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b(Hilb)s(ert)f(space)h(of)g(pure)g(state)g(v)m(ectors)i(of)d (the)h(system)h(just)f(describ)s(ed)h(is)257 4466 y(giv)m(en)33 b(b)m(y)1524 4586 y Fn(H)62 b Fq(:=)e Fn(H)1917 4601 y Fm(el)1998 4586 y Fn(\012)22 b(F)32 b Fo(;)1080 b Fq(\(I.1\))257 4741 y(where)33 b Fn(H)622 4756 y Fm(el)709 4741 y Fq(=)28 b Fo(L)879 4705 y Fl(2)918 4741 y Fq(\()p Fk(R)1022 4705 y Fl(3)1068 4741 y Fq(\))j(is)h(the)g(Hilb)s(ert)e(space)j(appropriate) e(for)g(the)i(description)e(of)g(a)257 4861 y(single)24 b(electron,)i(disregarding)d(its)h(spin,)j(and)d Fn(F)34 b Fq(is)24 b(the)h(F)-8 b(o)s(c)m(k)25 b(space)h(used)f(to)g(describ)s (e)257 4982 y(the)j(states)f(of)g(the)g(transv)m(erse)i(mo)s(des)d(of)h (the)g(quan)m(tized)g(electromagnetic)f(\014eld,)h(i.e.,)257 5102 y(the)33 b Fj(photons)p Fq(.)43 b(More)33 b(explicitly)-8 b(,)1176 5329 y Fn(F)69 b Fq(:=)1522 5221 y Fi(1)1494 5246 y Fh(M)1481 5430 y Fm(N)7 b Fl(=0)1650 5329 y Fn(F)1732 5288 y Fl(\()p Fm(N)g Fl(\))1876 5329 y Fo(;)159 b Fn(F)2144 5288 y Fl(\(0\))2265 5329 y Fq(=)28 b Fk(C)37 b Fq(\012)28 b Fo(;)731 b Fq(\(I.2\))p eop %%Page: 2 3 2 2 bop 257 266 a Fr(BFP-GrSt,)32 b(04-F)-8 b(eb-2005)2178 b Fq(2)257 573 y(where)47 b(\012)f(is)e(the)i(v)-5 b(acuum)45 b(v)m(ector,)50 b(i.e.,)e(the)d(state)h(of)f(the)h(electromagnetic)d (\014eld)257 693 y(without)32 b(an)m(y)i(excited)f(mo)s(des,)f(and)1222 982 y Fn(F)1304 941 y Fl(\()p Fm(N)7 b Fl(\))1486 982 y Fq(:=)60 b Fn(S)1709 997 y Fm(N)1841 874 y(N)1810 899 y Fh(O)1811 1081 y Fm(j)t Fl(=1)1952 982 y Fg(h)22 b Fo(;)158 b(N)38 b Fn(\025)29 b Fq(1)22 b Fo(;)777 b Fq(\(I.3\))257 1280 y(where)34 b(the)f(Hilb)s(ert)e(space)j Fg(h)e Fq(of)g(state)h(v)m (ectors)h(of)f(a)f(single)f(photon)i(is)1465 1500 y Fg(h)60 b Fq(:=)g Fo(L)1806 1459 y Fl(2)1846 1500 y Fq([)p Fk(R)1939 1459 y Fl(3)2006 1500 y Fn(\002)23 b Fk(Z)2175 1515 y Fl(2)2212 1500 y Fq(])f Fo(;)1020 b Fq(\(I.4\))257 1720 y(with)30 b(photon)g(momen)m(tum)e(space)j(giv)m(en)f(b)m(y)h Fk(R)2023 1684 y Fl(3)2068 1720 y Fq(,)g(and)f Fk(Z)2382 1735 y Fl(2)2449 1720 y Fq(is)f(accoun)m(ting)h(for)f(the)h(t)m(w)m(o) 257 1840 y(indep)s(enden)m(t)k(transv)m(erse)g(p)s(olarizations,)c(or)h (helicities,)g(of)g(a)h(photon.)44 b(In)32 b(Eq.)h(\(I.3\),)257 1961 y Fn(S)317 1976 y Fm(N)417 1961 y Fq(denotes)f(the)g(orthogonal)e (pro)5 b(jection)32 b(on)m(to)f(the)h(subspace)i(of)2787 1894 y Fh(N)2879 1920 y Fm(N)2879 1985 y(j)t Fl(=1)3022 1961 y Fg(h)e Fq(of)f(totally)257 2081 y(symmetric)k Fo(N)10 b Fq(-photon)35 b(w)m(a)m(v)m(e)j(functions,)e(in)f(accordance) h(with)f(the)h(fact)f(that)h(pho-)257 2201 y(tons)27 b(satisfy)g(Bose-Einstein)g(statistics.)41 b(Th)m(us,)29 b Fn(F)2165 2165 y Fl(\()p Fm(N)7 b Fl(\))2314 2201 y Fq(is)26 b(the)h(subspace)i(of)d Fn(F)36 b Fq(of)27 b(state)257 2322 y(v)m(ectors)k(for)d(con\014gurations)g(of)g(exactly)h Fo(N)40 b Fq(photons.)i(It)29 b(is)f(con)m(v)m(enien)m(t)j(to)d (represen)m(t)257 2442 y(the)34 b(Hilb)s(ert)d(space)k Fn(H)f 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Fq(\(I.6\))257 3500 y(In)40 b(Eq.)g(\(I.6\),)834 3475 y Fo(~)824 3500 y Fn(r)906 3515 y Fm(~)-34 b(x)990 3500 y Fq(denotes)40 b(the)g(gradien)m(t)e(with)h(resp)s(ect)h(to)f (the)g(electron)g(p)s(osition)256 3629 y Fo(~)-48 b(x)43 b Fn(2)g Fk(R)530 3593 y Fl(3)575 3629 y Fq(,)h Fo(\013)751 3602 y Fn(\030)752 3633 y Fq(=)871 3629 y(1)p Fo(=)p Fq(137)c(is)g(the)i(feinstructure)g(constan)m(t,)2469 3604 y Fo(~)2451 3629 y(A)q Fq(\()n Fo(~)-47 b(x)p Fq(\))41 b(denotes)h(the)g(v)m(ector)257 3750 y(p)s(oten)m(tial)c(of)h(the)h (transv)m(erse)i(mo)s(des)d(of)g(the)h(quan)m(tized)g(electromagnetic)e (\014eld)h(in)257 3870 y(the)33 b Fj(Coulomb)h(gauge)p Fq(,)1537 3965 y Fo(~)1528 3990 y Fn(r)1610 4005 y Fm(~)-34 b(x)1677 3990 y Fn(\001)1744 3965 y Fo(~)1726 3990 y(A)q Fq(\()n Fo(~)-47 b(x)p Fq(\))60 b(=)g(0)22 b Fo(;)1083 b Fq(\(I.7\))257 4165 y Fo(V)56 b Fq(is)33 b(the)i(Coulom)m(b)e(p)s (oten)m(tial)f(of)i(electrostatic)f(attraction)g(of)h(the)g(electron)g (to)g(the)257 4285 y(n)m(ucleus,)49 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Fq(denotes)i(the)e(electron's)h(mass.)p eop %%Page: 3 4 3 3 bop 257 266 a Fr(BFP-GrSt,)32 b(04-F)-8 b(eb-2005)2178 b Fq(3)257 573 y(Then)37 b(the)e(comp)s(onen)m(ts)g(of)g(the)g(momen)m (tum)f(v)m(ector,)2364 548 y Fo(~)2346 573 y(K)7 b Fq(,)36 b(of)e(a)h(single)f(photon)h(ha)m(v)m(e)257 693 y(dimension)d(\(length) o(\))1060 657 y Fi(\000)p Fl(1)1154 693 y Fq(,)h(and)g(w)m(e)g(set)1545 906 y Fo(~)1550 932 y(k)63 b Fq(:=)1837 865 y Fo(r)1881 880 y Fl(Bohr)p 1837 909 201 4 v 1873 1001 a Fq(2)17 b Fo(\013)2082 907 y(~)2064 932 y(K)29 b(:)1105 b Fq(\(I.9\))257 1202 y(The)44 b(v)-5 b(ariables)39 b Fo(~)-48 b(x)43 b Fq(and)1173 1176 y Fo(~)1178 1202 y(k)j Fq(are)c Fj(dimensionless)p Fq(.)70 b(W)-8 b(e)43 b(c)m(ho)s(ose)g(units)f(for)f(the)i(energy)257 1323 y(suc)m(h)34 b(that)1281 1443 y(2)17 b Fo(m)g(\013)1512 1402 y Fl(2)1611 1443 y Fq(=)60 b(4)17 b(Rydb)s(erg)60 b(=)g(1)22 b Fo(:)789 b Fq(\(I.10\))404 1616 y(With)32 b(these)h(con)m(v)m(en)m(tions)1042 1826 y Fo(~)1023 1851 y(K)d Fn(\001)1206 1826 y Fo(~)1185 1851 y(X)68 b Fq(=)60 b Fo(\013)1544 1825 y(~)1550 1851 y(k)25 b Fn(\001)20 b Fo(~)-47 b(x)22 b(;)112 b Fq(and)94 b Fo(V)22 b Fq(\()p Fo(x)p Fq(\))60 b(=)2590 1784 y(1)p 2559 1828 111 4 v 2559 1920 a Fn(j)n Fo(~)-47 b(x)p Fn(j)2702 1851 y Fo(:)531 b Fq(\(I.11\))257 2118 y(F)-8 b(urthermore,)1129 2237 y(\024)1103 2263 y Fo(H)68 b Fq(:=)1431 2180 y Fh(X)1415 2364 y Fm(\025)p Fl(=)p Fi(\006)1583 2145 y Fh(Z)1682 2263 y Fo(d)1733 2221 y Fl(3)1773 2263 y Fo(k)30 b(a)1905 2221 y Fi(\003)1945 2263 y Fq(\()1978 2236 y Fo(~)1983 2263 y(k)s(;)17 b(\025)p Fq(\))g Fn(j)2216 2236 y Fo(~)2221 2263 y(k)r Fn(j)g Fo(a)p Fq(\()2403 2236 y Fo(~)2408 2263 y(k)r(;)g(\025)p Fq(\))22 b Fo(;)611 b Fq(\(I.12\))257 2537 y(where)42 b Fo(a)598 2501 y Fi(\003)637 2537 y Fq(\()670 2511 y Fo(~)675 2537 y(k)s(;)17 b(\025)p Fq(\))40 b(and)g Fo(a)p Fq(\()1189 2511 y Fo(~)1194 2537 y(k)t(;)17 b(\025)p Fq(\))39 b(are)i(the)f(usual)g(photon)g(creation-)f(and)i (annihilation)257 2657 y(op)s(erators,)33 b(whic)m(h)g(satisfy)g(the)g (canonical)e(comm)m(utation)f(relations)860 2874 y([)p Fo(a)938 2837 y Fi(\003)978 2874 y Fq(\()1011 2847 y Fo(~)1016 2874 y(k)s(;)17 b(\025)p Fq(\))g Fo(;)32 b(a)1336 2837 y Fi(\003)1376 2874 y Fq(\()1409 2847 y Fo(~)1414 2874 y(k)1468 2837 y Fi(0)1491 2874 y Fo(;)17 b(\025)1592 2837 y Fi(0)1615 2874 y Fq(\)])56 b(=)f([)p Fo(a)p Fq(\()1978 2847 y Fo(~)1983 2874 y(k)s(;)17 b(\025)p Fq(\))g Fo(;)32 b(a)p Fq(\()2336 2847 y Fo(~)2341 2874 y(k)2395 2837 y Fi(0)2419 2874 y Fo(;)17 b(\025)2520 2837 y Fi(0)2543 2874 y Fq(\)])55 b(=)h(0)21 b Fo(;)368 b Fq(\(I.13\))1096 3029 y([)p Fo(a)p Fq(\()1207 3003 y Fo(~)1212 3029 y(k)s(;)17 b(\025)p Fq(\))g Fo(;)32 b(a)1532 2993 y Fi(\003)1572 3029 y Fq(\()1605 3003 y Fo(~)1610 3029 y(k)1664 2993 y Fi(0)1687 3029 y Fo(;)17 b(\025)1788 2993 y Fi(0)1811 3029 y Fq(\)])56 b(=)f Fo(\016)2106 3044 y Fm(\025\025)2188 3025 y Fd(0)2232 3029 y Fo(\016)t Fq(\()2312 3003 y Fo(~)2317 3029 y(k)25 b Fn(\000)2487 3003 y 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v 111 x Fq(2)17 b Fn(j)1225 4048 y Fo(~)1230 4074 y(k)r Fn(j)1354 3970 y Fq(\003\()1455 3943 y Fo(~)1460 3970 y(k)r Fq(\))1568 3873 y Fh(n)1622 3970 y Fo(~)-48 b(")p Fq(\()1702 3943 y Fo(~)1707 3970 y(k)s(;)17 b(\025)p Fq(\))p Fo(e)1945 3929 y Fi(\000)p Fm(i)2021 3911 y(~)2024 3929 y(k)q Fi(\001)o Fm(~)-34 b(x)2126 3970 y Fo(a)2177 3929 y Fi(\003)2216 3970 y Fq(\()2249 3943 y Fo(~)2254 3970 y(k)s(;)17 b(\025)p Fq(\))38 b(+)g Fo(~)-48 b(")p Fq(\()2679 3943 y Fo(~)2684 3970 y(k)s(;)17 b(\025)p Fq(\))2877 3929 y Fi(\003)2916 3970 y Fo(e)2961 3929 y Fm(i)2982 3911 y(~)2985 3929 y(k)q Fi(\001)o Fm(~)-34 b(x)3087 3970 y Fo(a)p Fq(\()3171 3943 y Fo(~)3176 3970 y(k)s(;)17 b(\025)p Fq(\))3369 3873 y Fh(o)3446 3970 y Fo(;)257 4311 y Fq(where)35 b(\003\()641 4284 y Fo(~)646 4311 y(k)s Fq(\))e(is)g(the)g(c)m(haracteristic)g (function)g(of)g(the)h(ball)d Fn(f)2537 4284 y Fo(~)2542 4311 y(k)h Fn(2)d Fk(R)2786 4274 y Fl(3)2848 4311 y Fn(j)g(j)2928 4284 y Fo(~)2933 4311 y(k)r Fn(j)g(\024)g Fo(\024)p 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Fo(~)-47 b(")o Fq(\()1624 4980 y Fo(~)1629 5007 y(k)s(;)17 b(\025)p Fq(\))30 b(=)f(0)34 b(implies)d(the)k(Coulom)m(b)d(gauge)i(condi-)257 5137 y(tion.)59 b(W)-8 b(e)38 b(note)g(that,)h(b)m(y)g(a)e(famous)g(theorem) h(due)g(to)g(Hopf,)h(the)f(v)m(ectors)g Fo(~)-48 b(")p Fq(\()3277 5111 y Fo(~)3282 5137 y(k)s(;)17 b Fn(\006)p Fq(\))257 5268 y(cannot)25 b(b)s(e)g(c)m(hosen)h(to)e(b)s(e)h(con)m (tin)m(uous)g(functions)g(of)2225 5241 y Fo(~)2230 5268 y(k)s(=)p Fn(j)2356 5241 y Fo(~)2361 5268 y(k)r Fn(j)p Fq(;)i(but)e(this)f(is)h(of)f(no)g(concern)257 5388 y(for)32 b(us.)p eop %%Page: 4 5 4 4 bop 257 266 a Fr(BFP-GrSt,)32 b(04-F)-8 b(eb-2005)2178 b Fq(4)404 573 y(The)34 b(function)f(\003\()1089 547 y Fo(~)1094 573 y(k)r Fq(\))g(insures)h(that)f(mo)s(des)g(of)g(the)h (electromagnetic)d(\014eld)i(corre-)257 703 y(sp)s(onding)k(to)g(w)m(a) m(v)m(e)i(v)m(ectors)1375 677 y Fo(~)1380 703 y(k)h Fq(with)d Fn(j)1721 677 y Fo(~)1726 703 y(k)s Fn(j)e(\025)h Fo(\024)h Fq(do)h(not)f(in)m(teract)g(with)f(the)i(electron;)257 824 y(i.e.,)31 b(\003)f(represen)m(ts)k(an)c Fj(ultr)-5 b(aviolet)34 b(cuto\013)51 b Fq(that)31 b(will)d(b)s(e)j(k)m(ept)h (\014xed)g(throughout)e(our)257 944 y(analysis.)57 b(The)39 b(v)m(ector)f(p)s(oten)m(tial)e(de\014ned)i(in)f(\(I.16\))g(is)g(th)m (us)h(cut)g(o\013)f(in)f(the)i(ultra-)257 1065 y(violet)33 b(-)g(the)i(true)f(v)m(ector)h(p)s(oten)m(tial)d(of)i(the)g (electromagnetic)f(\014eld)g(corresp)s(onds)j(to)257 1185 y(the)46 b(c)m(hoice)g(\003)j Fn(\021)g Fq(1.)81 b(In)46 b(our)f(analysis)g(of)f(groundstates)j(of)d Fj(light)55 b Fq(atoms)44 b(and)i(of)257 1305 y(lo)m(w-energy)38 b(Ra)m(yleigh)f(scattereing,)i(an)f(ultra)m(violet)d(cuto\013)j(with)f (5)f Fn(\024)h Fo(\024)g Fn(\024)g Fq(10)g(can)257 1426 y(b)s(e)k(exp)s(ected)i(to)d(yield)g(reliable)f(results.)68 b(F)-8 b(or)40 b(con)m(v)m(enience,)45 b(w)m(e)d(shall)d(henceforth)257 1546 y(assume)e(that)f Fo(\024)f Fn(\025)f Fq(1.)55 b(Of)36 b(course,)j(it)c(w)m(ould)h(b)s(e)h(desirable)f(to)g(carry)h(out)f(a)g (mathe-)257 1666 y(matically)e(precise)j(analysis)f(of)g(ultra)m (violet)f(renormalization)e(\(renormalization)g(of)257 1787 y(the)i(c)m(hemical)f(p)s(oten)m(tial)e(and)j(of)f(the)h(mass)f (of)h(an)f(electron\))g(and)h(to)f(construct)i(the)257 1907 y(limiting)30 b(dynamics,)35 b(as)g Fo(\024)c Fn(!)g(1)p Fq(.)48 b(But)35 b(this)f(problem)g(is)g(outside)g(the)h(scop)s(e)g(of) f(our)257 2028 y(pap)s(er.)404 2148 y(In)f(the)g(follo)m(wing,)d(w)m(e) j(simplify)d(our)j(notations)e(b)m(y)j(setting)257 2390 y Fo(k)52 b Fq(:=)c(\()544 2364 y Fo(~)549 2390 y(k)s(;)17 b(\025)p Fq(\))k Fo(;)111 b(!)t Fq(\()p Fo(k)s Fq(\))48 b Fn(\021)h(j)p Fo(k)s Fn(j)f Fq(:=)g Fn(j)1602 2364 y Fo(~)1607 2390 y(k)s Fn(j)21 b Fo(;)111 b Fq(and)2116 2273 y Fh(Z)2216 2390 y Fo(f)11 b Fq(\()p Fo(k)s Fq(\))20 b Fo(dk)52 b Fq(:=)2746 2307 y Fh(X)2730 2492 y Fm(\025)p Fl(=)p Fi(\006)2898 2273 y Fh(Z)2997 2390 y Fo(f)11 b Fq(\()3089 2364 y Fo(~)3094 2390 y(k)s(;)17 b(\025)p Fq(\))g Fo(d)3355 2349 y Fl(3)3393 2390 y Fo(k)24 b(;)3260 2590 y Fq(\(I.18\))257 2710 y(for)32 b(an)m(y)i(in)m(tegrable)d(\(p)s (ossibly)h(v)m(ector-v)-5 b(alued\))33 b(function)f Fo(f)11 b Fq(.)404 2831 y(Next,)33 b(w)m(e)h(summarize)d(the)i(main)e(results)i (of)f(this)g(pap)s(er.)404 2951 y(In)46 b(Sect.)h(I)s(I,)i(w)m(e)f(pro) m(v)m(e)f(that)f(the)g(Hamiltonian)c Fo(H)54 b Fq(in)m(tro)s(duced)46 b(in)f(Eq.)i(\(I.6\))257 3072 y(is)42 b(selfadjoin)m(t)f(on)h(the)h (domain)d(of)i(the)g(op)s(erator)g Fn(\000)p Fq(\001)2382 3087 y Fm(x)2455 3072 y Fq(+)2585 3046 y(\024)2560 3072 y Fo(H)7 b Fq(,)45 b(pro)m(vided)e(the)f(fein-)257 3192 y(structure)35 b(constan)m(t)f Fo(\013)q Fq(,)f(whic)m(h)h(w)m(e)g (consider)g(a)f(p)s(erturbation)g(parameter,)g(is)f(small,)257 3312 y(dep)s(ending)26 b(on)g(the)g(v)-5 b(alue)25 b(of)g(the)h(ultra)m (violet)e(cuto\013)i Fo(\024)p Fq(.)41 b(\(Actually)-8 b(,)26 b(it)e(is)i(not)f(di\016cult)257 3433 y(to)33 b(construct)g(a)g(canonical)e(selfadjoin)m(t)g(extension)j(of)e Fo(H)8 b Fq(,)32 b(for)g Fj(arbitr)-5 b(ary)41 b Fq(v)-5 b(alues)33 b(of)f Fo(\024)257 3553 y Fq(and)40 b Fo(\013)q Fq(,)g(using)f(Bro)m(wnian)g(motion)f(and)h(a)g(F)-8 b(eynman-Kac)39 b(form)m(ula)e(for)i(exp)q([)p Fn(\000)p Fo(tH)8 b Fq(];)257 3674 y(see,)34 b(e.g.,)f([5].