)1250 1161 y Ft(\()p Fi(p)g Fm(\000)g Fi(q)p Fo(;)c Fi(q)p Ft(\))p Fm(j)1663 1105 y Fo(p)1705 1075 y Fp(2)p 1663 1142 80 4 v 1664 1218 a Fo(q)1704 1194 y Fp(2)1770 1161 y Fm(\001)k Fo(q)1851 1127 y Fp(1)p Fn(\000)p Fl(\016)2019 1161 y Fm(\024)46 b Fo(p)2172 1127 y Fn(\000)p Fl(\016)2274 1048 y Fh(Z)2371 1161 y Fo(d)p Fi(q)23 b Fm(j)2522 1134 y Fh(b)2510 1161 y Ft([)p Fm(\001)p Ft(])2579 1186 y Fl(n)2625 1161 y Ft(\()p Fi(p)18 b Fm(\000)g Fi(q)p Fo(;)c Fi(q)p Ft(\))p Fm(j)3037 1105 y Fo(p)3079 1075 y Fp(2+)p Fl(\016)p 3037 1142 163 4 v 3038 1218 a Fo(q)3078 1194 y Fp(2+)p Fl(\016)1642 1406 y Fm(\024)45 b Fo(c)23 b(p)1853 1372 y Fp(1)p Fn(\000)p Fl(n\017)p Fn(\000)p Fl(\016)3074 1406 y Ft(\(3.18\))451 1559 y(if)28 b(1)16 b(+)691 1526 y Fl(\016)p 691 1540 34 4 v 691 1588 a Fp(2)771 1559 y Fo(<)869 1526 y Fp(3)p 869 1540 V 869 1588 a(2)912 1559 y Fo(:)41 b Ft(Then)27 b(with)g Fo(\016)f Ft(:=)d Fo(m\017;)97 b(I)1817 1571 y Fp(01)1924 1559 y Fm(\024)25 b Ft(~)-44 b Fo(c)14 b(p)2104 1529 y Fp(1)p Fn(\000)p Fp(\()p Fl(k)q Fn(\000)p Fl(m)p Fp(+1\))p Fl(\017)2518 1559 y Fo(p)2560 1529 y Fn(\000)p Fl(m\017)2748 1559 y Ft(=)48 b(~)-44 b Fo(c)13 b(p)2950 1529 y Fp(1)p Fn(\000)p Fp(\()p Fl(k)q Fp(+1\))p Fl(\017)3240 1559 y Fo(:)451 1660 y Ft(In)21 b Fo(I)584 1672 y Fp(00)675 1660 y Ft(w)n(e)f(estimate) h(in)f(the)h(denominator)f Fm(j)p Fi(q)t Fm(\000)t Fi(p)2018 1630 y Fn(0)2041 1660 y Fm(j)t Ft(+)t Fo(p)2179 1630 y Fn(0)2206 1660 y Ft(+)t(1)g(b)n(y)g Fo(p)2487 1630 y Fn(0)2531 1660 y Ft(and)g(subsequen)n(tly)g(use)451 1760 y(\(3.15\))i(to)g(estimate)g(the)h Fi(q)p Ft(-in)n(tegral)e(b)n(y) h Fo(c)14 b(p)1824 1704 y Ff(0)1846 1729 y Fn(\000)p Fp(\()p Fl(k)q Fn(\000)p Fl(m)p Fp(+1\))p Fl(\017)2214 1760 y Fo(:)45 b Ft(With)23 b Fo(\025)h Ft(:=)36 b(1)8 b(+)2820 1727 y Fl(k)q Fn(\000)p Fl(m)p Fp(+1)p 2820 1741 232 4 v 2919 1788 a(2)3075 1760 y Fo(\017)45 b Ft(\(for)451 1870 y(\()p Fo(k)5 b Fm(\000)r Fo(m)r Ft(+)r(1\))p Fo(\017)36 b(<)22 b Ft(1\))43 b(in)19 b(\(3.15\))g(w)n(e)g(obtain)g Fo(I)1808 1882 y Fp(00)1916 1870 y Fm(\024)38 b Ft(~)-44 b Fo(c)14 b(p)2109 1840 y Fn(\000)p Fp(\()p Fl(k)q Fn(\000)p Fl(m)p Fp(+1\))p Fl(\017)2478 1870 y Fm(\001)r Fo(p)2545 1840 y Fp(1)p Fn(\000)p Fl(m\017)2757 1870 y Ft(=)39 b(~)-44 b Fo(c)13 b(p)2950 1840 y Fp(1)p Fn(\000)p Fp(\()p Fl(k)q Fp(+1\))p Fl(\017)3240 1870 y Fo(:)451 1970 y Ft(Therefore)1079 2073 y Fm(j)p Ft(\()p Fo(';)h Ft([[)p Fm(\001)p Ft(])1317 2085 y Fl(m)1381 2073 y Fo(;)g(B)1481 2085 y Fl(k)q Fn(\000)p Fl(m)p Fp(+1)1717 2073 y Ft(])g Fo(')p Ft(\))p Fm(j)46 b(\024)g Fo(c)2056 2085 y Fp(0)2116 2073 y Ft(\()p Fo(';)14 b(p)2281 2039 y Fp(1)p Fn(\000)p Fp(\()p Fl(k)q Fp(+1\))p Fl(\017)2571 2073 y Fo(')p Ft(\))417 b(\(3.19\))451 2226 y(whic)n(h)55 b(pro)n(v)n(es)f(\(3.15\))g(for)h Fo(k)40 b Ft(+)c(1.)120 b(The)55 b(same)g(estimate)g(can)g(b)r(e)h(sho) n(wn)e(for)451 2326 y Fm(j)p Ft(\()p Fo(';)14 b Ft([)p Fo(V)5 b(;)14 b(B)773 2338 y Fl(k)815 2326 y Ft(])g Fo(')p Ft(\))p Fm(j)p Fo(:)51 b Ft(Hence)564 2550 y Fm(j)p Ft(\()p Fo(';)14 b(V)758 2562 y Fl(k)q Fp(+1)884 2550 y Fo(')p Ft(\))p Fm(j)47 b(\024)f Fo(c)1187 2516 y Fn(0)1233 2550 y Ft(\()p Fo(';)14 b(p)1398 2516 y Fp(1)p Fn(\000)p Fp(\()p Fl(k)q Fp(+1\))p Fl(\017)1688 2550 y Fo(')p Ft(\))46 b Fm(\024)23 b Fo(c)1944 2516 y Fn(0)1981 2550 y Fo(\016)2070 2494 y Ft(2)p 2045 2531 91 4 v 2045 2607 a Fo(\026)2095 2619 y Fl(k)2169 2550 y Fm(j)p