\))43 b(If)33 b Fo(\033)t Fq(\()p Fo(H)8 b Fq(\))32 b(denotes)i(the)f(sp)s (ectrum)g(of)f Fo(H)8 b Fq(,)32 b(w)m(e)i(pro)m(v)m(e)f(that)1329 3892 y Fo(E)1401 3907 y Fl(gs)1529 3892 y Fq(:=)60 b(inf)23 b Fo(\033)t Fq(\()p Fo(H)8 b Fq(\))59 b Fo(>)h Fn(\0001)837 b Fq(\(I.19\))257 4109 y(is)29 b(a)g Fj(simple)i(eigenvalue)k Fq(of)29 b Fo(H)37 b Fq(corresp)s(onding)29 b(to)g(a)g Fj(gr)-5 b(oundstate)31 b(eigenve)-5 b(ctor)39 b Fo(\036)3334 4124 y Fl(gs)3429 4109 y Fn(2)257 4230 y(H)34 b Fq(of)e Fo(H)8 b Fq(,)32 b(pro)m(vided)h Fo(\013)g Fq(is)g(su\016cien)m(tly)g (small)d(\(dep)s(ending)j(on)f Fo(\024)p Fq(\).)44 b(F)-8 b(urthermore,)1446 4448 y Fo(\033)t Fq(\()p Fo(H)8 b Fq(\))59 b(=)h([)p Fo(E)1964 4463 y Fl(gs)2054 4448 y Fo(;)39 b Fn(1)p Fq(\))22 b Fo(;)953 b Fq(\(I.20\))257 4666 y(in)23 b(fact,)i Fo(E)646 4681 y Fl(gs)736 4666 y Fq(is)e(a)f(threshold)h(for)g(con)m(tin)m(uous)h(sp)s(ectrum)f(of)g Fo(H)30 b Fq(of)23 b(in\014nite)f(m)m(ultiplicit)m(y)-8 b(.)257 4786 y(Stated)33 b(without)g(more)f(precision,)g(these)i (results)f(are)g(not)f(new.)45 b(The)34 b(\014rst)f(result)g(of)257 4907 y(this)40 b(t)m(yp)s(e)g(for)g(a)f(sligh)m(tly)f(simpli\014ed)g (mo)s(del)g(app)s(eared)i(in)f([1])h(and)f(for)h(the)g(mo)s(del)257 5027 y(describ)s(ed)32 b(ab)s(o)m(v)m(e)g(in)e([3];)i(a)f(stronger)g (result)g(v)-5 b(alid)29 b(for)i Fj(arbitr)-5 b(ary)34 b(values)39 b Fq(of)30 b Fo(\013)i Fq(and)f Fo(\024)257 5147 y Fq(w)m(as)j(pro)m(v)m(en)g(in)e([4,)h(7)o(].)44 b(The)34 b(p)s(oin)m(t)e(of)g(our)g(analysis)g(is)h(that)f(w)m(e)i (presen)m(t)g(an)f Fj(explicit)257 5268 y(inductive)d(c)-5 b(onstruction)34 b Fq(of)26 b(\()p Fo(E)1442 5283 y Fl(gs)1510 5268 y Fo(;)17 b(\036)1612 5283 y Fl(gs)1678 5268 y Fq(\),)29 b(based)e(on)g(a)g(decomp)s(osition)e(of)i(the)g(terms)g(in)257 5388 y(the)37 b(Hamiltonian)32 b Fo(H)43 b Fq(that)35 b(describ)s(e)i(the)f(in)m(teractions)f(b)s(et)m(w)m(een)j(the)e (electron)g(and)p eop %%Page: 5 6 5 5 bop 257 266 a Fr(BFP-GrSt,)32 b(04-F)-8 b(eb-2005)2178 b Fq(5)257 573 y(the)34 b(photons)g(in)m(to)e(a)h(sum)g(of)g(terms)g (with)g(the)g(prop)s(ert)m(y)h(that)f(at)g(least)g(one)g(photon)257 693 y(creation-)f(or)g(annihilation)d(op)s(erator)j(is)g(lo)s(calized)e (in)i(a)g(shell)287 817 y Fh(n)342 913 y Fo(k)424 814 y Fh(\014)424 863 y(\014)424 913 y(\014)g Fo(\024\014)601 872 y Fm(n)p Fl(+1)766 913 y Fn(\024)c(j)p Fo(k)s Fn(j)f(\024)h Fo(\024\014)1230 872 y Fm(n)1277 817 y Fh(o)1354 913 y Fo(;)88 b(\024)28 b Fn(\025)g Fq(1)22 b Fo(;)87 b Fq(0)28 b Fo(<)f(\014)34 b(<)27 b(\024)2271 872 y Fi(\000)p Fl(1)2388 913 y Fo(;)87 b(n)28 b Fn(2)g Fk(N)2748 928 y Fl(0)2821 913 y Fq(=)g Fn(f)p Fq(0)p Fo(;)17 b Fq(1)p Fo(;)g Fq(2)p Fo(;)g(:)g(:)g(:)n Fn(g)22 b Fo(;)3260 1046 y Fq(\(I.21\))257 1166 y(of)39 b(photon)h(momen)m(tum)e(space.)65 b(Our)40 b(inductiv)m(e)f(construction)h(yields)f(a)g(sequence)257 1287 y(\()p Fo(E)367 1302 y Fm(n)414 1287 y Fo(;)17 b(\036)516 1302 y Fm(n)563 1287 y Fq(\))601 1251 y Fi(1)601 1311 y Fm(n)p Fl(=0)760 1287 y Fq(of)k(appro)m(ximate)g(groundstate)h (energies)g(and)g(groundstates)h(constructed)257 1407 y(from)32 b(Hamiltonians)d Fo(H)1168 1422 y Fm(n)1247 1407 y Fq(with)k(an)f(infrared)g(cuto\013)h(at)f Fn(j)p Fo(k)s Fn(j)27 b Fq(=)g Fo(\024\014)2721 1371 y Fm(n)2768 1407 y Fq(.)44 b(W)-8 b(e)33 b(pro)m(v)m(e)g(that)1208 1627 y(lim)1184 1677 y Fm(n)p Fi(!1)1384 1627 y Fo(E)1456 1642 y Fm(n)1531 1627 y Fq(=)28 b Fo(E)1707 1642 y Fl(gs)1796 1627 y Fo(;)159 b Fq(lim)1958 1677 y Fm(n)p Fi(!1)2159 1627 y Fo(\036)2217 1642 y Fm(n)2291 1627 y Fq(=)28 b Fo(\036)2453 1642 y Fl(gs)2542 1627 y Fo(:)691 b Fq(\(I.22\))257 1868 y(Our)40 b(construction)f(is)g(inspired)g(b)m(y)i(and)e(similar)d (to)j(one)h(used)h(b)m(y)f(Pizzo)f([8])h(in)e(an)257 1988 y(analysis)g(of)g(Nelson's)h(mo)s(del.)59 b(The)39 b(main)e(new)i(c)m(hallenge)f(w)m(e)i(are)e(coping)g(with)g(is)257 2109 y(that,)c(while)f(the)h(in)m(teraction)f(term)g(in)g Fo(H)41 b Fq(sup)s(er\014cially)33 b(seems)i(to)e(b)s(e)h Fj(mar)-5 b(ginal)43 b Fq(\(in)257 2229 y(the)38 b(sense)h(of)d(p)s(o)m (w)m(er)i(coun)m(ting\))f(in)f(the)h Fj(infr)-5 b(ar)g(e)g(d)p Fq(,)38 b(it)e(is)g(actually)g Fj(irr)-5 b(elevant)46 b Fq(in)36 b(the)257 2350 y(infrared)k(on)h(the)h(subspace)g(of)f(all)e (those)i(states)h(where)g(the)f(electron)g(is)f(lo)s(calized)257 2470 y(near)35 b(the)h(n)m(ucleus)g(in)e(p)s(osition)f(space.)52 b(This)35 b(will)d(b)s(e)j(made)g(precise)h(with)e(the)h(help)257 2590 y(of)d(the)h(iden)m(tit)m(y)1533 2711 y Fo(i)17 b Fq([)p Fo(H)24 b(;)32 b(~)-48 b(x)q Fq(])60 b(=)g(2)13 b Fo(~)-45 b(v)25 b(;)1041 b Fq(\(I.23\))257 2894 y(where)39 b Fo(~)-45 b(v)46 b 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Fk(N)1798 2872 y Fl(0)1844 2857 y Fq(,)j(w)m(e)g(ha)m(v)m (e)g(that)f Fo(\033)2549 2872 y Fm(n)p Fl(+1)2719 2857 y Fn(\024)d Fo(\024\014)38 b Fn(\024)33 b Fq(1.)51 b(T)-8 b(o)35 b(cut)257 2977 y(the)e(in)m(teraction)f(in)m(to)g(slices)g(at)g (ev)m(er)i(lo)m(w)m(er)f(energy)h(scales,)f(w)m(e)g(in)m(tro)s(duce)318 3172 y Fo(~)303 3197 y(G)380 3212 y Fm(n)427 3197 y Fq(\()p Fo(k)s Fq(\))60 b(:=)g Fc(1)836 3101 y Fh(\020)886 3197 y Fo(\033)941 3212 y Fm(n)1016 3197 y Fn(\024)28 b(j)p Fo(k)s Fn(j)1231 3101 y Fh(\021)1317 3172 y Fo(~)1302 3197 y(G)p Fq(\()p Fo(k)s Fq(\))22 b Fo(;)112 b Fq(and)1936 3172 y Fo(~)1921 3197 y(G)1998 3156 y Fm(m)1998 3222 y(n)2065 3197 y Fq(\()p Fo(k)s Fq(\))60 b(:=)g Fc(1)2474 3101 y Fh(\020)2524 3197 y Fo(\033)2579 3212 y Fm(n)2654 3197 y Fn(\024)28 b(j)p Fo(k)s Fn(j)f Fo(<)h(\033)3055 3212 y Fm(m)3122 3101 y Fh(\021)3208 3172 y Fo(~)3193 3197 y(G)p Fq(\()p Fo(k)s Fq(\))22 b Fo(;)3270 3330 y Fq(\(I)s(I.9\))257 3459 y(for)35 b(all)d Fo(k)j Fn(2)d Fk(R)796 3423 y Fl(3)865 3459 y Fn(\002)24 b Fk(Z)1035 3474 y Fl(2)1106 3459 y Fq(and)35 b Fo(m;)17 b(n)32 b Fn(2)g Fk(N)1681 3474 y Fl(0)1726 3459 y Fq(,)k(with)e Fo(m)e(<)f(n)p Fq(.)51 b(Note)34 b(that)2839 3434 y Fo(~)2824 3459 y(G)d Fq(=)3040 3393 y Fh(P)3127 3419 y Fi(1)3127 3484 y Fm(n)p Fl(=0)3296 3434 y Fo(~)3281 3459 y(G)3358 3423 y Fm(n)3358 3484 y(n)p Fl(+1)257 3596 y Fq(and)41 b(that)689 3570 y Fo(~)674 3596 y(G)751 3611 y Fm(n)839 3596 y Fq(is)f(the)h(coupling)e(function)h(of)g(the)h(in)m(teraction,)g (regularized)f(in)g(the)257 3716 y(infrared)e(region.)61 b(W)-8 b(e)40 b(also)d(decomp)s(ose)j(the)f(F)-8 b(o)s(c)m(k)39 b(space)g Fn(F)48 b Fq(=)38 b Fn(F)10 b Fq(\()p Fg(h)p Fq(\))38 b(in)m(to)g(energy)257 3836 y(scales)33 b(b)m(y)h(in)m(tro)s (ducing)1029 4056 y Fg(h)1081 4071 y Fm(n)1188 4056 y Fq(:=)60 b Fo(L)1417 4015 y Fl(2)1457 3960 y Fh(h)1496 4056 y Fn(K)1572 4071 y Fm(n)1619 3960 y Fh(i)1753 4056 y Fq(and)94 b Fg(h)2056 4015 y Fm(m)2056 4081 y(n)2183 4056 y Fq(:=)60 b Fo(L)2412 4015 y Fl(2)2452 3960 y Fh(h)2491 4056 y Fn(K)2568 4015 y Fm(m)2567 4081 y(n)2635 3960 y Fh(i)2697 4056 y Fo(;)498 b Fq(\(I)s(I.10\))257 4276 y(where)506 4496 y(for)32 b(0)c Fn(\024)g Fo(n)g Fn(\024)g(1)p Fq(:)93 b Fn(K)1324 4511 y Fm(n)1465 4496 y Fq(:=)1655 4400 y Fh(n)1711 4496 y Fq(\()1744 4470 y Fo(~)1749 4496 y(k)s(;)17 b(\034)11 b Fq(\))28 b Fn(2)g Fk(R)2126 4455 y Fl(3)2193 4496 y Fn(\002)23 b Fk(Z)2362 4511 y Fl(2)2426 4397 y Fh(\014)2426 4446 y(\014)2426 4496 y(\014)33 b Fo(\033)2542 4511 y Fm(n)2617 4496 y Fn(\024)28 b Fo(!)t Fq(\()2820 4470 y Fo(~)2825 4496 y(k)r Fq(\))2916 4400 y Fh(o)2994 4496 y Fo(;)201 b Fq(\(I)s(I.11\))290 4679 y(for)32 b(0)27 b Fn(\024)h Fo(m)g(<)g(n)g Fn(\024)g(1)p Fq(:)83 b Fn(K)1315 4643 y Fm(m)1314 4704 y(n)1465 4679 y Fq(:=)1655 4583 y Fh(n)1711 4679 y Fq(\()1744 4653 y Fo(~)1749 4679 y(k)s(;)17 b(\034)11 b Fq(\))28 b Fn(2)g Fk(R)2126 4638 y Fl(3)2193 4679 y Fn(\002)23 b Fk(Z)2362 4694 y Fl(2)2426 4579 y Fh(\014)2426 4629 y(\014)2426 4679 y(\014)33 b Fo(\033)2542 4694 y Fm(n)2617 4679 y Fn(\024)28 b Fo(!)t Fq(\()2820 4653 y Fo(~)2825 4679 y(k)r Fq(\))g Fo(<)f(\033)3102 4694 y Fm(m)3169 4583 y Fh(o)3247 4679 y Fo(:)-52 b Fq(\(I)s(I.12\))257 4916 y(Note)31 b(that)g Fn(K)778 4879 y Fl(0)777 4940 y Fm(n)852 4916 y Fn(\032)d(K)1033 4931 y Fm(n)1110 4916 y Fq(is)j(a)f(prop)s(er)h (subset.)45 b(F)-8 b(or)30 b(in)m(tegers)h(1)c Fn(\024)i Fo(m)f(<)f(n)h(<)g(`)f Fn(\024)h(1)p Fq(,)j(w)m(e)257 5036 y(ha)m(v)m(e)26 b(the)f(disjoin)m(t)e(decomp)s(osition)f Fn(K)1683 5051 y Fm(`)1744 5036 y Fq(=)27 b Fn(K)1923 5051 y Fm(m)1995 5036 y Fn([)5 b(K)2143 5000 y Fm(m)2142 5061 y(n)2216 5036 y Fn([)g(K)2364 5000 y Fm(n)2363 5061 y(`)2436 5036 y Fq(and)25 b(hence)g(the)g(orthogonal)257 5156 y(sum)1421 5277 y Fg(h)1473 5292 y Fm(`)1566 5249 y Fn(\030)1566 5281 y Fq(=)1703 5277 y Fg(h)1755 5292 y Fm(m)1844 5277 y Fn(\010)e Fg(h)1996 5236 y Fm(m)1996 5301 y(n)2084 5277 y Fn(\010)g Fg(h)2236 5236 y Fm(n)2236 5301 y(`)2305 5277 y Fo(;)890 b Fq(\(I)s(I.13\))p eop %%Page: 9 10 9 9 bop 257 266 a Fr(BFP-GrSt,)32 b(04-F)-8 b(eb-2005)2178 b Fq(9)257 573 y(whic)m(h)33 b(giv)m(es)g(rise)g(to)f(the)h (isomorphism)1371 793 y Fn(F)1443 808 y Fm(`)1536 765 y Fn(\030)1537 797 y Fq(=)1674 793 y Fn(F)1746 808 y Fm(m)1834 793 y Fn(\012)23 b(F)2016 752 y Fm(m)2006 817 y(n)2104 793 y Fn(\012)g(F)2286 752 y Fm(n)2276 817 y(`)2354 793 y Fo(;)841 b Fq(\(I)s(I.14\))257 1013 y(with)33 b Fn(F)552 1028 y Fm(n)626 1013 y Fq(:=)27 b Fn(F)10 b Fq(\()p Fg(h)928 1028 y Fm(n)975 1013 y Fq(\))32 b(and)h Fn(F)1317 977 y Fm(m)1307 1037 y(n)1411 1013 y Fq(:=)27 b Fn(F)10 b Fq(\()p Fg(h)1713 977 y Fm(m)1713 1037 y(n)1779 1013 y Fq(\).)44 b(In)33 b(particular,)d(for)i(an)m(y)i Fo(n)28 b Fn(2)g Fk(N)9 b Fq(,)1146 1233 y Fn(F)70 b Fq(=)60 b Fn(F)1496 1248 y Fi(1)1630 1205 y Fn(\030)1631 1237 y Fq(=)1768 1233 y Fn(F)1840 1248 y Fm(n)1909 1233 y Fn(\012)22 b(F)2090 1192 y Fm(n)2080 1258 y(n)p Fl(+1)2239 1233 y Fn(\012)h(F)2421 1192 y Fm(n)p Fl(+1)2411 1258 y Fi(1)2580 1233 y Fo(:)615 b Fq(\(I)s(I.15\))257 1453 y(Setting)916 1573 y Fn(H)1000 1588 y Fm(n)1107 1573 y Fq(:=)60 b Fn(H)1354 1588 y Fm(el)1435 1573 y Fn(\012)23 b(F)1607 1588 y Fm(n)1748 1573 y Fq(and)95 b Fn(H)2085 1532 y Fm(m)2084 1598 y(n)2212 1573 y Fq(:=)60 b Fn(H)2459 1588 y Fm(el)2540 1573 y Fn(\012)22 b(F)2721 1532 y Fm(m)2711 1598 y(n)2810 1573 y Fo(;)385 b Fq(\(I)s(I.16\))257 1748 y(and)44 b(giv)m(en)g(energy)h(scale)e(indices)h Fo(m;)17 b(n)47 b Fn(2)g Fk(N)2036 1763 y Fl(0)2081 1748 y Fq(,)g Fo(m)f(<)h(n)p Fq(,)g(w)m(e)d(de\014ne)h(the)f(v)m(elo)s(cit)m(y)257 1868 y(op)s(erator)28 b Fo(~)-45 b(v)697 1883 y Fm(n)744 1868 y Fq(,)33 b(the)g(\014eld)g(energy)g(op)s(erators)1952 1843 y(\024)1927 1868 y Fo(H)2008 1883 y Fm(n)2055 1868 y Fq(,)2140 1843 y(\024)2114 1868 y Fo(H)2203 1832 y Fm(m)2195 1893 y(n)2270 1868 y Fq(,)f(and)h(the)g(Hamiltonian)c Fo(H)3329 1883 y Fm(n)3409 1868 y Fq(at)257 1988 y(those)34 b(scales)f(b)m(y)890 2208 y Fo(~)-45 b(v)941 2223 y Fm(n)1071 2208 y Fq(:=)83 b Fn(\000)p Fo(i)1376 2183 y(~)1367 2208 y Fn(r)1450 2223 y Fm(x)1534 2208 y Fq(+)38 b Fo(a)1699 2167 y Fi(\003)1739 2208 y Fq(\()1792 2183 y Fo(~)1777 2208 y(G)1854 2223 y Fm(n)1901 2208 y Fq(\))g(+)h Fo(a)p Fq(\()2196 2183 y Fo(~)2181 2208 y(G)2258 2223 y Fm(n)2305 2208 y Fq(\))22 b Fo(;)830 b Fq(\(I)s(I.17\))886 2358 y(\024)860 2384 y Fo(H)941 2399 y Fm(n)1071 2384 y Fq(:=)1257 2267 y Fh(Z)1357 2384 y Fc(1)p Fq(\()p Fo(\033)1506 2399 y Fm(n)1581 2384 y Fn(\024)28 b(j)p Fo(k)s Fn(j)p Fq(\))21 b Fo(!)t Fq(\()p Fo(k)s Fq(\))h Fo(a)2123 2343 y Fi(\003)2163 2384 y Fq(\()p Fo(k)s Fq(\))p Fo(a)p Fq(\()p Fo(k)s Fq(\))g Fo(dk)j(;)572 b Fq(\(I)s(I.18\))858 2576 y(\024)833 2601 y Fo(H)922 2560 y Fm(m)914 2626 y(n)1071 2601 y Fq(:=)1257 2484 y Fh(Z)1357 2601 y Fc(1)p Fq(\()p Fo(\033)1506 2616 y Fm(n)1581 2601 y Fn(\024)28 b(j)p Fo(k)s Fn(j)f Fo(<)h(\033)1982 2616 y Fm(m)2049 2601 y Fq(\))22 b Fo(!)t Fq(\()p Fo(k)s Fq(\))f Fo(a)2376 2560 y Fi(\003)2416 2601 y Fq(\()p Fo(k)s Fq(\))p Fo(a)p Fq(\()p Fo(k)s Fq(\))h Fo(dk)j(;)319 b Fq(\(I)s(I.19\))860 2790 y Fo(H)941 2805 y Fm(n)1071 2790 y Fq(:=)79 b Fo(~)-45 b(v)1319 2749 y Fl(2)1304 2814 y Fm(n)1398 2790 y Fn(\000)39 b Fo(V)22 b Fq(\()p Fo(x)p Fq(\))39 b(+)1903 2765 y(\024)1877 2790 y Fo(H)1958 2805 y Fm(n)2027 2790 y Fo(;)1168 b Fq(\(I)s(I.20\))257 3010 y(as)45 b(op)s(erators)g(on)g Fn(H)1065 3025 y Fm(n)1157 3010 y Fq(and)g Fn(H)1444 2974 y Fm(m)1443 3034 y(n)1511 3010 y Fq(,)j(resp)s(ectiv)m(ely)-8 b(.)81 b(W)-8 b(e)45 b(in)m(tro)s(duce)g(the)h(groundstate)257 3130 y(energy)34 b(at)e(scales)h Fo(n)g Fq(and)g(their)f(di\013erences)h(for)g (di\013eren)m(t)f(scales)h Fo(m)28 b(<)g(n)33 b Fq(b)m(y)935 3350 y Fo(E)1007 3365 y Fm(n)1114 3350 y Fq(:=)60 b(inf)23 b Fo(\033)t Fq(\()p Fo(H)1590 3365 y Fm(n)1637 3350 y Fq(\))94 b(and)h Fo(E)2099 3309 y Fm(m)2093 3375 y(n)2226 3350 y Fq(:=)60 b Fo(E)2461 3365 y Fm(m)2550 3350 y Fn(\000)22 b Fo(E)2721 3365 y Fm(n)2791 3350 y Fo(:)404 b Fq(\(I)s(I.21\))257 3570 y(T)-8 b(o)47 b(compare)g(Hamiltonians)c Fo(H)1514 3585 y Fm(n)1608 3570 y Fq(and)j Fo(H)1892 3585 y Fm(n)p Fl(+1)2076 3570 y Fq(at)h(successiv)m(e)i(energy)f(scales,)j(it)46 b(is)257 3690 y(con)m(v)m(enien)m(t)e(to)d(de\014ne)i(the)e(p)s(ositiv) m(e)g(op)s(erators)h Fo(H)2244 3654 y Fl(+)2236 3715 y Fm(n)2344 3690 y Fq(and)2550 3658 y Fh(f)2542 3690 y Fo(H)2631 3654 y Fl(+)2623 3715 y Fm(n)2731 3690 y Fq(on)g Fn(H)2960 3705 y Fm(n)3048 3690 y Fq(and)g Fn(H)3331 3705 y Fm(n)p Fl(+1)3468 3690 y Fq(,)257 3811 y(resp)s(ectiv)m(ely)-8 b(,)34 b(b)m(y)1109 4031 y Fo(H)1198 3990 y Fl(+)1190 4056 y Fm(n)1339 4031 y Fq(:=)83 b Fo(H)1606 4046 y Fm(n)1692 4031 y Fn(\000)39 b Fo(E)1880 4046 y Fm(n)1949 4031 y Fo(;)1246 b Fq(\(I)s(I.22\))1117 4144 y Fh(f)1109 4176 y Fo(H)1198 4135 y Fl(+)1190 4201 y Fm(n)1339 4176 y Fq(:=)83 b Fo(H)1614 4135 y Fl(+)1606 4201 y Fm(n)1695 4176 y Fn(\012)22 b Fc(1)1850 4135 y Fm(n)1850 4201 y(n)p Fl(+1)2032 4176 y Fq(+)44 b Fc(1)2208 4191 y Fm(n)2277 4176 y Fn(\012)2402 4151 y Fq(\024)2377 4176 y Fo(H)2466 4135 y Fm(n)2458 4201 y(n)p Fl(+1)2617 4176 y Fo(;)578 b Fq(\(I)s(I.23\))257 4396 y(where)32 b(w)m(e)f(denote)g(the)g(iden)m (tit)m(y)f(op)s(erator)g(on)g Fn(H)2122 4411 y Fm(n)2199 4396 y Fq(and)h Fn(F)2469 4360 y Fm(m)2459 4421 y(n)2565 4396 y Fq(b)m(y)g Fc(1)2754 4411 y Fm(n)2832 4396 y Fq(and)f Fc(1)3075 4360 y Fm(m)3075 4421 y(n)3142 4396 y Fq(,)g(resp)s(ec-)257 4517 y(tiv)m(ely)-8 b(.)50 b(One)35 b(resolv)m(en)m(t)h(on)f Fn(H)1404 4532 y Fm(n)p Fl(+1)1576 4517 y Fq(that)g(w)m(e)h(use)f(in)f (estimates)h(o)m(v)m(er)h(and)f(o)m(v)m(er)h(again)257 4637 y(is)c(the)h(follo)m(wing)d(one,)1358 4748 y Fh(e)1336 4781 y Fo(R)1410 4796 y Fm(n)1517 4781 y Fq(:=)1680 4684 y Fh(\020)1738 4748 y(f)1730 4781 y Fo(H)1819 4740 y Fl(+)1811 4805 y Fm(n)1916 4781 y Fq(+)39 b Fo(\033)2086 4796 y Fm(n)p Fl(+1)2224 4684 y Fh(\021)2273 4707 y Fi(\000)p Fl(1)2390 4781 y Fo(:)805 b Fq(\(I)s(I.24\))257 4985 y(Note)32 b(that)e(0)e Fn(\024)906 4953 y Fh(e)883 4985 y Fo(R)957 5000 y Fm(n)1032 4985 y Fn(\024)g Fo(\033)1196 4944 y Fi(\000)p Fl(1)1192 5007 y Fm(n)p Fl(+1)1330 4985 y Fq(.)43 b(W)-8 b(e)31 b(iden)m(tify)c Fo(~)-45 b(v)1967 5000 y Fm(n)2045 4985 y Fq(with)27 b Fo(~)-46 b(v)2312 5000 y Fm(n)2379 4985 y Fn(\012)19 b Fc(1)2531 4949 y Fm(n)2531 5010 y(n)p Fl(+1)2668 4985 y Fq(,)32 b(whic)m(h)f(is)g (acting)f(on)257 5106 y Fn(H)341 5121 y Fm(n)p Fl(+1)479 5106 y Fq(.)43 b(Note)33 b(that,)f(for)g Fo(n)c Fn(2)g Fk(N)1418 5121 y Fl(0)1464 5106 y Fq(,)1096 5326 y Fo(~)-45 b(v)1147 5341 y Fm(n)p Fl(+1)1345 5326 y Fq(=)56 b Fo(~)-45 b(v)1528 5341 y Fm(n)1614 5326 y Fq(+)38 b Fo(a)1779 5284 y Fi(\003)1819 5326 y Fq(\()1872 5300 y Fo(~)1857 5326 y(G)1934 5284 y Fm(n)1934 5350 y(n)p Fl(+1)2071 5326 y Fq(\))g(+)h Fo(a)p Fq(\()2366 5300 y Fo(~)2351 5326 y(G)2428 5284 y Fm(n)2428 5350 y(n)p Fl(+1)2565 5326 y Fq(\))22 b Fo(:)570 b Fq(\(I)s(I.25\))p eop %%Page: 10 11 10 10 bop 257 266 a Fr(BFP-GrSt,)32 b(04-F)-8 b(eb-2005)2130 b Fq(10)257 573 y(Similarly)-8 b(,)29 b(giv)m(en)k Fo( )1009 588 y Fm(n)1084 573 y Fn(2)28 b(H)1262 588 y Fm(n)1309 573 y Fq(,)33 b(w)m(e)g(de\014ne)h(a)e(v)m(ector)1286 744 y(~)1266 770 y Fo( )1329 785 y Fm(n)1437 770 y Fq(:=)60 b Fo( )1663 785 y Fm(n)1732 770 y Fn(\012)23 b Fq(\012)1902 729 y Fm(n)1902 795 y(n)p Fl(+1)2100 770 y Fn(2)50 b(H)2300 785 y Fm(n)p Fl(+1)2459 770 y Fo(;)736 b Fq(\(I)s(I.26\))257 967 y(where)42 b(\012)617 982 y Fm(n)704 967 y Fq(and)e(\012)971 931 y Fm(m)971 992 y(n)1079 967 y Fq(denote)g(the)h(v)-5 b(acuum)40 b(v)m(ectors)i(in)d Fn(F)2476 982 y Fm(n)2563 967 y Fq(and)h Fn(F)2842 931 y Fm(m)2832 992 y(n)2908 967 y Fq(,)i(resp)s(ectiv)m(ely)-8 b(.)257 1088 y(With)32 b(these)i(notations,)e(w)m(e)i(ha)m(v)m(e)f(that)380 1285 y Fo(H)469 1244 y Fl(+)461 1309 y Fm(n)p Fl(+1)695 1285 y Fq(=)875 1252 y Fh(f)867 1285 y Fo(H)956 1244 y Fl(+)948 1309 y Fm(n)1054 1285 y Fq(+)38 b Fo(W)1274 1244 y Fm(n)1260 1309 y(n)p Fl(+1)1436 1285 y Fq(+)h Fo(E)1629 1244 y Fm(n)1623 1309 y(n)p Fl(+1)1782 1285 y Fo(;)1413 b Fq(\(I)s(I.27\))369 1430 y Fo(W)475 1389 y Fm(n)461 1455 y(n)p Fl(+1)681 1430 y Fq(:=)83 b(\()l Fo(~)-45 b(v)952 1445 y Fm(n)p Fl(+1)1090 1430 y Fq(\))1128 1389 y Fl(2)1206 1430 y Fn(\000)39 b Fq(\()l Fo(~)-45 b(v)1407 1445 y Fm(n)1454 1430 y Fq(\))1492 1389 y Fl(2)3222 1430 y Fq(\(I)s(I.28\))695 1611 y(=)96 b(2)17 b Fo(a)984 1570 y Fi(\003)1023 1611 y Fq(\()1076 1586 y Fo(~)1061 1611 y(G)1138 1570 y Fm(n)1138 1635 y(n)p Fl(+1)1275 1611 y Fq(\))22 b Fn(\001)c Fo(~)-45 b(v)1432 1626 y Fm(n)1529 1611 y Fq(+)50 b(2)13 b Fo(~)-45 b(v)1768 1626 y Fm(n)1837 1611 y Fn(\001)21 b Fo(a)p Fq(\()1990 1586 y Fo(~)1975 1611 y(G)2052 1570 y Fm(n)2052 1635 y(n)p Fl(+1)2190 1611 y Fq(\))49 b(+)2403 1515 y Fh(\020)2453 1611 y Fo(a)2504 1570 y Fi(\003)2543 1611 y Fq(\()2596 1586 y Fo(~)2581 1611 y(G)2658 1570 y Fm(n)2658 1635 y(n)p Fl(+1)2795 1611 y Fq(\))22 b(+)g Fo(a)p Fq(\()3057 1586 y Fo(~)3042 1611 y(G)3119 1570 y Fm(n)3119 1635 y(n)p Fl(+1)3257 1611 y Fq(\))3295 1515 y Fh(\021)3344 1538 y Fl(2)695 1790 y Fq(=)96 b(2)17 b Fo(a)984 1749 y Fi(\003)1023 1790 y Fq(\()1076 1765 y Fo(~)1061 1790 y(G)1138 1749 y Fm(n)1138 1815 y(n)p Fl(+1)1275 1790 y Fq(\))22 b Fn(\001)c Fo(~)-45 b(v)1432 1805 y Fm(n)1529 1790 y Fq(+)50 b(2)13 b Fo(~)-45 b(v)1768 1805 y Fm(n)1837 1790 y Fn(\001)21 b Fo(a)p Fq(\()1990 1765 y Fo(~)1975 1790 y(G)2052 1749 y Fm(n)2052 1815 y(n)p Fl(+1)2190 1790 y Fq(\))49 b(+)h Fo(a)2454 1749 y Fi(\003)2494 1790 y Fq(\()2547 1765 y Fo(~)2532 1790 y(G)2609 1749 y Fm(n)2609 1815 y(n)p Fl(+1)2746 1790 y Fq(\))22 b Fn(\001)g Fo(a)2907 1749 y Fi(\003)2946 1790 y Fq(\()2999 1765 y Fo(~)2984 1790 y(G)3061 1749 y Fm(n)3061 1815 y(n)p Fl(+1)3198 1790 y Fq(\))895 1971 y(+)28 b Fo(a)p Fq(\()1103 1946 y Fo(~)1088 1971 y(G)1165 1930 y Fm(n)1165 1996 y(n)p Fl(+1)1302 1971 y Fq(\))22 b Fn(\001)g Fo(a)p Fq(\()1516 1946 y Fo(~)1501 1971 y(G)1578 1930 y Fm(n)1578 1996 y(n)p Fl(+1)1715 1971 y Fq(\))50 b(+)f(2)17 b Fo(a)2045 1930 y Fi(\003)2084 1971 y Fq(\()2137 1946 y Fo(~)2122 1971 y(G)2199 1930 y Fm(n)2199 1996 y(n)p Fl(+1)2336 1971 y Fq(\))22 b Fn(\001)g Fo(a)p Fq(\()2550 1946 y Fo(~)2535 1971 y(G)2612 1930 y Fm(n)2612 1996 y(n)p Fl(+1)2749 1971 y Fq(\))50 b(+)2963 1871 y Fh(\015)2963 1921 y(\015)2963 1971 y(\015)3024 1946 y Fo(~)3009 1971 y(G)3086 1930 y Fm(n)3086 1996 y(n)p Fl(+1)3223 1871 y Fh(\015)3223 1921 y(\015)3223 1971 y(\015)3269 1898 y Fl(2)3331 1971 y Fo(;)257 2222 y Fq(where)528 2122 y Fh(\015)528 2172 y(\015)528 2222 y(\015)589 2197 y Fo(~)574 2222 y(G)651 2186 y Fm(n)651 2247 y(n)p Fl(+1)789 2122 y Fh(\015)789 2172 y(\015)789 2222 y(\015)835 2149 y Fl(2)902 2222 y Fq(:=)1032 2151 y Fh(R)1104 2222 y Fn(j)1147 2197 y Fo(~)1132 2222 y(G)1209 2186 y Fm(n)1209 2247 y(n)p Fl(+1)1346 2222 y Fq(\()p Fo(k)s Fq(\))p Fn(j)1504 2186 y Fl(2)1560 2222 y Fo(dk)s Fq(.)39 b(In)22 b(Eq.)h(\(I)s(I.28\),)h(w)m(e)e(mak)m(e) g(use)h(of)e(the)h(Coulom)m(b)257 2376 y(gauge)33 b(condition)971 2351 y Fo(~)962 2376 y Fn(r)22 b(\001)1135 2351 y Fo(~)1117 2376 y(A)p Fq(\()p Fo(x)p Fq(\))28 b(=)g(0,)k(whic)m(h)h(implies)d (that)307 2582 y Fo(a)358 2541 y Fi(\003)398 2582 y Fq(\()451 2557 y Fo(~)436 2582 y(G)513 2541 y Fm(n)513 2607 y(n)p Fl(+1)650 2582 y Fq(\))19 b Fn(\001)c Fo(~)-45 b(v)801 2597 y Fm(n)907 2582 y Fq(=)55 b Fo(~)-45 b(v)1089 2597 y Fm(n)1155 2582 y Fn(\001)19 b Fo(a)1253 2541 y Fi(\003)1293 2582 y Fq(\()1346 2557 y Fo(~)1331 2582 y(G)1408 2541 y Fm(n)1408 2607 y(n)p Fl(+1)1545 2582 y Fq(\))94 b(and)h Fo(a)p Fq(\()2033 2557 y Fo(~)2018 2582 y(G)2095 2541 y Fm(n)2095 2607 y(n)p Fl(+1)2232 2582 y Fq(\))19 b Fn(\001)c Fo(~)-45 b(v)2383 2597 y Fm(n)2489 2582 y Fq(=)55 b Fo(~)-45 b(v)2671 2597 y Fm(n)2737 2582 y Fn(\001)19 b Fo(a)p Fq(\()2888 2557 y Fo(~)2873 2582 y(G)2950 2541 y Fm(n)2950 2607 y(n)p Fl(+1)3087 2582 y Fq(\))h Fo(:)50 b Fq(\(I)s(I.29\))257 2780 y(Finally)-8 b(,)38 b(it)g(is)g(con)m(v)m(enien)m(t)i(to)f (collect)e(the)i(v)-5 b(arious)38 b(assumptions)h(whic)m(h)g(en)m(ter)h (the)257 2900 y(deriv)-5 b(ations)32 b(of)g(the)h(estimates,)f(b)s(elo) m(w,)h(in)f(the)h(follo)m(wing)c(Hyp)s(othesis.)257 3084 y Fc(Hyp)s(othesis)41 b(1.)i Fj(Assume)38 b(that)g Fo(\024)33 b Fn(\025)g Fq(1)k Fj(and)g(that)h Fn(\006)p Fo(V)55 b Fn(\024)33 b Fo(")p Fq(\()p Fn(\000)p Fq(\001\))25 b(+)f Fo(b)2966 3099 y Fm(")3003 3084 y Fj(,)38 b(for)g Fo(")32 b(>)h Fq(0)p Fj(,)257 3204 y(and)i(set)1331 3325 y Fo(C)1401 3340 y Fl(1)1503 3325 y Fq(:=)62 b(8\()p Fo(\024)22 b Fq(+)g Fo(b)1972 3340 y Fl(1)p Fm(=)p Fl(8)2083 3325 y Fq(\))62 b Fn(\025)h Fq(8)22 b Fo(:)801 b Fq(\(I)s(I.30\))257 3489 y Fj(Assume)35 b(furthermor)-5 b(e)35 b(that)638 3725 y Fq(0)62 b Fn(\024)h Fo(\013)h Fn(\024)f Fo(\014)68 b Fn(\024)63 b Fo(\014)1473 3740 y Fl(1)1575 3725 y Fq(:=)f(min)1920 3604 y Fh(\032)1992 3658 y Fq(1)p 1992 3702 49 4 v 1992 3794 a(2)2067 3725 y Fo(;)45 b(\024)2195 3684 y Fi(\000)p Fl(1)2306 3725 y Fo(;)f Fq(\001)p Fo(e)2503 3740 y Fm(el)2579 3725 y Fo(\024)2635 3684 y Fi(\000)p Fl(1)2729 3604 y Fh(\033)2814 3725 y Fo(;)381 b Fq(\(I)s(I.31\))257 3954 y Fj(and)35 b(set)1496 4116 y Fo(g)1547 4065 y Fl(\(1\))1543 4141 y Fm(\013;\014)1718 4116 y Fq(:=)1893 4048 y Fo(C)1963 4063 y Fl(1)2024 4048 y Fo(\013)2087 4012 y Fl(3)p Fm(=)p Fl(2)p 1893 4093 305 4 v 1960 4184 a Fo(\014)2021 4155 y Fl(1)p Fm(=)p Fl(2)2229 4116 y Fo(:)966 b Fq(\(I)s(I.32\))257 4431 y Fb(I)t(I.2)135 b(Some)46 b(Basic)f(Estimates)257 4616 y Fq(W)-8 b(e)33 b(mak)m(e)g(use)h(of)e(the)h(notation)1293 4813 y Fn(k)p Fo(b)p Fn(k)1434 4772 y Fl(2)1534 4813 y Fq(:=)1697 4696 y Fh(Z)1797 4813 y Fn(k)p Fo(b)p Fq(\()p Fo(k)s Fq(\))p Fn(k)2068 4772 y Fl(2)2068 4837 y Fi(B)r Fl(\()p Fi(H)2203 4849 y Fa(el)2257 4837 y Fl(\))2305 4813 y Fo(dk)25 b(;)763 b Fq(\(I)s(I.33\))257 5062 y(for)32 b Fo(b)d Fn(2)f Fo(L)636 5026 y Fl(2)675 5062 y Fq([)p Fk(R)768 5026 y Fl(3)836 5062 y Fn(\002)23 b Fk(Z)1005 5077 y Fl(2)1042 5062 y Fq(;)17 b Fn(B)s Fq(\()p Fn(H)1276 5077 y Fm(el)1335 5062 y Fq(\)].)43 b(Moreo)m(v)m(er,)34 b(for)2063 5036 y Fo(~)2074 5062 y(b)28 b Fn(2)g Fo(L)2303 5026 y Fl(2)2343 5062 y Fq([)p Fk(R)2436 5026 y Fl(3)2504 5062 y Fn(\002)23 b Fk(Z)2673 5077 y Fl(2)2709 5062 y Fq(;)17 b Fn(B)s Fq(\()p Fn(H)2943 5077 y Fm(el)3002 5062 y Fq(\))3040 5026 y Fl(3)3080 5062 y Fq(],)32 b(w)m(e)i(set)522 5331 y Fn(k)561 5305 y Fo(~)572 5331 y(b)q Fq(\()p Fo(k)s Fq(\))p Fn(k)794 5290 y Fl(2)794 5356 y Fi(B)r Fl(\()p Fi(H)929 5368 y Fa(el)982 5356 y Fl(\))1042 5331 y Fq(:=)1219 5224 y Fl(3)1177 5248 y Fh(X)1172 5430 y Fm(\027)t Fl(=1)1318 5331 y Fn(k)p Fo(b)1409 5346 y Fm(\027)1453 5331 y Fq(\()p Fo(k)s Fq(\))p Fn(k)1633 5290 y Fl(2)1633 5356 y Fi(B)r Fl(\()p Fi(H)1768 5368 y Fa(el)1821 5356 y Fl(\))1947 5331 y Fq(and)95 b Fn(k)2238 5305 y Fo(~)2249 5331 y(b)p Fn(k)2340 5290 y Fl(2)2407 5331 y Fq(:=)2585 5224 y Fl(3)2542 5248 y Fh(X)2538 5430 y Fm(\027)t Fl(=1)2684 5331 y Fn(k)p Fo(b)2775 5346 y Fm(\027)2818 5331 y Fn(k)2868 5290 y Fl(2)2930 5331 y Fo(:)265 b Fq(\(I)s(I.34\))p eop %%Page: 11 12 11 11 bop 257 266 a Fr(BFP-GrSt,)32 b(04-F)-8 b(eb-2005)2130 b Fq(11)257 573 y(Using)33 b(this)f(notation,)f(w)m(e)j(ha)m(v)m(e,)g (for)e(all)e Fo(n)e Fn(2)g Fk(N)2065 588 y Fl(0)2143 573 y Fq(and)33 b(all)d Fo(k)h Fn(2)d Fk(R)2710 537 y Fl(3)2778 573 y Fn(\002)22 b Fk(Z)2946 588 y Fl(2)2983 573 y Fq(,)629 847 y Fn(k)694 821 y Fo(~)679 847 y(G)756 805 y Fm(n)756 871 y(n)p Fl(+1)893 847 y Fq(\()p Fo(k)s Fq(\))p Fn(k)1073 805 y Fl(2)1073 871 y Fi(B)r Fl(\()p Fi(H)1208 883 y Fa(el)1261 871 y Fl(\))1353 847 y Fn(\024)1624 779 y Fq(3)17 b Fo(\013)1753 743 y Fl(3)p 1501 823 415 4 v 1501 915 a Fq(\(2)p Fo(\031)t Fq(\))1685 886 y Fl(3)1740 915 y Fq(2)g Fn(j)p Fo(k)s Fn(j)1942 847 y Fc(1)1998 750 y Fh(\020)2047 847 y Fo(\033)2102 862 y Fm(n)p Fl(+1)2268 847 y Fn(\024)28 b Fo(!)t Fq(\()p Fo(k)s Fq(\))f