Ft(\()p Fo(';)14 b(V)2363 2562 y Fl(k)2405 2550 y Fo(')p Ft(\))p Fm(j)42 b Ft(+)f Fo(C)29 b Ft(\()p Fo(';)14 b(')p Ft(\))115 b(\(3.20\))451 2779 y(whic)n(h)28 b(completes)f(the)h(pro)r(of)f(of)h(Prop)r(osition)d (3.1)i(\(ii\).)451 2880 y(The)f(pro)r(of)g(of)f(\(i\))i(for)e Fo(k)h(>)d Ft(1)i(is)h(again)f(b)n(y)h(induction.)36 b(Assume)26 b Fo(V)2561 2892 y Fl(k)2629 2880 y Ft(is)f Fo(p)p Ft(-form)g(b)r(ounded)451 2980 y(with)j(form)g(b)r(ound)g Fo(c)1129 2992 y Fp(1)1189 2980 y Fo(<)22 b Ft(1)p Fo(:)51 b Ft(Then)27 b(w)n(e)h(ha)n(v)n(e)e(from)h(\(ii\))529 3165 y Fm(j)p Ft(\()p Fo(';)14 b(V)723 3177 y Fl(k)q Fp(+1)849 3165 y Fo(')p Ft(\))p Fm(j)24 b(\024)e Fo(\016)s Fm(j)p Ft(\()p Fo(';)14 b(V)1303 3177 y Fl(k)1345 3165 y Fo(')p Ft(\))p Fm(j)19 b Ft(+)f Fo(C)6 b Ft(\()p Fo(';)14 b(')p Ft(\))25 b Fm(\024)d Fo(\016)17 b Ft(\()q Fo(c)2065 3177 y Fp(1)2102 3165 y Ft(\()p Fo(';)d(p)g(')p Ft(\))19 b(+)f Fo(C)2528 3177 y Fp(1)2566 3165 y Ft(\()p Fo(';)c(')p Ft(\)\))19 b(+)f Fo(C)6 b Ft(\()p Fo(';)14 b(')p Ft(\))p Fo(:)3074 3264 y Ft(\(3.21\))451 3364 y(Since)35 b Fo(\016)i Ft(can)d(b)r(e)g(c)n(hosen)g(arbitrarily)e(small,)k(one)d(has)h Fo(\016)s(c)2344 3376 y Fp(1)2415 3364 y Fo(<)g Ft(1)p Fo(:)68 b Ft(A)35 b(consequence)e(of)451 3464 y(\(3.21\))e(is)g(the)g Fo(p)p Ft(-form)g(b)r(oundedness)g(\(with)h(form)f(b)r(ound)g Fo(<)e Ft(1\))60 b(of)31 b(ev)n(ery)f(\014nite)i(sum)451 3563 y Fo(V)499 3575 y Fp(1)555 3563 y Ft(+)18 b Fo(:::)h Ft(+)f Fo(V)857 3575 y Fl(n)903 3563 y Fo(:)451 3664 y Ft(F)-7 b(or)24 b(the)h(pro)r(of)f(of)g(\(iii\))i(w)n(e)e(ha)n(v)n(e) f(to)h(sho)n(w)g(that)h(all)f Fo(B)2161 3676 y Fl(k)2226 3664 y Ft(are)g(b)r(ounded)h(op)r(erators.)34 b(Then)451 3764 y(w)n(e)28 b(can)g(pro)r(ceed)g(as)g(in)h(section)f(2.4)f(to)i (sho)n(w)e(that)i(a)f(unitary)g(transform)g Fo(U)2942 3776 y Fl(k)3007 3764 y Ft(=)c Fo(e)3135 3734 y Fl(iB)3208 3743 y Fe(k)3244 3734 y Fl(\034)451 3863 y Ft(preserv)n(es)30 b(the)i Fo(p)1001 3833 y Fp(1)p Fn(\000)p Fp(\()p Fl(n)p Fp(+1\))p Fl(\017)1295 3863 y Ft(-form)f(b)r(oundedness)h(of)g(the)g (comm)n(utators)f(of)h(order)e Fo(n)21 b Ft(+)g(1)451 3963 y(in)29 b(the)f(p)r(oten)n(tial)g(strength)g(of)h(whic)n(h)f Fo(R)1770 3933 y Fp(\()p Fl(n)p Fp(+1\))1979 3963 y Ft(is)g (consisting.)38 b(Consequen)n(tly)-7 b(,)28 b(one)f(has)451 4063 y(with)h(\()p Fo(';)14 b(p)805 4032 y Fp(1)p Fn(\000)p Fl(k)q(\017)959 4063 y Fo(')p Ft(\))24 b Fo(<)1188 4030 y Fp(2)p 1166 4044 77 4 v 1166 4091 a Fl(\026)1206 4100 y Fe(k)1267 4063 y Fm(j)p Ft(\()p Fo(';)14 b(V)1461 4075 y Fl(k)1503 4063 y Fo(')p Ft(\))p Fm(j)51 b Ft(for)27 b Fo(k)f Ft(=)d Fo(n)18 b Ft(+)g(1)27 b(and)h(with)g(\(3.2\),)493 4274 y Fm(j)p Ft(\()p Fo(';)14 b(R)703 4240 y Fp(\()p Fl(n)p Fp(+1\))898 4274 y Fo(')p Ft(\))p Fm(j)47 b(\024)e Ft(const)q Fm(\001)q(j)p Ft(\()p Fo(';)14 b(V)1573 4286 y Fl(n)p Fp(+1)1703 4274 y Fo(')p Ft(\))p Fm(j)47 b(\024)f Ft(const)q Fm(\001)q Fo(\016)26 b Fm(j)p Ft(\()p Fo(';)14 b(V)2442 4286 y Fl(n)2488 4274 y Fo(')p Ft(\))p Fm(j)25 b Ft(+)e Fo(C)2775 4240 y Fn(0)2822 4274 y Ft(\()p Fo(';)14 b(')p Ft(\))43 b(\(3.22\))451 4459 y(with)33 b(const)14 b Fm(\001)g Fo(\016)32 b(<)e Ft(1)62 b(for)31 b(a)g(suitably)h(c)n (hosen)f Fo(\016)o(:)62 b Ft(This)32 b(sho)n(ws)e(the)j(sub)r(ordinacy) d(with)451 4559 y(resp)r(ect)e(to)f Fo(V)885 4571 y Fl(n)931 4559 y Fo(:)451 4660 y Ft(It)e(remains)e(to)i(pro)n(v)n(e)d(the)j(b)r (oundedness)g(of)f