Fo(<)g(\033)2753 862 y Fm(n)2801 750 y Fh(\021)3222 847 y Fq(\(I)s(I.35\))257 1117 y(and)1086 1237 y Fh(\015)1086 1287 y(\015)1086 1337 y(\015)1146 1312 y Fo(~)1132 1337 y(G)1209 1301 y Fm(n)1209 1362 y(n)p Fl(+1)1346 1237 y Fh(\015)1346 1287 y(\015)1346 1337 y(\015)1392 1264 y Fl(2)1491 1337 y Fn(\024)1639 1298 y Fl(3)12 b Fm(\013)1731 1274 y Fe(3)p 1639 1314 127 4 v 1640 1371 a Fl(2)g Fm(\031)1730 1352 y Fe(2)1793 1266 y Fh(R)1848 1292 y Fm(\033)1888 1300 y Fa(n)1832 1362 y Fl(0)1952 1337 y Fo(k)19 b(dk)63 b Fn(\024)2335 1298 y Fl(1)p 2335 1314 36 4 v 2335 1371 a(5)2397 1337 y Fo(\033)2456 1301 y Fl(2)2452 1362 y Fm(n)2516 1337 y Fo(\013)2579 1301 y Fl(3)2640 1337 y Fo(;)555 b Fq(\(I)s(I.36\))998 1436 y Fh(\015)998 1486 y(\015)998 1536 y(\015)p Fo(!)1109 1500 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)1304 1511 y Fo(~)1290 1536 y(G)1367 1500 y Fm(n)1367 1561 y(n)p Fl(+1)1504 1436 y Fh(\015)1504 1486 y(\015)1504 1536 y(\015)1550 1463 y Fl(2)1650 1536 y Fn(\024)1797 1497 y Fl(3)12 b Fm(\013)1889 1473 y Fe(3)p 1797 1513 127 4 v 1798 1570 a Fl(2)h Fm(\031)1889 1551 y Fe(2)1951 1465 y Fh(R)2006 1491 y Fm(\033)2046 1499 y Fa(n)1990 1561 y Fl(0)2110 1536 y Fo(dk)63 b Fn(\024)2422 1497 y Fl(1)p 2422 1513 36 4 v 2422 1570 a(5)2484 1536 y Fo(\033)2539 1551 y Fm(n)2603 1536 y Fo(\013)2666 1500 y Fl(3)2728 1536 y Fo(:)467 b Fq(\(I)s(I.37\))257 1773 y Fc(Lemma)49 b(I)s(I.2.)e Fj(L)-5 b(et)45 b Fo(f)5 b(;)17 b(g)48 b Fn(2)e Fo(L)1466 1737 y Fl(2)1506 1773 y Fq([)p Fk(R)1599 1737 y Fl(3)1673 1773 y Fn(\002)30 b Fk(Z)1849 1788 y Fl(2)1886 1773 y Fq(;)17 b Fn(B)s Fq(\()p Fn(H)2120 1788 y Fm(el)2179 1773 y Fq(\)])44 b Fj(b)-5 b(e)44 b(op)-5 b(er)g(ator-value)g(d)44 b(functions)257 1893 y(ob)-5 b(eying)35 b Fn(k)p Fq(\(1)21 b(+)h Fo(!)925 1857 y Fi(\000)p Fl(1)1019 1893 y Fq(\))1057 1857 y Fl(1)p Fm(=)p Fl(2)1167 1893 y Fo(f)11 b Fn(k)p Fo(;)17 b Fn(k)p Fq(\(1)k(+)h Fo(!)1641 1857 y Fi(\000)p Fl(1)1734 1893 y Fq(\))1772 1857 y Fl(1)p Fm(=)p Fl(2)1882 1893 y Fo(g)t Fn(k)27 b Fo(<)h Fn(1)p Fj(.)44 b(Then,)34 b(for)h(any)f Fo(\032)28 b(>)g Fq(0)587 2026 y Fh(\015)587 2076 y(\015)587 2126 y(\015)p Fo(a)p Fq(\()p Fo(f)11 b Fq(\))17 b(\()899 2100 y(\024)874 2126 y Fo(H)29 b Fq(+)22 b Fo(\032)p Fq(\))1170 2084 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)1336 2026 y Fh(\015)1336 2076 y(\015)1336 2126 y(\015)83 b Fn(\024)g(k)p Fo(!)1740 2084 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)1904 2126 y Fo(f)11 b Fn(k)1209 b Fq(\(I)s(I.38\))587 2209 y Fh(\015)587 2258 y(\015)587 2308 y(\015)p Fq(\()697 2283 y(\024)671 2308 y Fo(H)30 b Fq(+)22 b Fo(\032)p Fq(\))968 2267 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)1150 2308 y Fo(a)p Fq(\()p Fo(f)11 b Fq(\))1336 2209 y Fh(\015)1336 2258 y(\015)1336 2308 y(\015)83 b Fn(\024)g(k)p Fq(\()p Fo(\032)1763 2267 y Fi(\000)p Fl(1)1880 2308 y Fq(+)22 b Fo(!)2043 2267 y Fi(\000)p Fl(1)2136 2308 y Fq(\))2174 2267 y Fl(1)p Fm(=)p Fl(2)2284 2308 y Fo(f)11 b Fn(k)22 b Fo(;)780 b Fq(\(I)s(I.39\))464 2391 y Fh(\015)464 2441 y(\015)464 2491 y(\015)p Fo(a)p Fq(\()p Fo(f)11 b Fq(\))17 b Fo(a)p Fq(\()p Fo(g)t Fq(\))g(\()970 2466 y(\024)946 2491 y Fo(H)28 b Fq(+)22 b Fo(\032)p Fq(\))1241 2450 y Fi(\000)p Fl(1)1336 2391 y Fh(\015)1336 2441 y(\015)1336 2491 y(\015)83 b Fn(\024)g(k)p Fo(!)1740 2450 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)1904 2491 y Fo(f)11 b Fn(k)17 b(k)p Fo(!)2145 2450 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)2308 2491 y Fo(g)t Fn(k)22 b Fo(;)764 b Fq(\(I)s(I.40\))464 2574 y Fh(\015)464 2624 y(\015)464 2674 y(\015)p Fq(\()573 2648 y(\024)548 2674 y Fo(H)30 b Fq(+)22 b Fo(\032)p Fq(\))845 2632 y Fi(\000)p Fl(1)956 2674 y Fo(a)p Fq(\()p Fo(g)t Fq(\))17 b Fo(a)p Fq(\()p Fo(f)11 b Fq(\))1337 2574 y Fh(\015)1337 2624 y(\015)1337 2674 y(\015)82 b Fn(\024)h(k)p Fq(\()p Fo(\032)1763 2632 y Fi(\000)p Fl(1)1880 2674 y Fq(+)22 b Fo(!)2043 2632 y Fi(\000)p Fl(1)2136 2674 y Fq(\))2174 2632 y Fl(1)p Fm(=)p Fl(2)2284 2674 y Fo(f)11 b Fn(k)17 b(k)p Fq(\()p Fo(\032)2548 2632 y Fi(\000)p Fl(1)2664 2674 y Fq(+)22 b Fo(!)2827 2632 y Fi(\000)p Fl(1)2921 2674 y Fq(\))2959 2632 y Fl(1)p Fm(=)p Fl(2)3069 2674 y Fo(g)t Fn(k)f Fo(:)t Fq(\(I)s(I.41\))257 2906 y(The)44 b(pro)s(of)d(of)h(Lemma)f(I)s(I.2)h (can)h(b)s(e)g(found)f(in)g(man)m(y)g(texts,)k(e.g.,)f(in)d([1,)g(2].) 73 b(W)-8 b(e)257 3026 y(remark)39 b(that)g(the)g(relativ)m(e)f(b)s (ounds)i(asserted)g(in)e(Lemma)f(I)s(I.2)i(and)g(\(I)s(I.1\))g(are)g (the)257 3147 y(basic)28 b(input)g(to)f(pro)m(v)m(e)i(the)g(selfadjoin) m(tness)f(of)g Fo(H)35 b Fq(=)23 b Fo(~)-45 b(v)2316 3111 y Fl(2)2369 3147 y Fn(\000)13 b Fo(V)22 b Fq(\()p Fo(x)p Fq(\))13 b(+)2796 3122 y(\024)2771 3147 y Fo(H)35 b Fq(on)28 b(the)g(domain)257 3267 y(of)k Fn(\000)p Fq(\001)23 b(+)673 3242 y(\024)647 3267 y Fo(H)8 b Fq(,)33 b(pro)m(vided)g Fo(\013)g Fq(is)f(not)g(to)s(o)g(large.)257 3471 y Fc(Lemma)37 b(I)s(I.3.)42 b Fj(F)-7 b(or)34 b(al)5 b(l)34 b Fq(0)28 b Fn(\024)g Fo(\013)g Fn(\024)g Fo(\024)1655 3434 y Fi(\000)p Fl(1)p Fm(=)p Fl(3)1820 3471 y Fj(,)35 b(al)5 b(l)35 b Fo(n)28 b Fn(2)g Fk(N)2271 3486 y Fl(0)2316 3471 y Fj(,)35 b(and)f(any)h Fo(\032)28 b Fn(2)g Fq(\(0)p Fo(;)17 b(\033)3115 3486 y Fm(n)3162 3471 y Fq(])p Fj(,)836 3603 y Fh(\015)836 3653 y(\015)836 3703 y(\015)p Fq(\()p Fo(H)1009 3667 y Fl(+)1001 3728 y Fm(n)1089 3703 y Fq(+)22 b Fo(\032)p Fq(\))1275 3667 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)1457 3703 y Fq(\()l Fo(~)-45 b(v)1542 3718 y Fm(n)1589 3703 y Fq(\))1627 3667 y Fl(2)1683 3703 y Fq(\()p Fo(H)1810 3667 y Fl(+)1802 3728 y Fm(n)1891 3703 y Fq(+)22 b Fo(\032)p Fq(\))2077 3667 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)2242 3603 y Fh(\015)2242 3653 y(\015)2242 3703 y(\015)2288 3757 y Fi(H)2348 3765 y Fa(n)2457 3703 y Fn(\024)63 b Fo(C)2667 3718 y Fl(1)2723 3703 y Fo(\032)2773 3667 y Fi(\000)p Fl(1)2890 3703 y Fo(;)305 b Fq(\(I)s(I.42\))1147 3798 y Fh(\015)1147 3848 y(\015)1147 3898 y(\015)1216 3866 y(e)1193 3898 y Fo(R)1268 3862 y Fl(1)p Fm(=)p Fl(2)1267 3923 y Fm(n)1395 3898 y Fq(\()l Fo(~)-45 b(v)1480 3913 y Fm(n)1527 3898 y Fq(\))1565 3862 y Fl(2)1643 3866 y Fh(e)1621 3898 y Fo(R)1696 3862 y Fl(1)p Fm(=)p Fl(2)1695 3923 y Fm(n)1806 3798 y Fh(\015)1806 3848 y(\015)1806 3898 y(\015)1852 3952 y Fi(H)1912 3961 y Fa(n)p Fe(+1)2098 3898 y Fn(\024)63 b Fo(C)2308 3913 y Fl(1)2364 3898 y Fo(\033)2423 3857 y Fi(\000)p Fl(1)2419 3920 y Fm(n)p Fl(+1)2579 3898 y Fo(;)616 b Fq(\(I)s(I.43\))257 4151 y Fj(wher)-5 b(e)35 b Fo(C)603 4166 y Fl(1)677 4151 y Fj(is)f(de\014ne)-5 b(d)34 b(in)h(\(II.30\).)404 4354 y Fq(The)c(pro)s(of)f(of)h(Lemma)e(I)s(I.3)i(is)f(giv)m(en)h(in)f(App)s (endix)h(A.)43 b(In)31 b(the)g(next)h(lemma,)d(w)m(e)257 4475 y(deriv)m(e)f(a)f(quadratic)f(form)g(b)s(ound)h(on)g(the)h(in)m (teraction)e Fo(W)2449 4438 y Fm(n)2435 4499 y(n)p Fl(+1)2599 4475 y Fq(relativ)m(e)g(to)h(the)g(energy)257 4609 y(scale)36 b Fo(\033)549 4624 y Fm(n)p Fl(+1)721 4609 y Fq(whic)m(h)g(pro)m(v)m (es)h(that)e Fo(W)1627 4573 y Fm(n)1613 4634 y(n)p Fl(+1)1785 4609 y Fq(is)g(a)g(small)e(p)s(erturbation)h(of)2924 4577 y Fh(f)2916 4609 y Fo(H)3005 4573 y Fl(+)2997 4634 y Fm(n)3064 4609 y Fq(,)i(pro)m(vided)257 4750 y Fo(g)308 4699 y Fl(\(1\))304 4775 y Fm(\013;\014)444 4750 y Fq(:=)27 b Fo(C)644 4765 y Fl(1)684 4750 y Fo(\013)747 4714 y Fl(3)p Fm(=)p Fl(2)856 4750 y Fo(\014)917 4714 y Fl(1)p Fm(=)p Fl(2)1060 4750 y Fq(is)32 b(su\016cien)m(tly)h(small.)257 4953 y Fc(Lemma)k(I)s(I.4.)42 b Fj(Assume)35 b(Hyp)-5 b(othesis)35 b(H-1.)44 b(Then,)34 b(for)h(al)5 b(l)34 b Fo(n)28 b Fn(2)g Fk(N)2807 4968 y Fl(0)2853 4953 y Fj(,)1217 5073 y Fh(\015)1217 5123 y(\015)1217 5173 y(\015)1285 5141 y(e)1263 5173 y Fo(R)1338 5132 y Fl(1)p Fm(=)p Fl(2)1337 5198 y Fm(n)1465 5173 y Fo(W)1571 5132 y Fm(n)1557 5198 y(n)p Fl(+1)1733 5141 y Fh(e)1710 5173 y Fo(R)1785 5132 y Fl(1)p Fm(=)p Fl(2)1784 5198 y Fm(n)1895 5073 y Fh(\015)1895 5123 y(\015)1895 5173 y(\015)1942 5227 y Fi(H)2002 5236 y Fa(n)p Fe(+1)2188 5173 y Fn(\024)63 b Fo(g)2379 5122 y Fl(\(1\))2375 5198 y Fm(\013;\014)2509 5173 y Fo(:)686 b Fq(\(I)s(I.44\))p eop %%Page: 12 13 12 12 bop 257 266 a Fr(BFP-GrSt,)32 b(04-F)-8 b(eb-2005)2130 b Fq(12)257 573 y Fj(Pr)-5 b(o)g(of.)43 b Fq(By)33 b(Eq.)h(\(I)s (I.28\))e(and)h(Lemmata)e(I)s(I.2)h(and)h(I)s(I.3,)f(w)m(e)i(ha)m(v)m (e)636 706 y Fh(\015)636 756 y(\015)636 805 y(\015)705 773 y(e)682 805 y Fo(R)757 764 y Fl(1)p Fm(=)p Fl(2)756 830 y Fm(n)884 805 y Fo(W)990 764 y Fm(n)976 830 y(n)p Fl(+1)1152 773 y Fh(e)1130 805 y Fo(R)1205 764 y Fl(1)p Fm(=)p Fl(2)1204 830 y Fm(n)1315 706 y Fh(\015)1315 756 y(\015)1315 805 y(\015)1361 859 y Fi(H)1421 868 y Fa(n)p Fe(+1)719 1008 y Fn(\024)84 b Fq(4)946 909 y Fh(\015)945 958 y(\015)945 1008 y(\015)1014 976 y(e)991 1008 y Fo(R)1066 967 y Fl(1)p Fm(=)p Fl(2)1065 1033 y Fm(n)1172 1008 y Fo(~)-45 b(v)1223 1023 y Fm(n)1270 909 y Fh(\015)1270 958 y(\015)1270 1008 y(\015)1317 1062 y Fi(H)1377 1071 y Fa(n)p Fe(+1)1522 1008 y Fn(\001)1572 909 y Fh(\015)1572 958 y(\015)1572 1008 y(\015)p Fo(a)p Fq(\()1722 983 y Fo(~)1707 1008 y(G)1784 967 y Fm(n)1784 1033 y(n)p Fl(+1)1922 1008 y Fq(\))1999 976 y Fh(e)1976 1008 y Fo(R)2051 967 y Fl(1)p Fm(=)p Fl(2)2050 1033 y Fm(n)2161 909 y Fh(\015)2161 958 y(\015)2161 1008 y(\015)2207 1062 y Fi(H)2267 1071 y Fa(n)p Fe(+1)1026 1228 y Fq(+)1129 1128 y Fh(\015)1129 1178 y(\015)1129 1228 y(\015)1175 1131 y(\020)1225 1228 y Fo(a)1276 1187 y Fi(\003)1315 1228 y Fq(\()1368 1203 y Fo(~)1353 1228 y(G)1430 1187 y Fm(n)1430 1252 y(n)p Fl(+1)1567 1228 y Fq(\))23 b(+)f Fo(a)p Fq(\()1830 1203 y Fo(~)1815 1228 y(G)1892 1187 y Fm(n)1892 1252 y(n)p Fl(+1)2029 1228 y Fq(\))2067 1131 y Fh(\021)2156 1195 y(e)2133 1228 y Fo(R)2208 1187 y Fl(1)p Fm(=)p Fl(2)2207 1252 y Fm(n)2318 1128 y Fh(\015)2318 1178 y(\015)2318 1228 y(\015)2364 1154 y Fl(2)2364 1282 y Fi(H)2424 1291 y Fa(n)p Fe(+1)719 1499 y Fn(\024)890 1432 y Fq(4)17 b Fo(C)1033 1381 y Fl(1)p Fm(=)p Fl(2)1026 1454 y(1)p 890 1476 253 4 v 920 1587 a Fo(\033)979 1536 y Fl(1)p Fm(=)p Fl(2)975 1608 y Fm(n)p Fl(+1)1169 1499 y Fn(k)p Fo(!)1284 1458 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)1462 1474 y Fo(~)1448 1499 y(G)1525 1458 y Fm(n)1525 1524 y(n)p Fl(+1)1662 1499 y Fn(k)44 b Fq(+)g(4)17 b Fn(k)p Fq(\()p Fo(\033)2089 1458 y Fi(\000)p Fl(1)2085 1524 y Fm(n)p Fl(+1)2244 1499 y Fq(+)22 b Fo(!)2407 1458 y Fi(\000)p Fl(1)2501 1499 y Fq(\))2539 1458 y Fl(1)p Fm(=)p Fl(2)2680 1474 y Fo(~)2665 1499 y(G)2742 1458 y Fm(n)2742 1524 y(n)p Fl(+1)2879 1499 y Fn(k)2929 1458 y Fl(2)719 1817 y Fn(\024)890 1749 y Fq(4)17 b Fo(C)1033 1698 y Fl(1)p Fm(=)p Fl(2)1026 1771 y(1)1159 1749 y Fo(\013)1222 1713 y Fl(3)p Fm(=)p Fl(2)1348 1749 y Fo(\033)1407 1713 y Fl(1)p Fm(=)p Fl(2)1403 1774 y Fm(n)p 890 1793 628 4 v 1033 1822 a Fn(p)p 1116 1822 49 4 v 82 x Fq(5)f Fo(\033)1240 1853 y Fl(1)p Fm(=)p Fl(2)1236 1926 y Fm(n)p Fl(+1)1571 1817 y Fq(+)1701 1749 y(8)h Fo(\013)1830 1713 y Fl(3)1885 1749 y Fo(\033)1944 1713 y Fl(2)1940 1774 y Fm(n)p 1701 1793 287 4 v 1716 1885 a Fq(5)g Fo(\033)1837 1900 y Fm(n)p Fl(+1)2086 1817 y Fn(\024)2251 1695 y Fh(\022)2322 1749 y Fq(4)g Fo(C)2465 1698 y Fl(1)p Fm(=)p Fl(2)2458 1771 y(1)p 2322 1793 253 4 v 2382 1811 a Fn(p)p 2465 1811 49 4 v 82 x Fq(5)2607 1817 y(+)2715 1749 y(8)g Fo(\024)p 2715 1793 122 4 v 2751 1885 a Fq(5)2846 1695 y Fh(\023)2934 1749 y Fo(\013)2997 1713 y Fl(3)p Fm(=)p Fl(2)p 2934 1793 173 4 v 2935 1885 a Fo(\014)2996 1856 y Fl(1)p Fm(=)p Fl(2)719 2119 y Fn(\024)880 1998 y Fh(\022)1047 2052 y Fq(4)p 951 2096 242 4 v 951 2114 a Fn(p)p 1034 2114 159 4 v 76 x Fq(5)p Fo(C)1153 2205 y Fl(1)1224 2119 y Fq(+)1332 2052 y(1)p 1332 2096 49 4 v 1332 2188 a(5)1391 1998 y Fh(\023)1478 2052 y Fo(C)1548 2067 y Fl(1)1604 2052 y Fo(\013)1667 2016 y Fl(3)p Fm(=)p Fl(2)p 1478 2096 299 4 v 1542 2188 a Fo(\014)1603 2159 y Fl(1)p Fm(=)p Fl(2)1875 2119 y Fn(\024)2050 2052 y Fo(C)2120 2067 y Fl(1)2176 2052 y Fo(\013)2239 2016 y Fl(3)p Fm(=)p Fl(2)p 2050 2096 V 2114 2188 a Fo(\014)2175 2159 y Fl(1)p Fm(=)p Fl(2)2381 2119 y Fo(;)814 b Fq(\(I)s(I.45\))257 2392 y(where)47 b(w)m(e)g(use)g(Estimates)e(\(I)s(I.36\))g(and)h(\(I)s (I.37\),)j Fo(\024\014)56 b Fn(\024)50 b Fq(1,)f(and)d Fo(C)2920 2407 y Fl(1)3009 2392 y Fn(\025)51 b Fq(8)p Fo(\024)f Fn(\025)g Fq(8.)p 3425 2513 4 66 v 3429 2450 59 4 v 3429 2513 V 3487 2513 4 66 v 257 2836 a Fc(Corollary)39 b(I)s(I.5.)j