Fo(B)1942 4672 y Fl(k)1983 4660 y Fo(:)g Ft(W)-7 b(e)25 b(will)g(sho)n(w)e(this)i(directly)f(b)n(y)g (using)451 4759 y(the)37 b(algebra)e(of)i(sym)n(b)r(ol)f(estimates.)65 b(F)-7 b(or)36 b Fo(B)1947 4771 y Fp(1)1984 4759 y Ft(,)j(from)e (\(2.57\))f(with)h(the)g(substitution)451 4859 y Fi(q)501 4829 y Fn(0)548 4859 y Ft(:=)23 b Fi(q)18 b Fm(\000)g Fi(p)p Fo(;)822 5093 y(I)858 5105 y Fl(B)908 5113 y Fg(1)991 5093 y Fm(\024)1224 5036 y Fo(c)p 1112 5073 261 4 v 1112 5151 a Ft(\(2)p Fo(\031)s Ft(\))1268 5127 y Fp(3)p Fl(=)p Fp(2)1396 4980 y Fh(Z)1493 5093 y Fo(d)p Fi(q)1782 5036 y Ft(1)p 1620 5073 367 4 v 1620 5149 a Fm(j)p Fi(p)g Fm(\000)g Fi(q)p Fm(j)1870 5126 y Fp(1+)p Fl(\017)2327 5036 y Ft(1)p 2030 5073 638 4 v 2030 5149 a(\()p Fm(j)p Fi(p)g Fm(\000)g Fi(q)p Fm(j)h Ft(+)f Fo(q)k Ft(+)c(1\))2630 5126 y Fp(2)2695 5093 y Fm(\001)2747 5036 y Fo(p)2789 5006 y Fp(1)p Fn(\000)p Fl(\017)p 2747 5073 159 4 v 2748 5149 a Fo(q)2788 5126 y Fp(1)p Fn(\000)p Fl(\017)1108 5338 y Fu(Document)l(a)25 b(Ma)l(thema)l(tica)h(10)f(\(2005\))g (331{356)p eop %%Page: 352 22 352 21 bop 451 518 a Ft(352)727 b Fr(D.)30 b(H.)h(Jakubassa-Amundsen) 929 766 y Fm(\024)1162 710 y Fo(c)p 1050 747 261 4 v 1050 824 a Ft(\(2)p Fo(\031)s Ft(\))1206 800 y Fp(3)p Fl(=)p Fp(2)1334 653 y Fh(Z)1431 766 y Fo(d)p Fi(q)1524 732 y Fn(0)1650 710 y Ft(1)p 1581 747 179 4 v 1581 823 a Fo(q)1621 782 y Ff(0)1644 799 y Fp(1+)p Fl(\017)1936 710 y Ft(1)p 1803 747 309 4 v 1803 823 a(\()p Fo(q)1875 799 y Fn(0)1917 823 y Ft(+)18 b(1\))2074 799 y Fp(2)2139 766 y Fm(\001)2307 710 y Fo(p)2349 679 y Fp(1)p Fn(\000)p Fl(\017)p 2191 747 392 4 v 2191 823 a Fm(j)p Fi(p)g Ft(+)g Fi(q)2418 799 y Fn(0)2442 823 y Fm(j)2465 799 y Fp(1)p Fn(\000)p Fl(\017)2638 766 y Fm(\024)46 b Fo(c)2785 732 y Fn(0)3074 766 y Ft(\(3.23\))451 943 y(since)36 b(the)h(in)n(tegral)e (is)h(\014nite)h(for)f Fo(p)h Fm(!)g Ft(0)f(and)g(for)g Fo(p)h Fm(!)h(1)e Ft(and)g(the)h(singularit)n(y)e(of)451 1043 y(the)30 b(last)f(factor)g(at)h Fi(p)c Ft(=)g Fm(\000)p Fi(q)1384 1013 y Fn(0)1437 1043 y Ft(is)k(in)n(tegrable.)41 b(The)30 b(con)n(v)n(ergence)d(generating)h(function)451 1151 y Fo(f)9 b Ft(\()p Fo(p)p Ft(\))23 b(=)g Fo(p)770 1095 y Fg(1)p Ff(\000)p Fe(\017)p 770 1108 99 3 v 805 1141 a Fg(2)909 1151 y Ft(w)n(as)j(c)n(hosen)h(to)f(allo)n(w)g(for)h(a) g(\(2.60\)-t)n(yp)r(e)f(estimate)h(when)g(sho)n(wing)f(that)451 1251 y(the)e(presence)f(of)g Fo(U)1066 1263 y Fl(k)1131 1251 y Ft(pla)n(ys)f(no)h(role)g(\(but)i(to)e(pro)n(v)n(e)f(b)r (oundedness)h(of)h Fo(I)2716 1263 y Fl(B)2766 1271 y Fg(1)2803 1251 y Ft(,)h(one)e(can)g(also)451 1350 y(tak)n(e)k Fo(f)9 b Ft(\()p Fo(p)p Ft(\))23 b(=)g(1\).)451 1459 y(F)-7 b(or)21 b Fo(B)657 1471 y Fp(2)695 1459 y Ft(,)i(w)n(e)e(use)h (the)g(estimate)g(\(2.44\))f(of)1793 1437 y(^)1782 1459 y Fo(\036)1831 1471 y Fp(2)1891 1459 y Ft(b)n(y)38 b(^)-59 b Fo(w)2059 1471 y Fp(2)2119 1459 y Ft(and)21 b(recall)g(that)h Fo(W)2742 1471 y Fp(2)2802 1459 y Ft(is)g(determined)451 1558 y(from)27 b(the)h(comm)n(utator)f([)p Fo(V)5 b(;)14 b(B)1435 1570 y Fp(1)1472 1558 y Ft(])p Fo(:)51 b Ft(Consider)26 b(the)i(sym)n(b)r(ol)g(of)f Fo(V)19 b(B)2567 1570 y Fp(1)2632 1558 y Ft(via)27 b(\(2.17\),)653 1755 y Fm(j)681 1733 y Fh(c)676 1755 y Fo(v)s(\036)769 1775 y Fp(1)806 1755 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))p Fm(j)47 b(\024)1314 1698 y Fo(c)p 1201 1736 261 4 v 1201 1813 a Ft(\(2)p