Fj(Assume)35 b(Hyp)-5 b(othesis)36 b(H-1)g(and)f Fo(g)2290 2786 y Fl(\(1\))2286 2862 y Fm(\013;\014)2427 2836 y Fq(=)29 b Fo(C)2602 2851 y Fl(1)2641 2836 y Fo(\013)2704 2800 y Fl(3)p Fm(=)p Fl(2)2814 2836 y Fo(\014)2875 2800 y Fl(1)p Fm(=)p Fl(2)3014 2836 y Fn(\024)g Fq(1)p Fj(.)47 b(Then,)257 2957 y(for)35 b(al)5 b(l)35 b Fo(n)27 b Fn(2)i Fk(N)798 2972 y Fl(0)844 2957 y Fj(,)970 3177 y Fo(H)1051 3192 y Fm(n)p Fl(+1)1250 3177 y Fn(\025)63 b Fo(E)1462 3192 y Fm(n)1548 3177 y Fn(\000)40 b Fo(g)1716 3126 y Fl(\(1\))1712 3202 y Fm(\013;\014)1840 3177 y Fo(\033)1895 3192 y Fm(n)p Fl(+1)2071 3177 y Fq(+)f([1)22 b Fn(\000)g Fo(g)2434 3126 y Fl(\(1\))2430 3202 y Fm(\013;\014)2542 3177 y Fq(])2594 3145 y Fh(f)2586 3177 y Fo(H)2675 3136 y Fl(+)2667 3202 y Fm(n)2756 3177 y Fo(;)439 b Fq(\(I)s(I.46\))257 3424 y Fj(wher)-5 b(e)35 b Fo(g)584 3373 y Fl(\(1\))580 3449 y Fm(\013;\014)719 3424 y Fn(\025)28 b Fq(0)35 b Fj(is)f(de\014ne)-5 b(d)34 b(in)h(\(II.44\).)257 3627 y(Pr)-5 b(o)g(of.)43 b Fq(The)34 b(assertion)e(follo)m(ws)g(directly)g (from)f(Lemma)g(I)s(I.4:)257 3847 y Fo(H)338 3862 y Fm(n)p Fl(+1)559 3847 y Fq(=)84 b Fo(E)791 3862 y Fm(n)877 3847 y Fq(+)1000 3815 y Fh(f)991 3847 y Fo(H)1080 3806 y Fl(+)1072 3872 y Fm(n)1178 3847 y Fq(+)38 b Fo(W)1398 3806 y Fm(n)1384 3872 y(n)p Fl(+1)559 4011 y Fq(=)84 b Fo(E)791 4026 y Fm(n)877 4011 y Fn(\000)39 b Fo(\033)1048 4026 y Fm(n)p Fl(+1)1224 4011 y Fq(+)g(\()1385 3979 y Fh(f)1377 4011 y Fo(H)1466 3970 y Fl(+)1458 4036 y Fm(n)1547 4011 y Fq(+)22 b Fo(\033)1700 4026 y Fm(n)p Fl(+1)1837 4011 y Fq(\))1875 3970 y Fl(1)p Fm(=)p Fl(2)1985 3915 y Fh(h)2024 4011 y Fc(1)39 b Fq(+)2256 3979 y Fh(e)2234 4011 y Fo(R)2309 3970 y Fl(1)p Fm(=)p Fl(2)2308 4036 y Fm(n)2419 4011 y Fo(W)2525 3970 y Fm(n)2511 4036 y(n)p Fl(+1)2671 3979 y Fh(e)2648 4011 y Fo(R)2723 3970 y Fl(1)p Fm(=)p Fl(2)2722 4036 y Fm(n)2833 3915 y Fh(i)2872 4011 y Fq(\()2918 3979 y Fh(f)2910 4011 y Fo(H)2999 3970 y Fl(+)2991 4036 y Fm(n)3080 4011 y Fq(+)22 b Fo(\033)3233 4026 y Fm(n)p Fl(+1)3371 4011 y Fq(\))3409 3970 y Fl(1)p Fm(=)p Fl(2)558 4195 y Fn(\025)84 b Fo(E)791 4210 y Fm(n)877 4195 y Fn(\000)39 b Fo(\033)1048 4210 y Fm(n)p Fl(+1)1224 4195 y Fq(+)g([1)22 b Fn(\000)h Fo(g)1588 4144 y Fl(\(1\))1584 4220 y Fm(\013;\014)1695 4195 y Fq(])17 b(\()1785 4163 y Fh(f)1777 4195 y Fo(H)1866 4154 y Fl(+)1858 4220 y Fm(n)1947 4195 y Fq(+)22 b Fo(\033)2100 4210 y Fm(n)p Fl(+1)2237 4195 y Fq(\))559 4367 y(=)84 b Fo(E)791 4382 y Fm(n)877 4367 y Fn(\000)39 b Fo(g)1044 4316 y Fl(\(1\))1040 4392 y Fm(\013;\014)1168 4367 y Fo(\033)1223 4382 y Fm(n)p Fl(+1)1400 4367 y Fq(+)f([1)22 b Fn(\000)h Fo(g)1763 4316 y Fl(\(1\))1759 4392 y Fm(\013;\014)1871 4367 y Fq(])1923 4335 y Fh(f)1915 4367 y Fo(H)2004 4326 y Fl(+)1996 4392 y Fm(n)2084 4367 y Fo(:)1134 b Fq(\(I)s(I.47\))p 3425 4587 V 3429 4525 59 4 v 3429 4587 V 3487 4587 4 66 v 257 4996 a Fb(I)t(I.3)135 b(Energy)46 b(Shifts)f(and)f(Gaps)257 5181 y Fq(In)k(this)f(subsection)i(w)m(e)g(estimate)d(the)i (di\013erences)h Fo(E)2402 5145 y Fm(n)2396 5206 y(n)p Fl(+1)2586 5181 y Fq(=)54 b Fo(E)2788 5196 y Fm(n)2867 5181 y Fn(\000)33 b Fo(E)3049 5196 y Fm(n)p Fl(+1)3234 5181 y Fq(of)47 b(the)257 5301 y(groundstate)31 b(energies)h(at)e (scales)h Fo(n)g Fq(and)f Fo(n)19 b Fq(+)f(1,)30 b(see)i(\(I)s(I.21\).) 43 b(W)-8 b(e)31 b(inductiv)m(ely)f(pro)m(v)m(e)p eop %%Page: 13 14 13 13 bop 257 266 a Fr(BFP-GrSt,)32 b(04-F)-8 b(eb-2005)2130 b Fq(13)257 573 y(that)35 b Fo(E)543 588 y Fm(n)622 573 y Fq(=)c(inf)23 b Fo(\033)t Fq(\()p Fo(H)1042 588 y Fm(n)1089 573 y Fq(\))35 b(is)f(an)h(isolated)e(eigen)m(v)-5 b(alue)35 b(of)f(m)m(ultiplicit)m(y)d(one)36 b(and)f(deriv)m(e)257 693 y(a)e(lo)m(w)m(er)f(b)s(ound)h(on)g(the)g(size)f(of)h(the)g(gaps) 406 879 y(gap)558 903 y Fm(n)665 879 y Fq(:=)60 b(inf)963 783 y Fh(h)1002 879 y Fo(\033)t Fq(\()p Fo(H)1188 838 y Fl(+)1180 904 y Fm(n)1247 879 y Fq(\))22 b Fn(n)g(f)p Fq(0)p Fn(g)1528 783 y Fh(i)1661 879 y Fq(and)1928 872 y Fh(g)1913 879 y Fq(gap)2064 903 y Fm(n)2171 879 y Fq(:=)61 b(inf)2469 783 y Fh(h)2508 879 y Fo(\033)t Fq(\()2613 847 y Fh(f)2605 879 y Fo(H)2694 838 y Fl(+)2686 904 y Fm(n)2753 879 y Fq(\))22 b Fn(n)g(f)p Fq(0)p Fn(g)3034 783 y Fh(i)3222 879 y Fq(\(I)s(I.48\))257 1097 y(of)32 b Fo(H)457 1061 y Fl(+)449 1122 y Fm(n)549 1097 y Fq(and)746 1065 y Fh(f)738 1097 y Fo(H)827 1061 y Fl(+)819 1122 y Fm(n)918 1097 y Fq(ab)s(o)m(v)m(e)i(their)e(lo)m(w)m(est)h(sp)s (ectral)f(v)-5 b(alue)32 b(0.)404 1218 y(W)-8 b(e)41 b(\014rst)g(deriv)m(e)h(a)f(simple)e(estimate)h(on)h(the)g (di\013erence)h Fo(E)2729 1182 y Fm(n)2723 1243 y(n)p Fl(+1)2901 1218 y Fq(of)f(the)g(ground)257 1338 y(state)33 b(energies)g(of)g Fo(H)1058 1353 y Fm(n)1137 1338 y Fq(and)g Fo(H)1408 1353 y Fm(n)p Fl(+1)1544 1338 y Fq(.)257 1523 y Fc(Lemma)46 b(I)s(I.6.)g Fj(Assume)c(Hyp)-5 b(othesis)42 b(H-1)g(and)f Fo(g)2224 1472 y Fl(\(1\))2220 1548 y Fm(\013;\014)2373 1523 y Fq(=)g Fo(C)2560 1538 y Fl(1)2599 1523 y Fo(\013)2662 1487 y Fl(3)p Fm(=)p Fl(2)2771 1523 y Fo(\014)2832 1487 y Fl(1)p Fm(=)p Fl(2)2983 1523 y Fn(\024)g Fq(1)p Fj(.)66 b(Then,)257 1643 y(for)35 b(al)5 b(l)35 b Fo(n)27 b Fn(2)i Fk(N)798 1658 y Fl(0)844 1643 y Fj(,)1434 1764 y Fn(j)p Fo(E)1540 1723 y Fm(n)1534 1788 y(n)p Fl(+1)1671 1764 y Fn(j)62 b(\024)h Fo(g)1952 1713 y Fl(\(1\))1948 1789 y Fm(\013;\014)2077 1764 y Fo(\033)2132 1779 y Fm(n)p Fl(+1)2291 1764 y Fo(;)904 b Fq(\(I)s(I.49\))257 1950 y Fj(wher)-5 b(e)35 b Fo(g)584 1899 y Fl(\(1\))580 1975 y Fm(\013;\014)719 1950 y Fn(\025)28 b Fq(0)35 b Fj(is)f(de\014ne)-5 b(d)34 b(in)h(\(II.44\).)257 2124 y(Pr)-5 b(o)g(of.)46 b Fq(Giv)m(en)33 b Fo(")c(>)g Fq(0,)34 b(supp)s(ose)h(that)e Fo( )1776 2139 y Fm(n)1852 2124 y Fn(2)d(H)2032 2139 y Fm(n)2113 2124 y Fq(is)j(an)g Fo(")p Fq(-appro)m(ximate)f (groundstate)257 2245 y(of)i Fo(H)451 2260 y Fm(n)498 2245 y Fq(,)g(i.e.,)g Fn(h)p Fo( )847 2260 y Fm(n)894 2245 y Fn(j)p Fo(H)1003 2260 y Fm(n)1066 2245 y Fo( )1129 2260 y Fm(n)1176 2245 y Fn(i)c(\024)g Fo(E)1424 2260 y Fm(n)1494 2245 y Fq(+)23 b Fo(")p Fq(.)47 b(The)35 b(v)-5 b(ariational)30 b(principle)j(yields)g(the)i(upp)s(er)257 2365 y(b)s(ound)257 2551 y Fo(E)329 2566 y Fm(n)p Fl(+1)550 2551 y Fn(\024)83 b(h)769 2525 y Fq(~)749 2551 y Fo( )812 2566 y Fm(n)882 2551 y Fn(j)27 b Fo(H)1018 2566 y Fm(n)p Fl(+1)1202 2525 y Fq(~)1183 2551 y Fo( )1246 2566 y Fm(n)1293 2551 y Fn(i)60 b Fq(=)g Fo(E)1600 2566 y Fm(n)1686 2551 y Fq(+)38 b Fo(")h Fq(+)f Fn(h)2058 2525 y Fq(~)2038 2551 y Fo( )2101 2566 y Fm(n)2171 2551 y Fn(j)27 b Fo(W)2332 2510 y Fm(n)2318 2576 y(n)p Fl(+1)2503 2525 y Fq(~)2483 2551 y Fo( )2546 2566 y Fm(n)2593 2551 y Fn(i)621 b Fq(\(I)s(I.50\))551 2747 y(=)83 b Fo(E)782 2762 y Fm(n)868 2747 y Fq(+)39 b Fo(")f Fq(+)h Fn(k)1247 2722 y Fo(~)1232 2747 y(G)1309 2706 y Fm(n)1309 2772 y(n)p Fl(+1)1446 2747 y Fn(k)1496 2706 y Fl(2)1595 2747 y Fn(\024)61 b Fo(E)1805 2762 y Fm(n)1891 2747 y Fq(+)38 b Fo(")h Fq(+)2214 2680 y(1)p 2214 2724 49 4 v 2214 2816 a(5)2290 2747 y Fo(\013)2353 2706 y Fl(3)2408 2747 y Fo(\033)2467 2706 y Fl(2)2463 2772 y Fm(n)2571 2747 y Fn(\024)61 b Fo(E)2781 2762 y Fm(n)2867 2747 y Fq(+)38 b Fo(")h Fq(+)3180 2651 y Fh(\020)3230 2747 y Fo(g)3281 2697 y Fl(\(1\))3277 2773 y Fm(\013;\014)3389 2651 y Fh(\021)3438 2674 y Fl(2)3500 2747 y Fo(;)257 2969 y Fq(where)c(w)m(e)f(use)g(that)e Fo(C)1134 2984 y Fl(1)1202 2969 y Fn(\025)d Fo(\024)p Fq(,)k Fo(\014)1485 2933 y Fm(n)1560 2969 y Fn(\024)c Fq(1,)k(and)g Fo(\013)2028 2933 y Fl(3)2096 2969 y Fn(\024)c Fo(\014)38 b Fq(to)33 b(deriv)m(e)h(the)f(last)f(inequalit)m(y)-8 b(.)257 3089 y(In)34 b(the)g(limit)c Fo(")f Fn(!)f Fq(0,)34 b(w)m(e)g(obtain)e Fo(E)1616 3104 y Fm(n)p Fl(+1)1783 3089 y Fn(\024)d Fo(E)1961 3104 y Fm(n)2031 3089 y Fq(+)23 b(\()p Fo(g)2219 3038 y Fl(\(1\))2215 3114 y Fm(\013;\014)2327 3089 y Fq(\))2365 3053 y Fl(2)2437 3089 y Fq(from)33 b(\(I)s(I.50\).)45 b(Con)m(v)m(ersely)-8 b(,)257 3230 y(Eq.)34 b(\(I)s(I.46\))e(and)939 3198 y Fh(f)931 3230 y Fo(H)1020 3194 y Fl(+)1012 3255 y Fm(n)1106 3230 y Fn(\025)c Fq(0)k(imply)f(that)1327 3429 y Fo(E)1399 3444 y Fm(n)p Fl(+1)1597 3429 y Fn(\025)60 b Fo(E)1806 3444 y Fm(n)1892 3429 y Fn(\000)39 b Fo(g)2059 3379 y Fl(\(1\))2055 3455 y Fm(\013;\014)2184 3429 y Fo(\033)2239 3444 y Fm(n)p Fl(+1)2399 3429 y Fo(:)796 b Fq(\(I)s(I.51\))p 3425 3615 4 66 v 3429 3553 59 4 v 3429 3615 V 3487 3615 4 66 v 404 3856 a(In)25 b(the)g(next)h(lemma,)e (w)m(e)i(pro)m(v)m(e)h(that)d Fo(E)1892 3871 y Fm(n)1967 3856 y Fq(=)k(inf)22 b Fo(\033)t Fq(\()p Fo(H)2383 3871 y Fm(n)2430 3856 y Fq(\))j(is)f(an)h(isolated)f(eigen)m(v)-5 b(alue)257 3977 y(of)34 b(m)m(ultiplicit)m(y)d(one)k(and)f(deriv)m(e)h (a)g(lo)m(w)m(er)f(b)s(ound)h(on)f(the)h(size)g(of)e(the)i(sp)s(ectral) f(gap)257 4097 y(ab)s(o)m(v)m(e)46 b Fo(E)618 4112 y Fm(n)666 4097 y Fq(.)81 b(W)-8 b(e)46 b(use)g(a)f(v)-5 b(ariational)42 b(argumen)m(t)j(whic)m(h)h(w)m(ould)f(not)g(apply)g(in) g(the)257 4217 y(degenerate)31 b(case.)43 b(That)30 b(could)f(not)g(b)s (e)h(exp)s(ected,)i(either,)e(b)s(ecause)g(the)g(in)m(teraction)257 4338 y(with)42 b(the)h(quan)m(tized)g(radiation)e(\014eld)h(could)g (lift)e(the)j(degeneracy)h(at)e(an)m(y)h(energy)257 4458 y(scale,)33 b(and)g(there)g(is,)f(a)g(priori,)f(no)i(con)m(trol)f(on)g (the)h(resulting)f(splitting.)257 4632 y Fc(Lemma)39 b(I)s(I.7.)j Fj(Assume)36 b(Hyp)-5 b(othesis)36 b(H-1)g(and)f Fo(g)2189 4581 y Fl(\(1\))2185 4657 y Fm(\013;\014)2327 4632 y Fq(=)29 b Fo(C)2502 4647 y Fl(1)2542 4632 y Fo(\013)2605 4596 y Fl(3)p Fm(=)p Fl(2)2714 4632 y Fo(\014)2775 4596 y Fl(1)p Fm(=)p Fl(2)2915 4632 y Fn(\024)h Fq(1)p Fo(=)p Fq(6)p Fj(.)47 b(Then,)257 4753 y(for)39 b(al)5 b(l)39 b Fo(n)c Fn(2)h Fk(N)822 4768 y Fl(0)867 4753 y Fj(,)k(the)f(gr)-5 b(oundstate)39 b(ener)-5 b(gy)39 b Fo(E)2013 4768 y Fm(n)2095 4753 y Fq(=)c(inf)23 b Fo(\033)t Fq(\()p Fo(H)2519 4768 y Fm(n)2566 4753 y Fq(\))39 b Fj(is)f(a)h(simple,)g(isolate)-5 b(d)257 4873 y(eigenvalue)34 b(of)h Fo(H)924 4888 y Fm(n)970 4873 y Fj(,)g(with)g(c)-5 b(orr)g(esp)g(onding)33 b(normalize)-5 b(d)34 b(eigenve)-5 b(ctor)33 b Fo(\036)2921 4888 y Fm(n)2996 4873 y Fn(2)28 b(H)3174 4888 y Fm(n)3221 4873 y Fj(,)35 b(and)1354 5059 y Fq(gap)1506 5083 y Fl(0)1629 5059 y Fq(=)84 b(min)n Fn(f)p Fq(\001)p Fo(e)2127 5074 y Fm(el)2214 5059 y Fo(;)45 b(\024)p Fn(g)22 b Fo(;)781 b Fq(\(I)s(I.52\))1256 5218 y(gap)1408 5241 y Fm(n)p Fl(+1)1628 5218 y Fn(\025)84 b Fq([1)22 b Fn(\000)g Fq(3)p Fo(g)2086 5167 y Fl(\(1\))2082 5243 y Fm(\013;\014)2194 5218 y Fq(])17 b Fo(\033)2293 5233 y Fm(n)p Fl(+1)2452 5218 y Fo(;)743 b Fq(\(I)s(I.53\))1363 5356 y Fh(g)1347 5363 y Fq(gap)1498 5387 y Fm(n)1629 5363 y Fq(=)84 b Fo(\033)1844 5378 y Fm(n)p Fl(+1)2004 5363 y Fo(:)1191 b Fq(\(I)s(I.54\))p eop %%Page: 14 15 14 14 bop 257 266 a Fr(BFP-GrSt,)32 b(04-F)-8 b(eb-2005)2130 b Fq(14)257 573 y Fj(Pr)-5 b(o)g(of.)43 b Fq(W)-8 b(e)33 b(pro)s(ceed)g(b)m(y)h(induction)d(in)h Fo(n)c Fn(2)g Fk(N)2014 588 y Fl(0)2059 573 y Fq(.)44 b(Since)32 b Fo(H)2465 588 y Fl(0)2532 573 y Fq(=)c Fo(H)2717 588 y Fm(el)2797 573 y Fn(\012)23 b Fc(1)2953 588 y Fl(0)3014 573 y Fq(+)f Fc(1)3168 588 y Fm(el)3248 573 y Fn(\012)3373 548 y Fq(\024)3348 573 y Fo(H)3429 588 y Fl(0)3468 573 y Fq(,)257 693 y(w)m(e)34 b(ha)m(v)m(e)g(that)662 902 y Fo(\033)t Fq(\()p Fo(H)840 917 y Fl(0)879 902 y Fq(\))84 b(=)g Fo(\033)t Fq(\()p Fo(H)1339 917 y Fm(el)1397 902 y Fq(\))39 b(+)g Fo(\033)t Fq(\()1711 877 y(\024)1686 902 y Fo(H)1767 917 y Fl(0)1806 902 y Fq(\))60 b(=)g Fo(\033)t Fq(\()p Fo(H)2218 917 y Fm(el)2276 902 y Fq(\))39 b(+)2468 806 y Fh(\020)2517 902 y Fn(f)p Fq(0)p Fn(g)22 b([)g Fq([)p Fo(\024;)17 b Fn(1)p Fq(\))3041 806 y Fh(\021)1000 1085 y Fn(\022)84 b Fo(e)1206 1100 y Fm(el)1314 1085 y Fq(+)1440 989 y Fh(\020)1490 1085 y Fn(f)p Fq(0)p Fn(g)38 b([)h Fq([min)o Fn(f)p Fq(\001)p Fo(e)2148 1100 y Fm(el)2207 1085 y Fo(;)17 b(\024)p Fn(g)27 b Fo(;)45 b Fn(1)p Fq(\))2594 989 y Fh(\021)2665 1085 y Fo(;)530 b Fq(\(I)s(I.55\))257 1307 y(whic)m(h)41 b(implies)d(\(I)s(I.52\),)k(as)e(a)g(v)-5 b(ariational)37 b(argumen)m(t)i(sho)m(ws)j(that)e Fo(\033)t Fq(\()p Fo(H)3063 1322 y Fl(0)3102 1307 y Fq(\))h Fn(3)g Fo(e)3333 1322 y Fm(el)3419 1307 y Fq(+)257 1427 y(min)o Fn(f)p Fq(\001)p Fo(e)596 1442 y Fm(el)655 1427 y Fo(;)17 b(\024)p Fn(g)p Fq(.)65 b(Clearly)-8 b(,)40 b Fo(E)1335 1442 y Fl(0)1415 1427 y Fq(=)f Fo(e)1575 1442 y Fm(el)1674 1427 y Fq(is)g(an)h(eigen)m(v)-5 b(alue)39 b(of)g Fo(H)2594 1442 y Fl(0)2673 1427 y Fq(of)h(m)m(ultiplicit)m(y)c(one,)257 1548 y(the)d(corresp)s(onding)g(normalized)d(eigen)m(v)m(ector)k(b)s (eing)e Fo(\036)2378 1563 y Fl(0)2445 1548 y Fq(=)27 b Fo(')2612 1563 y Fm(el)2693 1548 y Fn(\012)c Fq(\012.)404 1668 y(A)g(similar)d(argumen)m(t)j(is)g(used)h(to)f(demonstrate)h(\(I)s (I.53\).)39 b(Namely)-8 b(,)25 b(due)f(to)f(\(I)s(I.23\),)257 1788 y(w)m(e)34 b(ha)m(v)m(e)g(that)360 1998 y Fo(\033)t Fq(\()465 1965 y Fh(f)457 1998 y Fo(H)546 1956 y Fl(+)538 2022 y Fm(n)605 1998 y Fq(\))60 b(=)g Fo(\033)t Fq(\()p Fo(H)1025 1956 y Fl(+)1017 2022 y Fm(n)1083 1998 y Fq(\))39 b(+)g Fo(\033)t Fq(\()1397 1972 y(\024)1372 1998 y Fo(H)1461 1956 y Fm(n)1453 2022 y(n)p Fl(+1)1590 1998 y Fq(\))60 b Fn(\022)g(f)p Fq(0)p Fn(g)39 b([)g Fq([gap)2297 2021 y Fm(n)2343 1998 y Fo(;)17 b Fn(1)p Fq(\))39 b Fn([)g Fq([)p Fo(\033)2751 2013 y Fm(n)p Fl(+1)2888 1998 y Fo(;)17 b Fn(1)p Fq(\))22 b Fo(;)103 b Fq(\(I)s(I.56\))257 2221 y(whic)m(h)23 b(implies)c(that)1062 2214 y Fh(g)1046 2221 y Fq(gap)1198 2244 y Fm(n)1273 2221 y Fn(\025)28 b Fq(min)o Fn(f)p Fq(gap)1742 2244 y Fm(n)1789 2221 y Fo(;)17 b(\033)1888 2236 y Fm(n)p Fl(+1)2025 2221 y Fn(g)p Fq(.)40 b(Since)22 b Fo(\033)t Fq(\()2491 2189 y Fh(f)2483 2221 y Fo(H)2572 2185 y Fl(+)2564 2246 y Fm(n)2630 2221 y Fq(\))28 b Fn(3)g Fq(min)o Fn(f)p Fq(gap)3154 2244 y Fm(n)3201 2221 y Fo(;)17 b(\033)3300 2236 y Fm(n)p Fl(+1)3438 2221 y Fn(g)p Fq(,)257 2341 y(w)m(e)34 b(ev)m(en)g(conclude) f(that)1280 2544 y Fh(g)1264 2551 y Fq(gap)1416 2574 y Fm(n)1523 2551 y Fq(=)60 b(min)1838 2454 y Fh(n)1894 2551 y Fq(gap)2045 2574 y Fm(n)2120 2551 y Fo(;)44 b(\033)2246 2566 y Fm(n)p Fl(+1)2384 2454 y Fh(o)2461 2551 y Fo(;)734 b Fq(\(I)s(I.57\))257 2772 y(for)39 b(all)f Fo(n)i Fn(2)g Fk(N)826 2787 y Fl(0)871 2772 y Fq(.)65 b(Since)40 b(Hyp)s(othesis)g (H-1)f(implies)e(that)i(gap)2636 2796 y Fl(0)2715 2772 y Fq(=)h(min)n Fn(f)p Fq(\001)p Fo(e)3169 2787 y Fm(el)3228 2772 y Fo(;)17 b(\024)p Fn(g)40 b(\025)257 2893 y Fo(\024\014)368 2908 y Fl(1)436 2893 y Fn(\025)28 b Fo(\024\014)33 b Fq(=)28 b Fo(\033)844 2908 y Fl(1)884 2893 y Fq(,)k(w)m(e)i(ha)m(v)m(e) g(that)1539 2886 y Fh(g)1523 2893 y Fq(gap)1675 2916 y Fl(0)1742 2893 y Fq(=)28 b Fo(\033)1901 2908 y Fl(1)1940 2893 y Fq(.)404 3025 y(Let)35 b(us)g(no)m(w)g(supp)s(ose)h(that)f Fo(n)d Fn(\025)g Fq(1)i(and)h(gap)2122 3049 y Fm(n)2200 3025 y Fn(\025)d Fq([1)23 b Fn(\000)h Fq(3)p Fo(g)2609 2974 y Fl(\(1\))2605 3050 y Fm(\013;\014)2717 3025 y Fq(])p Fo(\033)2799 3040 y Fm(n)2847 3025 y Fq(.)50 b(Since)34 b(3)p Fo(g)3280 2974 y Fl(\(1\))3276 3050 y Fm(\013;\014)3419 3025 y Fq(=)257 3157 y(3)p Fo(C)376 3172 y Fl(1)415 3157 y Fo(\013)478 3121 y Fl(3)p Fm(=)p Fl(2)588 3157 y Fo(\014)649 3121 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)866 3157 y Fn(\024)52 b Fq(1)p Fo(=)p Fq(2)46 b(and)h Fo(\014)58 b Fn(\024)53 b Fo(\014)1690 3172 y Fl(1)1781 3157 y Fn(\024)g Fq(1)p Fo(=)p Fq(2,)d(w)m(e)e(ha)m(v)m(e)g(that)f(gap)2909 3181 y Fm(n)3008 3157 y Fn(\025)53 b Fo(\033)3193 3172 y Fm(n)p Fl(+1)3330 3157 y Fq(,)e(so)273 3271 y Fh(g)257 3278 y Fq(gap)409 3301 y Fm(n)484 3278 y Fq(=)27 b Fo(\033)642 3293 y Fm(n)p Fl(+1)780 3278 y Fq(,)33 b(whic)m(h)g(establishes)g(\(I)s (I.54\).)404 3398 y(T)-8 b(o)37 b(p)s(erform)g(the)h(induction)f(step,) j(w)m(e)f(assume)f(that)f Fo(E)2557 3413 y Fm(n)2642 3398 y Fq(is)h(a)f(simple,)h(isolated)257 3519 y(eigen)m(v)-5 b(alue)36 b(of)g Fo(H)923 3534 y Fm(n)970 3519 y Fq(,)h(with)f(corresp) s(onding)g(normalized)f(eigen)m(v)m(ector)i Fo(\036)2961 3534 y Fm(n)3042 3519 y Fn(2)e(H)3227 3534 y Fm(n)3274 3519 y Fq(,)i(and)257 3639 y(that)490 3632 y Fh(g)474 3639 y Fq(gap)626 3663 y Fm(n)711 3639 y Fq(=)g Fo(\033)879 3654 y Fm(n)p Fl(+1)1017 3639 y Fq(.)60 b(It)39 b(remains)e(to)h(pro)m (v)m(e)i(that)e Fo(E)2268 3654 y Fm(n)p Fl(+1)2443 3639 y Fq(is)g(an)h(eigen)m(v)-5 b(alue)37 b(of)h Fo(H)3358 3654 y Fm(n)p Fl(+1)257 3759 y Fq(of)d(m)m(ultiplicit)m(y)d(one)k(and)f (that)g(the)h(gap)f(ab)s(o)m(v)m(e)h Fo(E)2194 3774 y Fm(n)p Fl(+1)2332 3759 y Fq(,)g(denoted)g(b)m(y)g(gap)3056 3783 y Fm(n)p Fl(+1)3193 3759 y Fq(,)g(ob)s(eys)257 3880 y(\(I)s(I.53\).)43 b(W)-8 b(e)33 b(observ)m(e)i(that)d(due)h(to)g(\(I)s (I.46\))f(and)g(\(I)s(I.49\),)471 4101 y Fo(X)552 4116 y Fm(n)p Fl(+1)772 4101 y Fq(:=)234 b(sup)958 4184 y Fm( )1004 4193 y Fe(0)1039 4184 y Fi(2H)1146 4193 y Fa(n)p Fe(+1)1265 4184 y Fi(nf)p Fl(0)p Fi(g)1601 4101 y Fq(inf)1451 4165 y Fm(\036)p Fi(?)p Fm( )1595 4174 y Fe(0)1628 4165 y Fm(;)19 b Fi(k)p Fm(\036)p Fi(k)p Fl(=1)1887 4005 y Fh(D)1937 4101 y Fo(\036)2023 4002 y Fh(\014)2023 4052 y(\014)2023 4101 y(\014)27 b Fo(H)2167 4060 y Fl(+)2159 4126 y Fm(n)p Fl(+1)2313 4101 y Fo(\036)2371 4005 y Fh(E)785 4342 y Fn(\025)248 b Fq(inf)958 4418 y Fm(\036)p Fi(?)1065 4401 y Fl(~)1056 4418 y Fm(\036)1098 4426 y Fa(n)1139 4418 y Fm(;)19 b Fi(k)p Fm(\036)p Fi(k)p Fl(=1)1398 4246 y Fh(D)1448 4342 y Fo(\036)1534 4243 y Fh(\014)1534 4292 y(\014)1534 4342 y(\014)27 b Fo(H)1678 4301 y Fl(+)1670 4367 y Fm(n)p Fl(+1)1824 4342 y Fo(\036)1882 4246 y Fh(E)3222 4342 y Fq(\(I)s(I.58\))785 4577 y Fn(\025)96 b Fo(E)1036 4536 y Fm(n)1030 4602 y(n)p Fl(+1)1206 4577 y Fn(\000)39 b Fo(g)1373 4526 y Fl(\(1\))1369 4602 y Fm(\013;\014)1498 4577 y Fo(\033)1553 4592 y Fm(n)p Fl(+1)1729 4577 y Fq(+)g([1)22 b Fn(\000)g Fo(g)2092 4526 y Fl(\(1\))2088 4602 y Fm(\013;\014)2200 4577 y Fq(])2260 4570 y Fh(g)2244 4577 y Fq(gap)2395 4600 y Fm(n)2503 4577 y Fn(\025)60 b Fq([1)22 b Fn(\000)h Fq(3)p Fo(g)2938 4526 y Fl(\(1\))2934 4602 y Fm(\013;\014)3046 4577 y Fq(])17 b Fo(\033)3145 4592 y Fm(n)p Fl(+1)257 4786 y Fq(holds)36 b(true.)56 b(Since)36 b Fo(X)1111 4801 y Fm(n)p Fl(+1)1282 4786 y Fo(>)e Fq(0,)j(the)g(min-max)d (principle)h(implies)f(that)i Fo(E)3117 4801 y Fm(n)p Fl(+1)3290 4786 y Fq(is)g(an)257 4907 y(isolated)28 b(p)s(oin)m(t)g(in) g(the)h(sp)s(ectrum)g(of)g Fo(H)1748 4922 y Fm(n)p Fl(+1)1913 4907 y Fq(of)g(m)m(ultiplicit)m(y)c(at)j(most)h(one.)42 b(The)30 b(m)m(ul-)257 5027 y(tiplicit)m(y)i(of)h Fo(E)806 5042 y Fm(n)p Fl(+1)977 5027 y Fq(is)g(at)h(least)f(one,)h(ho)m(w)m(ev) m(er,)j(since)d Fo(E)2348 5042 y Fm(n)p Fl(+1)2519 5027 y Fq(is,)g(b)m(y)h(de\014nition,)e(in)g(the)257 5147 y(sp)s(ectrum)i(of)f Fo(H)879 5162 y Fm(n)p Fl(+1)1016 5147 y Fq(.)50 b(Th)m(us,)36 b Fo(E)1441 5162 y Fm(n)p Fl(+1)1613 5147 y Fq(is)e(an)g(eigen)m(v)-5 b(alue)34 b(of)g Fo(H)2512 5162 y Fm(n)p Fl(+1)2684 5147 y Fq(of)g(m)m (ultiplicit)m(y)d(one,)257 5268 y(and)i(gap)599 5291 y Fm(n)p Fl(+1)764 5268 y Fq(=)27 b Fo(X)948 5283 y Fm(n)p Fl(+1)1085 5268 y Fq(.)p 3425 5268 4 66 v 3429 5205 59 4 v 3429 5268 V 3487 5268 4 66 v eop %%Page: 15 16 15 15 bop 257 266 a Fr(BFP-GrSt,)32 b(04-F)-8 b(eb-2005)2130 b Fq(15)257 573 y Fb(I)t(I.4)135 b(Ground)44 b(State)i(Pro)7 b(jections)257 758 y Fq(Lemma)23 b(I)s(I.7)h(enables)h(us)f(to)g(write) g(the)h(pro)5 b(jection)24 b Fo(P)2250 773 y Fm(n)2324 758 y Fq(=)k Fn(j)p Fo(\036)2514 773 y Fm(n)2560 758 y Fn(ih)p Fo(\036)2696 773 y Fm(n)2742 758 y Fn(j)c Fq(on)m(to)g(the)h (ground-)257 878 y(state)33 b Fo(\036)554 893 y Fm(n)634 878 y Fq(of)f Fo(H)826 893 y Fm(n)905 878 y Fq(as)h(a)f(con)m(tour)h (in)m(tegral)1342 1109 y Fo(P)1405 1124 y Fm(n)1512 1109 y Fq(=)1666 1042 y Fn(\000)p Fq(1)p 1658 1086 141 4 v 1658 1177 a(2)p Fo(\031)t(i)1826 992 y Fh(Z)1872 1181 y Fl(\000)1916 1189 y Fa(n)2099 1042 y Fo(dz)2195 1057 y Fm(n)p 1989 1086 362 4 v 1989 1177 a Fo(H)2078 1149 y Fl(+)2070 1202 y Fm(n)2159 1177 y Fn(\000)23 b Fo(z)2304 1192 y Fm(n)2383 1109 y Fo(;)812 b Fq(\(I)s(I.59\))257 1378 y(where)45 b(\000)611 1393 y Fm(n)705 1378 y Fq(:=)i Fn(f)915 1339 y Fl(1)p 915 1355 36 4 v 915 1412 a(4)960 1378 y Fo(\033)1015 1393 y Fm(n)1062 1378 y Fo(e)1107 1342 y Fm(i#)1225 1378 y Fn(2)g Fk(C)20 b Fn(j)p Fo(#)53 b Fn(2)47 b Fq([0)p Fo(;)17 b Fq(2)p Fo(\031)t Fq(\))p Fn(g)43 b Fq(and)h(pro)m(vided)g(that)2860 1339 y Fl(1)p 2860 1355 V 2860 1412 a(4)2952 1378 y Fo(<)j Fq(1)29 b Fn(\000)i Fq(3)p Fo(g)3361 1327 y Fl(\(1\))3357 1403 y Fm(\013;\014)3468 1378 y Fq(,)257 1498 y(for)38 b Fo(n)g Fn(2)f Fk(N)9 b Fq(,)46 b(and)955 1459 y Fl(1)p 955 1475 V 955 1533 a(4)1038 1498 y Fo(<)37 b Fq(min)n Fn(f)p Fq(\001)p Fo(e)1489 1513 y Fm(el)1548 1498 y Fo(=\024;)17 b Fq(1)p Fn(g)p Fq(,)39 b(for)f Fo(n)f Fq(=)g(0,)j(as)e(these)i (inequalities)c(insure)257 1642 y(that)h Fn(j)p Fo(z)546 1657 y Fm(n)593 1642 y Fn(j)d Fo(<)g Fq(gap)917 1665 y Fm(n)964 1642 y Fq(.)55 b(F)-8 b(or)1234 1603 y Fl(1)p 1234 1619 V 1234 1676 a(4)1314 1642 y Fo(<)34 b Fq(1)25 b Fn(\000)g Fq(3)p Fo(g)1700 1591 y Fl(\(1\))1696 1667 y Fm(\013;\014)1844 1642 y Fq(and)37 b Fo(z)2083 1657 y Fm(n)p Fl(+1)2255 1642 y Fn(2)d Fq(\000)2416 1657 y Fm(n)p Fl(+1)2553 1642 y Fq(,)k(w)m(e)g(also)d(observ)m(e)j(that)257 1762 y Fn(j)p Fo(z)330 1777 y Fm(n)p Fl(+1)467 1762 y Fn(j)28 b Fo(<)642 1755 y Fh(g)626 1762 y Fq(gap)778 1786 y Fm(n)858 1762 y Fq(and)k(th)m(us)1282 1967 y Fh(e)1259 1999 y Fo(P)1322 2014 y Fm(n)1429 1999 y Fq(=)1582 1931 y Fn(\000)p Fq(1)p 1575 1976 141 4 v 1575 2067 a(2)p Fo(\031)t(i)1742 1882 y Fh(Z)1789 2070 y Fl(\000)1833 2079 y Fa(n)p Fe(+1)2092 1931 y Fo(dz)2188 1946 y Fm(n)p Fl(+1)p 1983 1976 452 4 v 1991 2048 a Fh(f)1983 2080 y Fo(H)2072 2051 y Fl(+)2064 2105 y Fm(n)2153 2080 y Fn(\000)22 b Fo(z)2297 2095 y Fm(n)p Fl(+1)2467 1999 y Fo(;)728 b Fq(\(I)s(I.60\))257 2272 y(where)562 2240 y Fh(e)539 2272 y Fo(P)602 2287 y Fm(n)677 2272 y Fq(:=)27 b Fn(j)848 2246 y Fq(~)835 2272 y Fo(\036)893 2287 y Fm(n)940 2272 y Fn(ih)1031 2246 y Fq(~)1018 2272 y Fo(\036)1076 2287 y Fm(n)1122 2272 y Fn(j)g Fq(=)h Fo(P)1344 2287 y Fm(n)1412 2272 y Fn(\012)23 b Fo(P)1575 2287 y Fl(\012)1626 2265 y Fa(n)1626 2310 y(n)p Fe(+1)1781 2272 y Fq(is)32 b(the)h(pro)5 b(jection)32 b(on)m(to)2742 2246 y(~)2730 2272 y Fo(\036)2788 2287 y Fm(n)2834 2272 y Fq(.)44 b(Note)32 b(that,)h(for)257 2393 y Fo(n)28 b Fn(2)g Fk(N)503 2408 y Fl(0)549 2393 y Fq(,)k(w)m(e)i(ha)m(v)m(e)g(that)648 2619 y(gap)800 2643 y Fm(n)p Fl(+1)960 2619 y Fn(\000)22 b(j)p Fo(z)1132 2634 y Fm(n)p Fl(+1)1269 2619 y Fn(j)g(\000)h(j)p Fo(E)1525 2578 y Fm(n)1519 2644 y(n)p Fl(+1)1656 2619 y Fn(j)60 b(\025)1881 2498 y Fh(\022)1942 2619 y Fq(1)22 b Fn(\000)h Fq(4)p Fo(g)2213 2568 y Fl(\(1\))2209 2644 y Fm(\013;\014)2343 2619 y Fn(\000)2452 2552 y Fq(1)p 2452 2596 49 4 v 2452 2687 a(4)2511 2498 y Fh(\023)2589 2619 y Fo(\033)2644 2634 y Fm(n)p Fl(+1)2803 2619 y Fo(:)392 b Fq(\(I)s(I.61\))257 2870 y(Th)m(us,)35 b(if)c Fo(g)672 2819 y Fl(\(1\))668 2895 y Fm(\013;\014)807 2870 y Fo(<)d Fq(1)p Fo(=)p Fq(6)k(then)h Fo(g)1363 2819 y Fl(\(1\))1359 2895 y Fm(\013;\014)1499 2870 y Fo(<)1612 2830 y Fl(1)p 1612 2846 36 4 v 1612 2904 a(4)1685 2870 y Fo(<)28 b Fq([1)22 b Fn(\000)g Fq(4)p Fo(g)2086 2819 y Fl(\(1\))2082 2895 y Fm(\013;\014)2194 2870 y Fq(],)32 b(and)h(w)m(e)h(ha)m(v)m(e)g (that)319 3119 y Fo(P)382 3134 y Fm(n)p Fl(+1)580 3119 y Fq(=)733 3052 y Fn(\000)p Fq(1)p 726 3096 141 4 v 726 3188 a(2)p Fo(\031)t(i)893 3002 y Fh(Z)939 3191 y Fl(\000)983 3200 y Fa(n)p Fe(+1)1444 3052 y Fo(dz)1540 3067 y Fm(n)p Fl(+1)p 1134 3096 854 4 v 1134 3188 a Fo(H)1215 3203 y Fm(n)p Fl(+1)1374 3188 y Fn(\000)22 b Fo(E)1545 3203 y Fm(n)p Fl(+1)1705 3188 y Fn(\000)g Fo(z)1849 3203 y Fm(n)p Fl(+1)2057 3119 y Fq(=)2210 3052 y Fn(\000)p Fq(1)p 2203 3096 141 4 v 2203 3188 a(2)p Fo(\031)t(i)2371 3002 y Fh(Z)2417 3191 y Fl(\000)2461 3200 y Fa(n)p Fe(+1)2876 3052 y Fo(dz)2972 3067 y Fm(n)p Fl(+1)p 2611 3096 764 4 v 2611 3188 a Fo(H)2692 3203 y Fm(n)p Fl(+1)2851 3188 y Fn(\000)h Fo(E)3023 3203 y Fm(n)3092 3188 y Fn(\000)g Fo(z)3237 3203 y Fm(n)p Fl(+1)3406 3119 y Fo(;)3222 3293 y Fq(\(I)s(I.62\))257 3413 y(for)33 b(all)d Fo(n)e Fn(2)h Fk(N)789 3428 y Fl(0)834 3413 y Fq(,)k(b)s(ecause)h(in)e(b)s(oth)g(in)m (tegrals)g(in)g(\(I)s(I.62\))g(the)h(only)f(sp)s(ectral)h(p)s(oin)m(t)f (of)257 3533 y Fo(H)338 3548 y Fm(n)p Fl(+1)508 3533 y Fq(encircled)g(b)m(y)i(the)f(in)m(tegration)e(con)m(tours)i(is)f Fo(E)2277 3548 y Fm(n)p Fl(+1)2415 3533 y Fq(.)257 3719 y Fc(Lemma)i(I)s(I.8.)39 b Fj(Assume)33 b(Hyp)-5 b(othesis)32 b(H-1)g(and)g Fo(g)2167 3668 y Fl(\(1\))2163 3744 y Fm(\013;\014)2302 3719 y Fq(=)c Fo(C)2476 3734 y Fl(1)2515 3719 y Fo(\013)2578 3683 y Fl(3)p Fm(=)p Fl(2)2688 3719 y Fo(\014)2749 3683 y Fl(1)p Fm(=)p Fl(2)2886 3719 y Fn(\024)g Fq(1)p Fo(=)p