Fo(\031)s Ft(\))1357 1789 y Fp(3)p Fl(=)p Fp(2)1486 1642 y Fh(Z)1583 1755 y Fo(d)p Fi(p)1679 1720 y Fn(0)1909 1698 y Ft(1)p 1735 1736 391 4 v 1735 1812 a Fm(j)p Fi(q)19 b Fm(\000)f Fi(p)1963 1788 y Fn(0)1986 1812 y Fm(j)2009 1788 y Fp(2+)p Fl(\017)2154 1755 y Fm(\001)2501 1698 y Ft(1)p 2205 1736 634 4 v 2205 1812 a Fo(p)2247 1771 y Ff(0)2269 1788 y Fp(1+)p Fl(\017)2385 1812 y Ft(\()p Fo(p)2459 1788 y Fn(0)2501 1812 y Ft(+)g Fo(p)g Ft(+)g(1\))2801 1788 y Fp(2)2848 1755 y Fo(:)203 b Ft(\(3.24\))451 1980 y(It)38 b(is)f(found)h(that)g Fm(j)1105 1959 y Fh(c)1100 1980 y Fo(v)s(\036)1193 2001 y Fp(1)1230 1980 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))p Fm(j)55 b Ft(=)37 b(const)88 b(for)37 b Fo(p)j 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1536 2535 a(\()p Fo(p)1610 2511 y Fn(0)1652 2535 y Ft(+)18 b Fo(p)g Ft(+)g(1\))1952 2511 y Fp(2)2013 2361 y Fh(\022)2206 2422 y Ft(1)p 2084 2459 285 4 v 2084 2535 a Fm(j)p Fo(q)k Fm(\000)c Fo(p)2291 2511 y Fn(0)2314 2535 y Fm(j)2337 2511 y Fl(\017)2420 2478 y Fm(\000)2658 2422 y Ft(1)p 2536 2459 V 2536 2535 a Fm(j)p Fo(q)k Ft(+)c Fo(p)2743 2511 y Fn(0)2766 2535 y Fm(j)2789 2511 y Fl(\017)2831 2361 y Fh(\023)2906 2478 y Fo(:)145 b Ft(\(3.25\))451 2677 y(Since)32 b(the)g(div)n(ergence)e (at)i Fo(q)g Ft(=)e(0)h(results)g(from)g(the)h(b)r(eha)n(viour)f(of)g (the)h(in)n(tegral)f(near)451 2777 y Fo(p)493 2747 y Fn(0)551 2777 y Ft(=)k(0,)h(it)f(is)g(su\016cien)n(t)f(to)h(reduce)f (the)h(in)n(tegration)f(region)f(to)i([0)p Fo(;)14 b Ft(1])33 b(and)i(estimate)451 2877 y(\()p Fo(p)525 2846 y Fn(0)568 2877 y Ft(+)18 b Fo(p)h Ft(+)g(1\))870 2846 y Fn(\000)p Fp(2)984 2877 y Fm(\024)25 b Ft(1)p Fo(:)53 b Ft(The)29 b(resulting)f(in)n(tegral)f(can)i(b)r(e)g(p)r(erformed)f (analytically)f(with)451 2976 y(the)38 b(help)g(of)g(h)n(yp)r (ergeometric)e(functions)i([7,)i(p.284],)g(and)d(it)h(b)r(eha)n(v)n(es) f Fm(\030)j Fo(q)2999 2946 y Fp(1)p Fn(\000)p Fp(2)p Fl(\017)3186 2976 y Ft(for)451 3076 y Fo(q)26 b Fm(!)d Ft(0)p Fo(:)69 b(B)817 3088 y Fp(1)854 3076 y Fo(V)47 b Ft(is)27 b(in)h(the)g(same)f(op)r(erator)f(class)h(suc)n(h)g(that)h (w)n(e)f(obtain)1299 3266 y Fm(j)17 b Ft(^)-59 b Fo(w)1381 3278 y Fp(2)1418 3266 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))p Fm(j)48 b(\024)d Fo(c)2031 3210 y Ft(1)18 b(+)g Fo(q)2214 3179 y Fl(\017)p 1872 3247 534 4 v 1872 3323 a Fo(q)1912 3299 y Fp(2)p Fl(\017)1977 3323 y Ft(\()p Fo(q)k Ft(+)c Fo(p)g Ft(+)g(1\))2368 3299 y Fp(2)2415 3266 y Fo(:)636 b Ft(\(3.26\))451 3468 y(By)30 b(induction,)i(one)d (can)h(sho)n(w)g(that)g(for)g Fo(k)g(>)d Ft(2,)k(one)f(has)f Fm(j)18 b Ft(^)-60 b Fo(w)2458 3480 y Fl(k)2500 3468 y Ft(\()p Fi(q)p Fo(;)14 b Fi(p)p Ft(\))p Fm(j)42 b(\024)3033 3435 y Fl(c)p 2871 3449 355 4 v 2871 3497 a Fp(\()p Fl(q)r Fp(+)p Fl(p)p Fp(+1\))3124 3480 y Fg(2+)p Fe(\017)3235 3468 y Fo(:)451 3577 y Ft(Th)n(us)27 b(one)h(obtains)f(regularisation)e (up)r(on)i(increasing)g Fo(k)s Ft(,)g(resulting)g(in)h(b)r(ounded)g(op) r(er-)451 3677 y(ators)e Fo(B)722 3689 y Fl(k)763 3677 y Fo(;)60 b(k)26 b(>)d Ft(1)p Fo(:)2153 b Fb(\003)451 3923 y Ft(Prop)r(osition)35 b(3.1)g(pro)n(vides)g(justi\014cation)i (for)f(represen)n(ting)e(the)j(transformed)e(Dirac)451 4023 y(op)r(erator)18 b(in)i(terms)f(of)h(a)f(series)g(expansion)f(in)i (the)g(p)r(oten)n(tial)g(strength.)34 b(Note,)21 b(ho)n(w)n(ev)n(er,) 451 4131 y(that)30 b(the)g(limit)g Fo(\017)c Fm(!)g Ft(0)j(cannot)g(b)r (e)h(carried)e(out)h(since)g(in)h(\(3.7\),)2662 4098 y Fl(\017)p 2599 4112 154 4 v 2599 4159 a Fp(1)p Fn(\000)p Fl(n\017)2776 4131 y Fo(\016)2816 4101 y Fn(\000)2878 4075 y Fg(1)p Ff(\000)p Fe(n\017)p 2878 4088 136 3 v 2933 4121 a(\017)3068 4131 y Fm(!)c(1)451 4230 y Ft(as)i Fo(\017)c Fm(!)h Ft(0)p Fo(;)j Ft(whic)n(h)g(implies)h Fo(C)i Fm(!)25 b(1)j Ft(in)h(\(3.3\).)39 b(Therefore,)28 b(this)h(limit)g(cannot)f(b)r(e)h(used)451 4330 y(to)e(pro)n(v)n(e)e (for)h(the)h(Coulom)n(b)f(p)r(oten)n(tial)h(the)g Fo(p)p Ft(-form)f(b)r(oundedness)g(of)h Fo(V)2780 4342 y Fl(k)2821 4330 y Fo(;)60 b Ft(1)23 b Fm(\024)f Fo(k)k Fm(\024)d Fo(n)451 4430 y Ft(with)34 b(form)f(b)r(ound)g Fo(<)f Ft(1)p Fo(:)66 b Ft(On)33 b(the)g(other)g(hand,)i(it)e(has)g(b)r(een)h (sho)n(wn)e(with)i(di\013eren)n(t)451 4529 y(to)r(ols)28 b(that)h(this)g(prop)r(ert)n(y)f(holds)g(for)h Fo(V)1733 4541 y Fp(1)1799 4529 y Ft(and)g Fo(V)2010 4541 y Fp(1)2067 4529 y Ft(+)18 b Fo(V)2198 4541 y Fp(2)2265 4529 y Ft(in)29 b(case)f(of)h(the)g(Coulom)n(b)f(\014eld)451 4629 y([2,)f(8].)451 4849 y Fr(4)92 b(Pr)n(oof)31 b(of)h(Theorem)f(1.2)451 5031 y Ft(The)f(link)g(b)r(et)n(w)n(een)g(the)g(t)n(w)n(o)f (transformation)f(sc)n(hemes)h(under)h(consideration)e(is)i(pro-)451 5130 y(vided)e(b)n(y)f(the)h(follo)n(wing)f(lemma.)1108 5338 y Fu(Document)l(a)e(Ma)l(thema)l(tica)h(10)f(\(2005\))g(331{356)p eop 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b(the)h(uniqueness)f(of)g(the)g(solution)g(it)h(follo)n (ws)e(from)h(\(4.7\))g(and)g(\(1.10\))g(that)451 2292 y Fo(U)517 2262 y Fn(0)508 2313 y Fp(0)545 2292 y Fo(B)608 2304 y Fp(1)645 2292 y Fo(U)711 2232 y Ff(0)734 2257 y Fn(\000)p Fp(1)702 2314 y(0)860 2292 y Ft(=)22 b Fo(S)998 2304 y Fp(1)1077 2292 y Ft(and)27 b(hence)h(the)g(uniqueness)f(of)h (the)g(op)r(erator)e Fo(B)2666 2304 y Fp(1)2703 2292 y Fo(:)451 2510 y Ft(b\))i(Pro)r(of)f(of)g(\(4.1\))h(b)n(y)f(induction) h(for)f(arbitrary)e(order)i Fo(n)g Ft(in)h Fo(\015)451 2657 y Ft(W)-7 b(e)32 b(assume)f(that)h(to)f(order)g Fo(n)21 b Fm(\000)f Ft(1)31 b(the)h(assertion)f(of)g(Lemma)g(4.1)g (holds.)49 b(Then,)33 b(the)451 2756 y(relation)25 b(b)r(et)n(w)n(een)h (the)g(Dirac)f(op)r(erators)f(transformed)h(to)g Fo(n)15 b Fm(\000)f Ft(1st)26 b(order,)f(asserted)f(b)n(y)451 2856 y(Theorem)j(1.2,)g(is)g(also)g(true,)g(i.e.)37 b(\(including)28 b(the)g Fo(n)p Ft(-th)g(order)e(terms\))1070 3030 y Fo(\014)t(E)1182 3042 y Fl(p)1263 3030 y Ft(+)41 b Fm(E)1413 3042 y Fp(1)1492 3030 y Ft(+)g Fm(E)1642 3042 y Fp(2)1720 3030 y Ft(+)18 b Fo(:::)h Ft(+)41 b Fm(E)2041 3042 y Fl(n)p Fn(\000)p Fp(1)2213 3030 y Ft(+)g Fm(E)2363 3042 y Fl(n)2449 3030 y Ft(+)g Fm(O)2621 3042 y Fl(n)950 3219 y Ft(=)46 b Fo(U)1127 3185 y Fn(0)1118 3240 y Fp(0)1192 3219 y Ft(\()p Fo(D)1293 3231 y Fp(0)1362 3219 y Ft(+)41 b Fo(V)1516 3231 y Fp(1)1596 3219 y Ft(+)18 b Fo(:::)g Ft(+)g Fo(V)1897 3231 y Fl(n)p Fn(\000)p Fp(1)2069 3219 y Ft(+)41 b Fo(V)2223 3231 y Fl(n)2311 3219 y Ft(+)g Fo(W)2495 3231 y Fl(n)2540 3219 y Ft(\))c Fo(U)2675 3159 y Ff(0)2698 3184 y Fn(\000)p Fp(1)2666 3241 y(0)3115 3219 y Ft(\(4.8\))451 3363 y(since)29 b Fm(E)700 3375 y Fl(n)775 3363 y Ft(and)g Fm(O)1004 3375 y Fl(n)1079 3363 y Ft(only)g(dep)r(end)h(on)f Fo(\014)t(E)1782 3375 y Fl(p)1821 3363 y Fo(;)40 b Fm(E)1928 3375 y Fl(j)1963 3363 y Fo(;)g Fm(O)2092 3375 y Fl(j)2127 3363 y Fo(;)g(S)2241 3375 y Fl(j)2276 3363 y Fo(;)66 