Fq(6)p Fj(.)43 b(F)-7 b(or)32 b(al)5 b(l)257 3840 y Fo(n)28 b Fn(2)g Fk(N)503 3855 y Fl(0)549 3840 y Fj(,)35 b(we)f(have)g(that) 1358 4096 y Fq(sup)1308 4175 y Fm(z)s Fi(2)p Fl(\000)1435 4184 y Fa(n)p Fe(+1)1571 3947 y Fh(\015)1571 3997 y(\015)1571 4047 y(\015)1571 4096 y(\015)1571 4146 y(\015)1635 3997 y(f)1627 4029 y Fo(H)1716 3993 y Fl(+)1708 4054 y Fm(n)1797 4029 y Fq(+)22 b Fo(\033)1950 4044 y Fm(n)p Fl(+1)p 1627 4073 461 4 v 1706 4145 a Fh(f)1698 4178 y Fo(H)1787 4149 y Fl(+)1779 4202 y Fm(n)1868 4178 y Fn(\000)g Fo(z)2098 3947 y Fh(\015)2098 3997 y(\015)2098 4047 y(\015)2098 4096 y(\015)2098 4146 y(\015)2206 4096 y Fn(\024)63 b Fq(4)22 b Fo(:)778 b Fq(\(I)s(I.63\))257 4352 y Fj(Pr)-5 b(o)g(of.)75 b Fq(Eq.)44 b(\(I)s(I.63\))e(follo)m(ws)g(directly)g(from) g(the)h(sp)s(ectral)g(theorem)g(and)3191 4345 y Fh(g)3175 4352 y Fq(gap)3327 4376 y Fm(n)3419 4352 y Fq(=)257 4473 y Fo(\033)312 4488 y Fm(n)p Fl(+1)450 4473 y Fq(,)33 b(namely)722 4582 y Fh(\015)722 4632 y(\015)722 4682 y(\015)722 4732 y(\015)722 4781 y(\015)786 4632 y(f)778 4664 y Fo(H)867 4628 y Fl(+)859 4689 y Fm(n)947 4664 y Fq(+)22 b Fo(\033)1100 4679 y Fm(n)p Fl(+1)p 778 4708 V 856 4781 a Fh(f)848 4813 y Fo(H)937 4784 y Fl(+)929 4838 y Fm(n)1018 4813 y Fn(\000)h Fo(z)1248 4582 y Fh(\015)1248 4632 y(\015)1248 4682 y(\015)1248 4732 y(\015)1248 4781 y(\015)1377 4732 y Fn(\024)84 b Fq(max)1736 4586 y Fh(\()1803 4732 y Fq(4)21 b Fo(;)50 b Fq(sup)1961 4808 y Fm(r)r(>)p Fl(0)2113 4607 y Fh(\014)2113 4657 y(\014)2113 4707 y(\014)2113 4757 y(\014)2151 4664 y Fo(r)25 b Fq(+)2334 4657 y Fh(g)2318 4664 y Fq(gap)2469 4688 y Fm(n)2538 4664 y Fq(+)d Fo(\033)2691 4679 y Fm(n)p Fl(+1)p 2151 4708 679 4 v 2221 4800 a Fo(r)j Fq(+)2404 4793 y Fh(g)2388 4800 y Fq(gap)2540 4824 y Fm(n)2609 4800 y Fn(\000)e Fo(z)2839 4607 y Fh(\014)2839 4657 y(\014)2839 4707 y(\014)2839 4757 y(\014)2867 4586 y(\))3222 4732 y Fq(\(I)s(I.64\))1377 5020 y Fn(\024)84 b Fq(max)1736 4874 y Fh(\()1803 5020 y Fq(4)21 b Fo(;)50 b Fq(sup)1961 5096 y Fm(r)r(>)p Fl(0)2113 4899 y Fh(\022)2187 4953 y Fo(r)25 b Fq(+)d(2)p Fo(\033)2458 4968 y Fm(n)p Fl(+1)p 2184 4997 415 4 v 2184 5089 a Fo(r)j Fq(+)2361 5050 y Fl(3)p 2361 5066 36 4 v 2361 5123 a(4)2406 5089 y Fo(\033)2461 5104 y Fm(n)p Fl(+1)2609 4899 y Fh(\023)2670 4874 y(\))2797 5020 y Fq(=)60 b(4)22 b Fo(:)p 3425 5268 4 66 v 3429 5205 59 4 v 3429 5268 V 3487 5268 4 66 v eop %%Page: 16 17 16 16 bop 257 266 a Fr(BFP-GrSt,)32 b(04-F)-8 b(eb-2005)2130 b Fq(16)257 574 y Fc(Lemma)39 b(I)s(I.9.)j Fj(Assume)36 b(Hyp)-5 b(othesis)36 b(H-1)g(and)f Fo(g)2189 523 y Fl(\(1\))2185 599 y Fm(\013;\014)2327 574 y Fq(=)29 b Fo(C)2502 589 y Fl(1)2542 574 y Fo(\013)2605 538 y Fl(3)p Fm(=)p Fl(2)2714 574 y Fo(\014)2775 538 y Fl(1)p Fm(=)p Fl(2)2915 574 y Fn(\024)h Fq(1)p Fo(=)p Fq(8)p Fj(.)47 b(Then,)257 694 y(for)35 b(al)5 b(l)35 b Fo(n)27 b Fn(2)i Fk(N)798 709 y Fl(0)844 694 y Fj(,)35 b(the)g(series)f(exp)-5 b(ansion)913 966 y Fo(P)976 981 y Fm(n)p Fl(+1)1209 966 y Fq(=)1399 899 y Fn(\000)p Fq(1)p 1392 943 141 4 v 1392 1035 a(2)p Fo(\031)t(i)1588 859 y Fi(1)1564 883 y Fh(X)1559 1065 y Fm(\027)t Fl(=0)1705 849 y Fh(Z)1751 1038 y Fl(\000)1795 1047 y Fa(n)p Fe(+1)1919 966 y Fq(\()p Fn(\000)p Fq(1\))2121 925 y Fm(\027)2186 966 y Fo(Y)2264 916 y Fl(\()p Fm(\027)t Fl(\))2243 988 y Fm(n)p Fl(+1)2380 966 y Fq(\()p Fo(z)2463 981 y Fm(n)p Fl(+1)2600 966 y Fq(\))22 b Fo(dz)2756 981 y Fm(n)p Fl(+1)2916 966 y Fo(;)279 b Fq(\(I)s(I.65\))793 1238 y Fo(Y)871 1187 y Fl(\()p Fm(\027)t Fl(\))850 1260 y Fm(n)p Fl(+1)987 1238 y Fq(\()p Fo(z)t Fq(\))84 b(:=)1527 1170 y(1)p 1392 1214 320 4 v 1400 1287 a Fh(f)1392 1319 y Fo(H)1481 1290 y Fl(+)1473 1344 y Fm(n)1561 1319 y Fn(\000)23 b Fo(z)1721 1117 y Fh(\022)1782 1238 y Fo(W)1888 1197 y Fm(n)1874 1262 y(n)p Fl(+1)2173 1170 y Fq(1)p 2038 1214 V 2046 1287 a Fh(f)2038 1319 y Fo(H)2127 1290 y Fl(+)2119 1344 y Fm(n)2207 1319 y Fn(\000)g Fo(z)2367 1117 y Fh(\023)2428 1139 y Fm(\027)2493 1238 y Fo(;)702 b Fq(\(I)s(I.66\))257 1521 y Fj(is)35 b(norm-c)-5 b(onver)g(gent,)33 b(with)918 1790 y Fq(sup)847 1873 y Fi(j)p Fm(z)s Fi(j)p Fl(=)p Fm(\033)1018 1882 y Fa(n)p Fe(+1)1153 1691 y Fh(\015)1153 1740 y(\015)1153 1790 y(\015)p Fo(Y)1278 1739 y Fl(\()p Fm(\027)t Fl(\))1256 1812 y Fm(n)p Fl(+1)1393 1790 y Fq(\()p Fo(z)t Fq(\))1518 1691 y Fh(\015)1518 1740 y(\015)1518 1790 y(\015)63 b Fn(\024)1849 1723 y Fq(4)p 1777 1767 193 4 v 1777 1859 a Fo(\033)1832 1874 y Fm(n)p Fl(+1)2002 1694 y Fh(\020)2052 1790 y Fq(4)17 b Fo(g)2169 1739 y Fl(\(1\))2165 1815 y Fm(\013;\014)2276 1694 y Fh(\021)2325 1717 y Fm(\027)2431 1790 y Fn(\024)2581 1723 y Fq(\(1)p Fo(=)p Fq(2\))2804 1687 y Fm(\027)p 2581 1767 266 4 v 2617 1859 a Fo(\033)2672 1874 y Fm(n)p Fl(+1)2879 1790 y Fo(;)316 b Fq(\(I)s(I.67\))257 2075 y Fj(for)35 b(al)5 b(l)35 b Fo(\027)f Fn(2)28 b Fk(N)795 2090 y Fl(0)840 2075 y Fj(.)257 2279 y(Pr)-5 b(o)g(of.)43 b Fq(W)-8 b(e)33 b(rewrite)g Fo(Y)1134 2228 y Fl(\()p Fm(\027)t Fl(\))1113 2300 y Fm(n)p Fl(+1)1250 2279 y Fq(\()p Fo(z)t Fq(\))g(as)385 2577 y Fo(Y)463 2526 y Fl(\()p Fm(\027)t Fl(\))442 2599 y Fm(n)p Fl(+1)579 2577 y Fq(\()p Fo(z)t Fq(\))61 b(=)923 2545 y Fh(e)901 2577 y Fo(R)976 2536 y Fl(1)p Fm(=)p Fl(2)975 2602 y Fm(n)1121 2478 y Fh(f)1112 2510 y Fo(H)1201 2474 y Fl(+)1193 2534 y Fm(n)1282 2510 y Fq(+)22 b Fo(\033)1435 2525 y Fm(n)p Fl(+1)p 1112 2554 461 4 v 1191 2626 a Fh(f)1183 2659 y Fo(H)1272 2630 y Fl(+)1264 2683 y Fm(n)1353 2659 y Fn(\000)g Fo(z)1583 2431 y Fh( )1671 2545 y(e)1648 2577 y Fo(R)1723 2536 y Fl(1)p Fm(=)p Fl(2)1722 2602 y Fm(n)1850 2577 y Fo(W)1956 2536 y Fm(n)1942 2602 y(n)p Fl(+1)2118 2545 y Fh(e)2096 2577 y Fo(R)2171 2536 y Fl(1)p Fm(=)p Fl(2)2170 2602 y Fm(n)2321 2478 y Fh(f)2313 2510 y Fo(H)2402 2474 y Fl(+)2394 2534 y Fm(n)2483 2510 y Fq(+)g Fo(\033)2636 2525 y Fm(n)p Fl(+1)p 2313 2554 V 2392 2626 a Fh(f)2384 2659 y Fo(H)2473 2630 y Fl(+)2465 2683 y Fm(n)2553 2659 y Fn(\000)h Fo(z)2783 2431 y Fh(!)2849 2454 y Fm(\027)2931 2545 y Fh(e)2909 2577 y Fo(R)2984 2536 y Fl(1)p Fm(=)p Fl(2)2983 2602 y Fm(n)3222 2577 y Fq(\(I)s(I.68\))257 2860 y(and)33 b(obtain)f(the)h(claim)d(from)h (Lemmata)g(I)s(I.4)h(and)h(I)s(I.8.)p 3425 2860 4 66 v 3429 2798 59 4 v 3429 2860 V 3487 2860 4 66 v 404 3101 a(W)-8 b(e)29 b(remark)h(that,)g(b)m(y)g(\(I)s(I.60\),)g(the)f(0)1811 3065 y Fm(th)1881 3101 y Fq(-order)g(term)g(in)g(the)h(series)g (expansion)g(for)257 3221 y Fo(P)320 3236 y Fm(n)p Fl(+1)490 3221 y Fq(equals)809 3189 y Fh(e)786 3221 y Fo(P)849 3236 y Fm(n)896 3221 y Fq(,)i(so)h(that)645 3496 y Fo(P)708 3511 y Fm(n)p Fl(+1)867 3496 y Fn(\000)990 3464 y Fh(e)967 3496 y Fo(P)1030 3511 y Fm(n)1137 3496 y Fq(=)1290 3429 y Fn(\000)p Fq(1)p 1283 3473 141 4 v 1283 3564 a(2)p Fo(\031)t(i)1480 3388 y Fi(1)1455 3413 y Fh(X)1450 3594 y Fm(\027)t Fl(=1)1596 3379 y Fh(Z)1642 3568 y Fl(\000)1686 3577 y Fa(n)p Fe(+1)1810 3496 y Fq(\()p Fn(\000)p Fq(1\))2012 3455 y Fm(\027)2077 3496 y Fo(Y)2156 3445 y Fl(\()p Fm(\027)t Fl(\))2134 3518 y Fm(n)p Fl(+1)2271 3496 y Fq(\()p Fo(z)2354 3511 y Fm(n)p Fl(+1)2492 3496 y Fq(\))22 b Fo(dz)2648 3511 y Fm(n)p Fl(+1)2807 3496 y Fo(;)388 b Fq(\(I)s(I.69\))257 3780 y(and)33 b(hence)560 4041 y Fn(k)p Fo(P)673 4056 y Fm(n)p Fl(+1)832 4041 y Fn(\000)954 4009 y Fh(e)931 4041 y Fo(P)994 4056 y Fm(n)1041 4041 y Fn(k)60 b(\024)h Fq(2)1400 3933 y Fi(1)1375 3958 y Fh(X)1371 4139 y Fm(\027)t Fl(=1)1517 3945 y Fh(\020)1566 4041 y Fq(4)17 b Fo(g)1683 3990 y Fl(\(1\))1679 4066 y Fm(\013;\014)1790 3945 y Fh(\021)1840 3968 y Fm(\027)1943 4041 y Fn(\024)61 b Fq(4)p Fo(g)2181 3990 y Fl(\(1\))2177 4066 y Fm(\013;\014)2349 4041 y Fn(\024)f Fq(4)17 b Fo(C)2622 4056 y Fl(1)2688 3974 y Fo(\013)2751 3937 y Fl(3)p Fm(=)p Fl(2)p 2688 4018 173 4 v 2689 4109 a Fo(\014)2750 4081 y Fl(1)p Fm(=)p Fl(2)2892 4041 y Fo(;)303 b Fq(\(I)s(I.70\))257 4341 y(whic)m(h)30 b(is)g(of)f(order)g Fo(\013)1051 4305 y Fl(3)p Fm(=)p Fl(2)1161 4341 y Fo(\014)1222 4305 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)1387 4341 y Fq(.)42 b(This)30 b(yields)f(an)g (alternativ)m(e)g(pro)s(of)g(for)g(the)h(fact)f(that)257 4462 y(if)38 b Fo(P)416 4477 y Fm(n)502 4462 y Fq(has)h(rank)g(one,)i (so)e(do)1418 4429 y Fh(e)1395 4462 y Fo(P)1458 4477 y Fm(n)1544 4462 y Fq(and)g(hence)h(also)e Fo(P)2282 4477 y Fm(n)p Fl(+1)2419 4462 y Fq(,)j(pro)m(vided)e Fo(\013)2957 4426 y Fl(3)p Fm(=)p Fl(2)3067 4462 y Fo(\014)3128 4426 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)3331 4462 y Fn(\025)f Fq(0)257 4582 y(is)d(su\016cien)m(tly)h(small.)48 b(Eq.)36 b(\(I)s(I.70\))f(is)g(not)g(su\016cien)m(t,)i(ho)m(w)m(ev)m(er,)h(to)d (pro)m(v)m(e)h(that)f(the)257 4702 y(sequence)k(\()p Fo(P)766 4717 y Fm(n)813 4702 y Fq(\))851 4666 y Fi(1)851 4727 y Fm(n)p Fl(=0)1024 4702 y Fq(of)d(pro)5 b(jections)37 b(con)m(v)m(erge.)56 b(Indeed,)38 b(this)e(cannot)h(b)s(e)f(exp)s (ected,)257 4823 y(in)k(general,)h(b)s(ecause)h(the)f(p)s(erturbation)e Fo(W)54 b Fq(is)39 b(marginal,)g(in)h(the)h(sense)h(of)d(p)s(o)m(w)m (er)257 4943 y(coun)m(ting.)64 b(T)-8 b(o)39 b(pro)m(v)m(e)i(con)m(v)m (ergence)h(of)d(\()p Fo(P)1899 4958 y Fm(n)1946 4943 y Fq(\))1984 4907 y Fi(1)1984 4968 y Fm(n)p Fl(=0)2121 4943 y Fq(,)i(w)m(e)f(therefore)g(need)g(re\014ned)h(esti-)257 5064 y(mates)d(that)f(tak)m(e)h(the)g(particularities)c(of)j(the)h(mo)s (del)e(in)m(to)h(accoun)m(t)h(-)f(mere)g(p)s(o)m(w)m(er)257 5184 y(coun)m(ting)c(cannot)f(ac)m(hiev)m(e)i(this)e(goal.)p eop %%Page: 17 18 17 17 bop 257 266 a Fr(BFP-GrSt,)32 b(04-F)-8 b(eb-2005)2130 b Fq(17)257 573 y Fb(I)t(I.5)135 b(Con)l(v)l(ergence)47 b(of)e(the)g(Ground)f(State)i(Pro)7 b(jections)257 758 y Fq(Our)36 b(pro)s(of)f(of)h(the)g(con)m(v)m(ergence)j(of)c(\()p Fo(P)1763 773 y Fm(n)1810 758 y Fq(\))1848 721 y Fi(1)1848 782 y Fm(n)p Fl(=0)2021 758 y Fq(is)h(based)h(on)f(a)f(virial)f (argumen)m(t)h(that)257 878 y(exploits)24 b(Iden)m(tit)m(y)i(\(I)s (I.7\))e(and)h(uses)h(the)f(b)s(oundedness)h(of)f Fn(h)p Fo(\036)2526 893 y Fm(n)2572 878 y Fn(j)p Fq(\()p Fo(x)2693 842 y Fl(2)2738 878 y Fn(\012)6 b Fc(1)p Fq(\))p Fo(\036)2973 893 y Fm(n)3020 878 y Fn(i)24 b Fq(uniformly)257 998 y(in)32 b Fo(n)p Fq(.)257 1202 y Fc(Theorem)49 b(I)s(I.10.)e Fj(Assume)d(Hyp)-5 b(othesis)45 b(H-1.)73 b(Then)42 b Fo(~)-48 b(x)q(\036)2558 1217 y Fm(n)2649 1202 y Fn(2)46 b Fq(dom)o(\()p Fo(H)3064 1217 y Fm(n)3111 1202 y Fq(\))p Fj(,)h(for)d(al)5 b(l)257 1322 y Fo(n)28 b Fn(2)g Fk(N)9 b Fj(,)1415 1443 y Fq(2)k Fo(~)-45 b(v)1528 1458 y Fm(n)1592 1443 y Fo(\036)1650 1458 y Fm(n)1759 1443 y Fq(=)62 b Fo(i)17 b(H)2036 1401 y Fl(+)2028 1467 y Fm(n)2110 1443 y Fo(~)-48 b(x)17 b(\036)2241 1458 y Fm(n)2310 1443 y Fo(;)885 b Fq(\(I)s(I.71\))257 1617 y Fj(and)42 b(ther)-5 b(e)42 b(exists)g(a)f(universal)h(c)-5 b(onstant)42 b Fo(C)1960 1632 y Fl(2)2040 1617 y Fo(<)f Fn(1)p Fj(,)i(dep)-5 b(ending)41 b(only)g(on)h Fo(V)22 b Fj(,)43 b(such)257 1737 y(that)1423 1858 y Fq(sup)1411 1937 y Fm(n)p Fi(2)p Ff(N)1544 1946 y Fe(0)1600 1758 y Fh(\015)1600 1808 y(\015)1600 1858 y(\015)n Fo(~)-47 b(x)22 b(\036)1781 1873 y Fm(n)1828 1758 y Fh(\015)1828 1808 y(\015)1828 1858 y(\015)1874 1912 y Fi(H)1934 1920 y Fa(n)2043 1858 y Fn(\024)63 b Fo(C)2253 1873 y Fl(2)2315 1858 y Fo(:)880 b Fq(\(I)s(I.72\))257 2115 y(W)-8 b(e)48 b(remark)e(that)h(actually)e(a)i(m)m(uc)m(h)g (stronger)g(estimate)f(holds.)86 b(Namely)-8 b(,)49 b(from)257 2235 y(a)40 b(Com)m(b)s(es-Thomas)f(argumen)m(t,)i(or)e(Agmon)g (estimate,)i(it)d(follo)m(ws)h(that)g Fo(e)3162 2199 y Fm(\015)t Fi(j)o Fm(~)-34 b(x)p Fi(j)3285 2235 y Fo(\036)3343 2250 y Fm(n)3430 2235 y Fq(is)257 2355 y(univ)m(ersally)41 b(b)s(ounded)i(in)d(norm,)j(for)f(some)f Fo(\015)48 b(>)43 b Fq(0.)70 b(F)-8 b(or)41 b(our)g(purp)s(ose,)k(ho)m(w)m(ev)m(er,)257 2476 y(Theorem)33 b(I)s(I.10)g(is)f(su\016cien)m(t.)44 b(Its)33 b(pro)s(of)f(is)g(giv)m(en)h(in)f(App)s(endix)h(A.)404 2596 y(In)38 b(the)g(pro)s(of)f(of)g(con)m(v)m(ergence)k(of)c(the)i (sequence)h(\()p Fo(P)2425 2611 y Fm(n)2472 2596 y Fq(\))2510 2560 y Fi(1)2510 2621 y Fm(n)p Fl(=0)2685 2596 y Fq(of)d(pro)5 b(jections)38 b(on)m(to)257 2717 y(the)33 b(ground)g(state,)g(w)m(e)h (mak)m(e)e(use)i(of)e(the)h(follo)m(wing)d(lemma.)257 2920 y Fc(Lemma)48 b(I)s(I.11.)e Fj(L)-5 b(et)44 b Fo(a;)17 b(b)44 b Fn(2)g(H)h Fj(b)-5 b(e)43 b(two)h(normalize)-5 b(d)42 b(ve)-5 b(ctors)43 b(in)g(a)g(Hilb)-5 b(ert)44 b(sp)-5 b(ac)g(e)257 3040 y Fn(H)q Fj(,)45 b(and)d(denote)g(the)g(c)-5 