b(j)31 b Ft(=)26 b(1)p Fo(;)14 b(:::;)g(n)19 b Fm(\000)g Ft(1)p Fo(;)56 b Ft(with)30 b(the)451 3463 y(iden)n(tical)36 b(dep)r(endence)g(of)f Fo(V)1397 3475 y Fl(n)1479 3463 y Ft(and)g Fo(W)1726 3475 y Fl(n)1808 3463 y Ft(on)g Fo(D)2000 3475 y Fp(0)2037 3463 y Fo(;)50 b(V)2158 3475 y Fl(j)2194 3463 y Fo(;)g(W)2345 3475 y Fl(j)2381 3463 y Fo(;)g(B)2517 3475 y Fl(j)2552 3463 y Fo(;)87 b(j)41 b Ft(=)36 b(1)p Fo(;)14 b(:::;)g(n)23 b Fm(\000)h Ft(1)p Fo(:)451 3579 y Ft(F)-7 b(rom)34 b(\(4.1\))g(for)g Fo(j)39 b Ft(=)33 b(1)p Fo(;)14 b(:::;)g(n)22 b Fm(\000)h Ft(1)34 b(it)g(therefore)g(follo)n(ws)f(that)h Fm(E)2561 3591 y Fl(n)2654 3579 y Ft(=)g Fo(U)2819 3548 y Fn(0)2810 3599 y Fp(0)2847 3579 y Fo(V)2895 3591 y Fl(n)2940 3579 y Fo(U)3006 3518 y Ff(0)3029 3543 y Fn(\000)p Fp(1)2997 3601 y(0)3152 3579 y Ft(and)451 3693 y Fm(O)517 3705 y Fl(n)599 3693 y Ft(=)23 b Fo(U)753 3663 y Fn(0)744 3713 y Fp(0)781 3693 y Fo(W)859 3705 y Fl(n)905 3693 y Fo(U)971 3632 y Ff(0)993 3657 y Fn(\000)p Fp(1)962 3715 y(0)1082 3693 y Fo(:)51 b Ft(Carrying)25 b(out)j(the)g Fo(n)p Ft(-th)g(transformation)d(one)j(gets)451 3881 y Fo(U)517 3846 y Fn(0)508 3901 y Fl(n)567 3881 y Fm(\001)14 b(\001)g(\001)g Fo(U)744 3846 y Fn(0)735 3901 y Fp(0)772 3881 y Fo(H)7 b(U)914 3820 y Ff(0)935 3845 y Fn(\000)p Fp(1)905 3903 y(0)1038 3881 y Fm(\001)14 b(\001)g(\001)g Fo(U)1215 3821 y Ff(0)1237 3846 y Fn(\000)p Fp(1)1206 3901 y Fl(n)1350 3881 y Ft(=)9 b Fo(\014)t(E)1536 3893 y Fl(p)1588 3881 y Ft(+)14 b Fm(E)1711 3893 y Fp(1)1762 3881 y Ft(+)g Fo(:::)g Ft(+)g Fm(E)2047 3893 y Fl(n)p Fn(\000)p Fp(1)2177 3881 y Ft(+)g Fm(E)2300 3893 y Fl(n)2359 3881 y Ft(+)g Fm(O)2504 3893 y Fl(n)2563 3881 y Ft(+)g Fo(i)p Ft([)p Fo(\014)t(E)2806 3893 y Fl(p)2845 3881 y Fo(;)g(S)2933 3893 y Fl(n)2978 3881 y Ft(])g(+)g Fo(R)3157 3893 y Fl(n)p Fp(+1)493 4065 y Fo(U)559 4030 y Fn(\003)550 4085 y Fl(n)583 4065 y Fm(\001)g(\001)g(\001)f Fo(U)759 4030 y Fn(\003)750 4085 y Fp(1)797 4065 y Fo(H)7 b(U)930 4077 y Fp(1)967 4065 y Fm(\001)14 b(\001)g(\001)g Fo(U)1135 4077 y Fl(n)1175 4065 y Ft(=)9 b Fo(D)1318 4077 y Fp(0)1365 4065 y Ft(+)h Fo(V)1488 4077 y Fp(1)1534 4065 y Ft(+)g Fo(:::)g Ft(+)g Fo(V)1811 4077 y Fl(n)p Fn(\000)p Fp(1)1936 4065 y Ft(+)g Fo(V)2059 4077 y Fl(n)2114 4065 y Ft(+)g Fo(W)2267 4077 y Fl(n)2322 4065 y Ft(+)g Fo(i)p Ft([)p Fo(D)2518 4077 y Fp(0)2554 4065 y Fo(;)k(B)2654 4077 y Fl(n)2699 4065 y Ft(])c(+)g Fo(R)2871 4030 y Fp(\()p Fl(n)p Fp(+1\))3051 4065 y Fo(:)41 b Ft(\(4.9\))451 4209 y Fo(B)514 4221 y Fl(n)587 4209 y Ft(is)28 b(obtained)f(from)g Fo(W)1286 4221 y Fl(n)1369 4209 y Ft(=)22 b Fm(\000)p Fo(i)14 b Ft([)p Fo(D)1656 4221 y Fp(0)1692 4209 y Fo(;)g(B)1792 4221 y Fl(n)1837 4209 y Ft(])p Fo(;)51 b Ft(or)27 b(transformed)f(with) i Fo(U)2755 4178 y Fn(0)2746 4229 y Fp(0)2783 4209 y Ft(,)535 4395 y Fo(U)601 4361 y Fn(0)592 4416 y Fp(0)642 4395 y Fo(W)720 4407 y Fl(n)780 4395 y Fo(U)846 4335 y Ff(0)868 4360 y Fn(\000)p Fp(1)837 4417 y(0)994 4395 y Ft(=)46 b Fm(O)1171 4407 y Fl(n)1262 4395 y Ft(=)g Fm(\000)p Fo(iU)1533 4361 y Fn(0)1524 4416 y Fp(0)1583 4395 y Ft([)p Fo(D)1675 4407 y Fp(0)1712 4395 y Fo(;)14 b(B)1812 4407 y Fl(n)1857 4395 y Ft(])23 b Fo(U)1969 4335 y Ff(0)1991 4360 y Fn(\000)p Fp(1)1960 4417 y(0)2127 4395 y Ft(=)45 b Fm(\000)p Fo(i)23 b Ft([)p Fo(\014)t(E)2489 4407 y Fl(p)2528 4395 y Fo(;)14 b(U)2631 4361 y Fn(0)2622 4416 y Fp(0)2659 4395 y Fo(B)2722 4407 y Fl(n)2767 4395 y Fo(U)2833 4335 y Ff(0)2855 4360 y Fn(\000)p Fp(1)2824 4417 y(0)2944 4395 y Ft(])p Fo(:)84 b Ft(\(4.10\))451 4585 y(Since)31 b(the)f(solution)g Fo(S)1184 4597 y Fl(n)1259 4585 y Ft(to)h Fm(O)1430 4597 y Fl(n)1516 4585 y Ft(=)c Fm(\000)p Fo(i)p Ft([)p Fo(\014)t(E)1837 4597 y Fl(p)1876 4585 y Fo(;)14 b(S)1964 4597 y Fl(n)2009 4585 y Ft(])57 b(is)31 b(unique,)g(one)f(gets)f Fo(U)2866 4555 y Fn(0)2857 4605 y Fp(0)2894 4585 y Fo(B)2957 4597 y Fl(n)3003 4585 y Fo(U)3069 4524 y Ff(0)3091 4549 y Fn(\000)p Fp(1)3060 4607 y(0)3221 4585 y Ft(=)451 4684 y Fo(S)502 4696 y Fl(n)547 4684 y Fo(:)2624 b Fb(\003)451 4831 y Ft(F)-7 b(rom)26 b(the)g(corresp)r(ondence)e(of)i(the)g(t)n(w)n(o)f (transformation)g(sc)n(hemes)g(it)h(follo)n(ws)f(that)h(for)451 4931 y Fo(n)e Ft(=)f(2)p Fo(;)61 b(H)815 4901 y Fp(\(2\))932 4931 y Ft(from)28 b(\(2.12\))f(when)h(acting)f(on)h(the)h(p)r(ositiv)n (e)e(sp)r(ectral)h(subspace)f(of)h Fo(D)3226 4943 y Fp(0)3263 4931 y Ft(,)451 5031 y(reduces)f(to)h(\003)907 5043 y Fp(+)962 5031 y Ft(\()p Fo(D)1063 5043 y Fp(0)1119 5031 y Ft(+)18 b Fo(V)51 b Ft(+)1399 4998 y Fl(i)p 1395 5012 34 4 v 1395 5059 a Fp(2)1452 5031 y Ft([)p Fo(W)1553 5043 y Fp(1)1590 5031 y Fo(;)14 b(B)1690 5043 y Fp(1)1727 5031 y Ft(]\)\003)1840 5043 y Fp(+)1947 5031 y Ft(since)28 b Fo(i)p Ft([)p Fo(V)2251 5043 y Fp(1)2288 5031 y Fo(;)14 b(B)2388 5043 y Fp(1)2425 5031 y Ft(])28 b(corresp)r(onds)e(to)i(an)g (o)r(dd)451 5130 y(op)r(erator)e(whic)n(h)i(v)-5 b(anishes)27 b(up)r(on)h(pro)5 b(jection.)1108 5338 y Fu(Document)l(a)25 b(Ma)l(thema)l(tica)h(10)f(\(2005\))g(331{356)p eop %%Page: 355 25 355 24 bop 1291 518 a Fr(Pr)n(ojected)32 b(Dira)n(c)f(Opera)-6 b(tor)716 b Ft(355)451 725 y Fr(A)n(ckno)n(wledgment)451 873 y Ft(I)27 b(should)g(lik)n(e)g(to)g(thank)g(H.)g(Kalf)g(and)g(H.)g (Sieden)n(top)g(for)g(enligh)n(tening)f(discussions.)36 b(I)451 973 y(am)19 b(particularly)e(grateful)h(to)h(A.)g(Sob)r(olev)f (for)h(suggesting)e(the)i(unitary)g(transformation)451 1072 y(sequence)29 b(applied)g(in)h(this)f(w)n(ork)f(and)h(for)g(his)g (op)r(enness)g(and)g(patience)h(during)e(man)n(y)451 1172 y(discussions.)34 b(I)24 b(should)e(also)g(lik)n(e)h(to)g(thank)f (A.)i(Sob)r(olev)e(and)h(the)g(Univ)n(ersit)n(y)f(of)h(Sussex)451 1272 y(for)f(their)h(hospitalit)n(y)f(during)g(m)n(y)h(visits)f(in)h (the)g(course)f(of)g(this)h(w)n(ork.)34 b(P)n(artial)21 b(supp)r(ort)451 1371 y(b)n(y)27 b(the)h(V)-7 b(olksw)n(agen)26 b(F)-7 b(oundation)28 b(is)f(gratefully)g(ac)n(kno)n(wledged.)451 1839 y Fr(References)493 2023 y Ft([1])41 b(G.)35 b(E.)f(Bro)n(wn)f (and)h(D.)i(G.)e(Ra)n(v)n(enhall.)g(On)g(the)h(in)n(teraction)f(of)g(t) n(w)n(o)g(electrons.)622 2123 y Fd(Pr)l(o)l(c.)c(R)l(oy.)h(So)l(c.)f(L) l(ondon)g(A,)e Ft(208:)35 b(552-559,)25 b(1952.)493 2294 y([2])41 b(R.)28 b(Brummelh)n(uis,)f(H.)h(Sieden)n(top,)f(and)h(E.)f (Sto)r(c)n(kmey)n(er.)g(The)g(ground)g(state)g(en-)622 2394 y(ergy)d(of)h(relativistic)g(one-electron)e(atoms)i(according)f (to)h(Jansen)f(and)h(Hess.)g Fd(Do)l(c.)622 2493 y(Math.,)k Ft(7:)37 b(167-182,)24 b(2002.)493 2665 y([3])41 b(N.)26 b(Datta,)g(R.)g(F)-7 b(ern\023)-42 b(andez)25 b(and)g(J.)h(F)-7 b(r\177)-42 b(ohlic)n(h.)25 b(E\013ectiv)n(e)g(hamiltonians)g(and)g (phase)622 2764 y(diagrams)h(for)h(tigh)n(t-binding)g(mo)r(dels.)h Fd(J.)i(Stat.)f(Phys.,)h Ft(96:)36 b(545-611,)24 b(1999.)493 2936 y([4])41 b(M.)36 b(Douglas)f(and)h(N.)g(M.)h(Kroll.)d(Quan)n(tum)i (electro)r(dynamical)f(corrections)f(to)622 3035 y(the)28 b(\014ne)g(structure)f(of)g(helium.)h Fd(A)n(nn.)i(Phys.)h(\(N.Y.