b(orr)g(esp)g(onding)41 b(r)-5 b(ank-one)42 b(pr)-5 b(oje)g(ctions)41 b(by)i Fo(P)2835 3055 y Fm(a)2918 3040 y Fq(:=)f Fn(j)p Fo(a)p Fn(ih)p Fo(a)p Fn(j)g Fj(and)257 3161 y Fo(P)320 3176 y Fm(b)382 3161 y Fq(:=)28 b Fn(j)p Fo(b)p Fn(ih)p Fo(b)p Fn(j)p Fj(.)44 b(Then)646 3405 y Fn(k)p Fo(P)759 3420 y Fm(a)822 3405 y Fn(\000)23 b Fo(P)985 3420 y Fm(b)1019 3405 y Fn(k)62 b Fq(=)1270 3300 y Fh(q)p 1353 3300 480 4 v 105 x Fq(1)22 b Fn(\000)g(jh)p Fo(a)p Fn(j)p Fo(b)p Fn(ij)1777 3376 y Fl(2)1895 3405 y Fq(=)2033 3300 y Fh(q)p 2116 3300 668 4 v 105 x Fn(h)p Fo(a)17 b Fn(j)p Fq(\()p Fo(P)2352 3420 y Fm(a)2415 3405 y Fn(\000)23 b Fo(P)2578 3420 y Fm(b)2612 3405 y Fq(\))28 b Fo(a)p Fn(i)38 b Fo(:)389 b Fq(\(I)s(I.73\))257 3625 y Fj(Pr)-5 b(o)g(of.)42 b Fq(W)-8 b(e)30 b(ma)m(y)f(assume)h Fn(f)p Fo(a;)17 b(b)p Fn(g)29 b Fq(to)g(b)s(e)h(linearly)d(indep)s(enden)m(t.)43 b Fo(P)2772 3640 y Fm(a)2829 3625 y Fn(\000)15 b Fo(P)2984 3640 y Fm(b)3049 3625 y Fq(is)29 b(a)g(rank-2)257 3746 y(op)s(erator)j(with)g(matrix)g(represen)m(tation)1340 4022 y Fo(M)71 b Fq(=)1641 3876 y Fh( )1862 3961 y Fq(1)196 b Fn(h)p Fo(a)p Fn(j)p Fo(b)p Fn(i)1749 4082 y(\000h)p Fo(b)p Fn(j)p Fo(a)p Fn(i)119 b(\000)p Fq(1)2346 3876 y Fh(!)3222 4022 y Fq(\(I)s(I.74\))257 4305 y(in)43 b(the)h(basis)g Fn(f)p Fo(a;)17 b(b)p Fn(g)p Fq(.)76 b(Since)44 b Fo(M)54 b Fq(is)43 b(traceless,)k(it)c(has)h(eigen)m(v)-5 b(alues)44 b Fn(\006)p Fo(\025)p Fq(,)i(for)d(some)257 4444 y Fo(\025)28 b(>)f Fq(0.)44 b(Hence,)34 b(\(I)s(I.73\))e(follo)m(ws)f(from)g Fn(k)p Fo(M)10 b Fn(k)29 b Fq(=)e Fn(j)p Fo(\025)p Fn(j)g Fq(=)2318 4344 y Fh(q)p 2401 4344 389 4 v 100 x Fn(\000)17 b Fq(det[)p Fo(M)10 b Fq(])q(.)p 3425 4444 4 66 v 3429 4382 59 4 v 3429 4444 V 3487 4444 4 66 v 257 4768 a Fc(Theorem)54 b(I)s(I.12.)49 b Fj(Assume)f(Hyp)-5 b(othesis)49 b(H-1)f(and)g Fo(g)2395 4717 y Fl(\(1\))2391 4793 y Fm(\013;\014)2555 4768 y Fq(=)53 b Fo(C)2754 4783 y Fl(1)2793 4768 y Fo(\013)2856 4732 y Fl(3)p Fm(=)p Fl(2)2966 4768 y Fo(\014)3027 4732 y Fl(1)p Fm(=)p Fl(2)3189 4768 y Fn(\024)g Fq(1)p Fo(=)p Fq(8)p Fj(.)257 4889 y(Then,)34 b(for)h(al)5 b(l)34 b Fo(n)28 b Fn(2)g Fk(N)1082 4904 y Fl(0)1128 4889 y Fj(,)1245 5109 y Fn(k)1318 5076 y Fh(e)1295 5109 y Fo(P)1358 5124 y Fm(n)1427 5109 y Fn(\000)23 b Fo(P)1590 5124 y Fm(n)p Fl(+1)1727 5109 y Fn(k)62 b(\024)h Fo(C)2049 5124 y Fl(3)2105 5109 y Fo(g)2156 5058 y Fl(\(1\))2152 5134 y Fm(\013;\014)2280 5109 y Fo(\014)2341 5067 y Fm(n=)p Fl(2)2480 5109 y Fo(;)715 b Fq(\(I)s(I.75\))257 5341 y Fj(wher)-5 b(e)35 b Fo(C)603 5356 y Fl(3)669 5341 y Fq(=)28 b Fo(\024)829 5304 y Fl(1)p Fm(=)p Fl(2)956 5341 y Fq(\()p Fo(C)1064 5356 y Fl(2)1125 5341 y Fq(+)22 b(1\))p Fj(.)p eop %%Page: 18 19 18 18 bop 257 266 a Fr(BFP-GrSt,)32 b(04-F)-8 b(eb-2005)2130 b Fq(18)404 573 y(By)38 b(Theorem)h(I)s(I.12,)g(the)f(sequence)j(\()p Fo(\036)1923 588 y Fm(n)1995 573 y Fn(\012)27 b Fq(\012)2169 537 y Fm(n)2169 597 y Fi(1)2244 573 y Fq(\))2282 537 y Fi(1)2282 597 y Fm(n)p Fl(=0)2457 573 y Fq(con)m(v)m(erges)40 b(to)e(the)g(ground)257 693 y(state)33 b Fo(\036)554 708 y Fl(gs)621 693 y Fq(.)257 886 y Fc(Corollary)52 b(I)s(I.13.)c Fj(Assume)f(Hyp)-5 b(othesis)46 b(H-1)h(and)f Fo(g)2409 835 y Fl(\(1\))2405 911 y Fm(\013;\014)2566 886 y Fq(=)j Fo(C)2761 901 y Fl(1)2800 886 y Fo(\013)2863 850 y Fl(3)p Fm(=)p Fl(2)2973 886 y Fo(\014)3034 850 y Fl(1)p Fm(=)p Fl(2)3192 886 y Fn(\024)h Fq(1)p Fo(=)p Fq(8)p Fj(.)257 1006 y(Then)c(the)h(gr)-5 b(oundstate)47 b(ener)-5 b(gy)46 b Fo(E)1622 1021 y Fl(gs)1739 1006 y Fq(=)j(inf)23 b Fo(\033)t Fq(\()p Fo(H)8 b Fq(\))46 b Fj(is)h(a)f(simple)g(eigenvalue)f(of)i Fo(H)257 1127 y Fj(c)-5 b(orr)g(esp)g(onding)34 b(to)h(a)f(gr)-5 b(ound)35 b(state)g(eigenve)-5 b(ctor)34 b Fo(\036)2196 1142 y Fl(gs)2291 1127 y Fn(2)28 b(H)36 b Fj(of)e Fo(H)8 b Fj(.)44 b(Mor)-5 b(e)g(over,)1007 1347 y Fq(1)22 b Fn(\000)g(jh)p Fo(\036)1302 1362 y Fl(gs)1369 1347 y Fn(j)p Fo(\036)1455 1362 y Fm(n)1523 1347 y Fn(\012)h Fq(\012)1693 1306 y Fm(n)1693 1372 y Fi(1)1768 1347 y Fn(ij)1835 1306 y Fl(2)1937 1347 y Fn(\024)63 b Fq(4)17 b Fo(C)2220 1306 y Fl(2)2213 1372 y(3)2275 1251 y Fh(\020)2325 1347 y Fo(g)2376 1296 y Fl(\(1\))2372 1372 y Fm(\013;\014)2483 1251 y Fh(\021)2533 1274 y Fl(2)2589 1347 y Fo(\014)2650 1306 y Fm(n)2719 1347 y Fo(;)476 b Fq(\(I)s(I.76\))257 1536 y Fj(for)35 b(al)5 b(l)35 b Fo(n)27 b Fn(2)i Fk(N)8 b Fj(.)257 1712 y(Pr)-5 b(o)g(of)35 b(of)g(The)-5 b(or)g(em)33 b(II.12.)42 b Fq(In)33 b(view)g(of)f(Lemma)f(I)s(I.11,)i(w)m(e)g(compute)462 1900 y Fn(h)514 1874 y Fq(~)501 1900 y Fo(\036)559 1915 y Fm(n)622 1900 y Fn(j)p Fq(\()711 1868 y Fh(e)688 1900 y Fo(P)751 1915 y Fm(n)820 1900 y Fn(\000)22 b Fo(P)982 1915 y Fm(n)p Fl(+1)1119 1900 y Fq(\))1192 1874 y(~)1179 1900 y Fo(\036)1237 1915 y Fm(n)1284 1900 y Fn(i)545 2090 y Fq(=)721 2023 y Fn(\000)p Fq(1)p 714 2067 141 4 v 714 2158 a(2)p Fo(\031)t(i)881 1973 y Fh(Z)927 2162 y Fl(\000)971 2171 y Fa(n)p Fe(+1)1112 1969 y Fh(\034)1187 2064 y Fq(~)1174 2090 y Fo(\036)1232 2105 y Fm(n)1306 1966 y Fh(\014)1306 2015 y(\014)1306 2065 y(\014)1306 2115 y(\014)1334 1969 y(\032)1608 2023 y Fq(1)p 1406 2067 452 4 v 1414 2139 a Fh(f)1406 2171 y Fo(H)1495 2143 y Fl(+)1487 2196 y Fm(n)1576 2171 y Fn(\000)h Fo(z)1721 2186 y Fm(n)p Fl(+1)1907 2090 y Fn(\000)2390 2023 y Fq(1)p 2033 2067 764 4 v 2033 2158 a Fo(H)2114 2173 y Fm(n)p Fl(+1)2273 2158 y Fn(\000)g Fo(E)2445 2173 y Fm(n)2514 2158 y Fn(\000)g Fo(z)2659 2173 y Fm(n)p Fl(+1)2806 1969 y Fh(\033)2881 2064 y Fq(~)2868 2090 y Fo(\036)2926 2105 y Fm(n)2973 1969 y Fh(\035)3058 2090 y Fo(dz)3154 2105 y Fm(n)p Fl(+1)545 2366 y Fq(=)760 2298 y(1)p 714 2343 141 4 v 714 2434 a(2)p Fo(\031)t(i)881 2249 y Fh(Z)927 2437 y Fl(\000)971 2446 y Fa(n)p Fe(+1)1122 2298 y Fo(dz)1218 2313 y Fm(n)p Fl(+1)p 1122 2343 234 4 v 1147 2434 a Fo(z)1192 2449 y Fm(n)p Fl(+1)1397 2245 y Fh(\034)1472 2340 y Fq(~)1460 2366 y Fo(\036)1518 2381 y Fm(n)1592 2241 y Fh(\014)1592 2291 y(\014)1592 2341 y(\014)1592 2391 y(\014)1987 2298 y Fq(1)p 1630 2343 764 4 v 1630 2434 a Fo(H)1711 2449 y Fm(n)p Fl(+1)1870 2434 y Fn(\000)f Fo(E)2041 2449 y Fm(n)2111 2434 y Fn(\000)g Fo(z)2255 2449 y Fm(n)p Fl(+1)2419 2366 y Fo(W)2525 2325 y Fm(n)2511 2390 y(n)p Fl(+1)2678 2340 y Fq(~)2665 2366 y Fo(\036)2723 2381 y Fm(n)2770 2245 y Fh(\035)545 2626 y Fq(=)721 2558 y Fn(\000)p Fq(1)p 714 2603 141 4 v 714 2694 a(2)p Fo(\031)t(i)881 2509 y Fh(Z)927 2697 y Fl(\000)971 2706 y Fa(n)p Fe(+1)1122 2558 y Fo(dz)1218 2573 y Fm(n)p Fl(+1)p 1122 2603 234 4 v 1147 2694 a Fo(z)1196 2660 y Fl(2)1192 2716 y Fm(n)p Fl(+1)1397 2505 y Fh(\034)1472 2600 y Fq(~)1460 2626 y Fo(\036)1518 2641 y Fm(n)1592 2501 y Fh(\014)1592 2551 y(\014)1592 2601 y(\014)1592 2651 y(\014)1647 2626 y Fo(W)1753 2585 y Fm(n)1739 2650 y(n)p Fl(+1)1906 2600 y Fq(~)1893 2626 y Fo(\036)1951 2641 y Fm(n)1998 2505 y Fh(\035)2110 2626 y Fq(+)1036 b(\(I)s(I.77\))760 2825 y(1)p 714 2869 141 4 v 714 2961 a(2)p Fo(\031)t(i)881 2776 y Fh(Z)927 2964 y Fl(\000)971 2973 y Fa(n)p Fe(+1)1122 2825 y Fo(dz)1218 2840 y Fm(n)p Fl(+1)p 1122 2869 234 4 v 1147 2961 a Fo(z)1196 2927 y Fl(2)1192 2983 y Fm(n)p Fl(+1)1397 2771 y Fh(\034)1460 2893 y Fo(W)1566 2852 y Fm(n)1552 2917 y(n)p Fl(+1)1718 2866 y Fq(~)1706 2893 y Fo(\036)1764 2908 y Fm(n)1838 2768 y Fh(\014)1838 2818 y(\014)1838 2868 y(\014)1838 2918 y(\014)2233 2825 y Fq(1)p 1876 2869 764 4 v 1876 2961 a Fo(H)1957 2976 y Fm(n)p Fl(+1)2116 2961 y Fn(\000)22 b Fo(E)2287 2976 y Fm(n)2357 2961 y Fn(\000)g Fo(z)2501 2976 y Fm(n)p Fl(+1)2665 2893 y Fo(W)2771 2852 y Fm(n)2757 2917 y(n)p Fl(+1)2924 2866 y Fq(~)2911 2893 y Fo(\036)2969 2908 y Fm(n)3016 2771 y Fh(\035)545 3159 y Fq(=)760 3092 y(1)p 714 3136 141 4 v 714 3228 a(2)p Fo(\031)t(i)881 3042 y Fh(Z)927 3231 y Fl(\000)971 3240 y Fa(n)p Fe(+1)1122 3092 y Fo(dz)1218 3107 y Fm(n)p Fl(+1)p 1122 3136 234 4 v 1147 3228 a Fo(z)1196 3194 y Fl(2)1192 3250 y Fm(n)p Fl(+1)1397 3038 y Fh(\034)1460 3159 y Fo(W)1566 3118 y Fm(n)1552 3184 y(n)p Fl(+1)1718 3133 y Fq(~)1706 3159 y Fo(\036)1764 3174 y Fm(n)1838 3035 y Fh(\014)1838 3085 y(\014)1838 3135 y(\014)1838 3184 y(\014)2233 3092 y Fq(1)p 1876 3136 764 4 v 1876 3228 a Fo(H)1957 3243 y Fm(n)p Fl(+1)2116 3228 y Fn(\000)g Fo(E)2287 3243 y Fm(n)2357 3228 y Fn(\000)g Fo(z)2501 3243 y Fm(n)p Fl(+1)2665 3159 y Fo(W)2771 3118 y Fm(n)2757 3184 y(n)p Fl(+1)2924 3133 y Fq(~)2911 3159 y Fo(\036)2969 3174 y Fm(n)3016 3038 y Fh(\035)3100 3159 y Fo(;)257 3408 y Fq(using)40 b(\(I)s(I.62\),)g (the)g(second)h(resolv)m(en)m(t)g(equation,)g(and)e(the)h(fact)g(that)f (the)h(con)m(tour)257 3528 y(in)m(tegral)612 3457 y Fh(R)652 3553 y Fl(\000)696 3562 y Fa(n)p Fe(+1)836 3528 y Fo(z)885 3492 y Fi(\000)p Fl(2)997 3528 y Fo(dz)32 b Fq(=)27 b(0)33 b(v)-5 b(anishes.)404 3672 y(Th)m(us,)34 b(with)924 3640 y Fh(e)900 3672 y Fo(Q)977 3687 y Fm(n)1052 3672 y Fq(:=)1205 3640 y Fh(e)1183 3672 y Fo(R)1258 3636 y Fi(\000)p Fl(1)1257 3697 y Fm(n)1380 3672 y Fq(=)1491 3640 y Fh(f)1483 3672 y Fo(H)1572 3636 y Fl(+)1564 3697 y Fm(n)1653 3672 y Fq(+)22 b Fo(\033)1806 3687 y Fm(n)p Fl(+1)1944 3672 y Fq(,)32 b(w)m(e)i(ha)m(v)m(e)352 3861 y Fn(jh)432 3835 y Fq(~)419 3861 y Fo(\036)477 3876 y Fm(n)540 3861 y Fn(j)p Fq(\()629 3828 y Fh(e)606 3861 y Fo(P)669 3876 y Fm(n)737 3861 y Fn(\000)23 b Fo(P)900 3876 y Fm(n)p Fl(+1)1037 3861 y Fq(\))1110 3835 y(~)1097 3861 y Fo(\036)1155 3876 y Fm(n)1202 3861 y Fn(ij)435 4035 y(\024)84 b Fo(\033)655 3994 y Fi(\000)p Fl(1)651 4060 y Fm(n)p Fl(+1)816 3935 y Fh(\015)816 3985 y(\015)816 4035 y(\015)885 4003 y(e)862 4035 y Fo(R)937 3994 y Fl(1)p Fm(=)p Fl(2)936 4060 y Fm(n)1064 4035 y Fo(W)1170 3994 y Fm(n)1156 4060 y(n)p Fl(+1)1322 4009 y Fq(~)1309 4035 y Fo(\036)1367 4050 y Fm(n)1414 3935 y Fh(\015)1414 3985 y(\015)1414 4035 y(\015)1460 3962 y Fl(2)1647 4035 y Fq(sup)1538 4114 y Fm(z)1571 4123 y Fa(n)p Fe(+1)1691 4114 y Fi(2)p Fl(\000)1782 4123 y Fa(n)p Fe(+1)1918 3935 y Fh(\015)1918 3985 y(\015)1918 4035 y(\015)1988 4003 y(e)1964 4035 y Fo(Q)2041 3994 y Fl(1)p Fm(=)p Fl(2)2041 4060 y Fm(n)2168 4035 y Fq(\()p Fo(H)2287 4050 y Fm(n)p Fl(+1)2446 4035 y Fn(\000)23 b Fo(E)2618 4050 y Fm(n)2687 4035 y Fn(\000)g Fo(z)2832 4050 y Fm(n)p Fl(+1)2969 4035 y Fq(\))3007 3994 y Fi(\000)p Fl(1)3142 4003 y Fh(e)3118 4035 y Fo(Q)3195 3994 y Fl(1)p Fm(=)p Fl(2)3195 4060 y Fm(n)3305 3935 y Fh(\015)3305 3985 y(\015)3305 4035 y(\015)f Fo(;)-178 b Fq(\(I)s(I.78\))257 4284 y(and)33 b(using)f(a)h(Neumann)f(series)h(expansion,)677 4440 y Fh(e)653 4472 y Fo(Q)730 4431 y Fl(1)p Fm(=)p Fl(2)730 4497 y Fm(n)857 4472 y Fq(\()p Fo(H)976 4487 y Fm(n)p Fl(+1)1135 4472 y Fn(\000)22 b Fo(E)1306 4487 y Fm(n)1376 4472 y Fn(\000)g Fo(z)1520 4487 y Fm(n)p Fl(+1)1658 4472 y Fq(\))1696 4431 y Fi(\000)p Fl(1)1830 4440 y Fh(e)1807 4472 y Fo(Q)1884 4431 y Fl(1)p Fm(=)p Fl(2)1884 4497 y Fm(n)736 4699 y Fq(=)924 4592 y Fi(1)899 4616 y Fh(X)895 4801 y Fm(k)r Fl(=0)1040 4578 y Fh(\022)1119 4600 y(f)1111 4632 y Fo(H)1200 4596 y Fl(+)1192 4657 y Fm(n)1281 4632 y Fq(+)g Fo(\033)1434 4647 y Fm(n)p Fl(+1)p 1111 4676 461 4 v 1124 4748 a Fh(f)1115 4781 y Fo(H)1204 4752 y Fl(+)1196 4805 y Fm(n)1285 4781 y Fn(\000)h Fo(z)1430 4796 y Fm(n)p Fl(+1)1582 4578 y Fh(\023)1659 4553 y(")1730 4667 y(e)1708 4699 y Fo(R)1783 4658 y Fl(1)p Fm(=)p Fl(2)1782 4724 y Fm(n)1909 4699 y Fo(W)2015 4658 y Fm(n)2001 4724 y(n)p Fl(+1)2178 4667 y Fh(e)2155 4699 y Fo(R)2230 4658 y Fl(1)p Fm(=)p Fl(2)2229 4724 y Fm(n)2357 4578 y Fh(\022)2436 4600 y(f)2428 4632 y Fo(H)2517 4596 y Fl(+)2509 4657 y Fm(n)2598 4632 y Fq(+)f Fo(\033)2751 4647 y Fm(n)p Fl(+1)p 2428 4676 V 2440 4748 a Fh(f)2432 4781 y Fo(H)2521 4752 y Fl(+)2513 4805 y Fm(n)2602 4781 y Fn(\000)h Fo(z)2747 4796 y Fm(n)p Fl(+1)2898 4578 y Fh(\023)2959 4553 y(#)3008 4576 y Fm(k)3073 4699 y Fo(;)122 b Fq(\(I)s(I.79\))257 4955 y(in)32 b(connection)h(with)f(Lemmata)f(I)s (I.4)h(and)h(I)s(I.8,)g(w)m(e)g(obtain)f(the)h(norm)f(b)s(ound)280 5096 y Fh(\015)280 5146 y(\015)280 5195 y(\015)350 5163 y(e)326 5195 y Fo(Q)403 5154 y Fl(1)p Fm(=)p Fl(2)403 5220 y Fm(n)530 5195 y Fq(\()p Fo(H)649 5210 y Fm(n)p Fl(+1)808 5195 y Fn(\000)22 b Fo(E)979 5210 y Fm(n)1049 5195 y Fn(\000)g Fo(z)1193 5210 y Fm(n)p Fl(+1)1331 5195 y Fq(\))1369 5154 y Fi(\000)p Fl(1)1503 5163 y Fh(e)1480 5195 y Fo(Q)1557 5154 y Fl(1)p Fm(=)p Fl(2)1557 5220 y Fm(n)1667 5096 y Fh(\015)1667 5146 y(\015)1667 5195 y(\015)60 b Fn(\024)h Fq(4)2005 5087 y Fi(1