\),)d Ft(82:)36 b(89-155,)25 b(1974.)493 3206 y([5])41 b(W.)22 b(D.)h(Ev)-5 b(ans,)22 b(P)-7 b(.)22 b(P)n(erry)-7 b(,)21 b(and)h(H.)h(Sieden)n(top.)f(The)g(sp)r(ectrum)g(of)g(relativistic)g (one-)622 3306 y(electron)30 b(atoms)h(according)e(to)i(Bethe)g(and)g (Salp)r(eter.)g Fd(Commun.)j(Math.)g(Phys.,)622 3406 y Ft(178:)h(733-746,)25 b(1996.)493 3577 y([6])41 b(L.)23 b(L.)g(F)-7 b(oldy)23 b(and)g(S.)g(A.)g(W)-7 b(outh)n(uysen.)23 b(On)g(the)h(Dirac)e(theory)g(of)h(spin)2915 3544 y Fp(1)p 2915 3558 34 4 v 2915 3605 a(2)2981 3577 y Ft(particles)622 3676 y(and)k(its)h(non-relativistic)e(limit.)j Fd(Phys.)i(R)l(ev.,)d Ft(78:)36 b(29-36,)26 b(1950.)493 3848 y([7])41 b(I.)22 b(S.)g(Gradsh)n(teyn)f(and)h(I.)g(M.)g(Ryzhik.)g Fd(T)-6 b(able)25 b(of)h(Inte)l(gr)l(als,)f(Series)g(and)g(Pr)l(o)l(ducts.)622 3947 y Ft(Academic)i(Press,)g(New)g(Y)-7 b(ork,)27 b(1965.)493 4118 y([8])41 b(D.)d(H.)f(Jakubassa-Am)n(undsen)10 b Fo(:)35 b Ft(The)i(essen)n(tial)g(sp)r(ectrum)g(of)g(relativistic)g (one-)622 4218 y(electron)e(ions)h(in)h(the)g(Jansen-Hess)e(mo)r(del.)h Fd(Math.)j(Phys.)h(Ele)l(ctr)l(on.)e(J.,)h Ft(8\(3\):)622 4318 y(1-30,)26 b(2002)493 4489 y([9])41 b(G.)29 b(Jansen)e(and)i(B.)g (A.)g(Hess.)f(Revision)g(of)h(the)g(Douglas-Kroll)d(transformation.)622 4589 y Fd(Phys.)31 b(R)l(ev.)f(A,)e Ft(39:)36 b(6016-6017,)23 b(1989.)451 4760 y([10])41 b(T.)23 b(Kato.)f Fd(Perturb)l(ation)j(The)l (ory)i(for)g(Line)l(ar)f(Op)l(er)l(ators,)e Ft(v)n(olume)f(132)e(of)i Fd(Grund-)622 4859 y(lehr)l(en)j(der)g(mathematischen)h (Wissenschaften.)e Ft(Springer-V)-7 b(erlag,)22 b(Berlin,)i(1966.)451 5031 y([11])41 b(E.)24 b(H.)g(Lieb)g(and)g(H.-T.)g(Y)-7 b(au.)24 b(The)g(stabilit)n(y)g(and)g(instabilit)n(y)g(of)g (relativistic)f(mat-)622 5130 y(ter.)k Fd(Commun.)j(Math.)h(Phys.,)f Ft(118:)35 b(177-213,)25 b(1988.)1108 5338 y Fu(Document)l(a)g(Ma)l (thema)l(tica)h(10)f(\(2005\))g(331{356)p eop %%Page: 356 26 356 25 bop 451 518 a Ft(356)727 b Fr(D.)30 b(H.)h(Jakubassa-Amundsen) 451 725 y Ft([12])41 b(P)-7 b(.)23 b(R.)h(Halmos)e(and)i(V.)g(S.)f (Sunder.)h Fd(Bounde)l(d)j(Inte)l(gr)l(al)e(Op)l(er)l(ators)h(on)g Fo(L)2964 695 y Fp(2)3027 725 y Fd(Sp)l(ac)l(es.)622 825 y Ft(Springer-V)-7 b(erlag,)25 b(Berlin,)i(1976.)451 991 y([13])41 b(A.)35 b(V.)g(Sob)r(olev.)f(In)n(tegrated)g(densit)n(y)h (of)f(states)h(for)f(the)h(p)r(erio)r(dic)g(Sc)n(hr\177)-42 b(odinger)622 1090 y(op)r(erator)26 b(in)i(dimension)f(t)n(w)n(o.)g Fd(A)n(nn.)i(Henri)h(Poinc)l(ar)n(\023)-40 b(e,)29 b Ft(6:)37 b(31-84,)25 b(2005.)451 1256 y([14])41 b(B.)27 b(Thaller.)g Fd(The)k(Dir)l(ac)f(Equation,)f Ft(Springer-V)-7 b(erlag,)25 b(Berlin,)i(1992.)451 1422 y([15])41 b(A.)26 b(W)-7 b(olf,)27 b(M.)e(Reiher,)h(and)g(B.)f(A.)i(Hess.)e(The)h (generalised)e(Douglas-Kroll)f(trans-)622 1522 y(formation.)k Fd(J.)i(Chem.)i(Phys.,)f Ft(117:)36 b(9215-9226,)23 b(2002.)1243 3111 y(D.)28 b(H.)g(Jakubassa-Am)n(undsen)1243 3210 y(Mathematics)f (Institute)1243 3310 y(Univ)n(ersit)n(y)f(of)i(Munic)n(h)1243 3409 y(Theresienstr.)35 b(39)1243 3509 y(80333)25 b(Munic)n(h)1243 3609 y(German)n(y)1243 3708 y(doris.jaku@lrz.uni-m)n(uenc)n(hen.de)1108 5338 y Fu(Document)l(a)g(Ma)l(thema)l(tica)h(10)f(\(2005\))g(331{356)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF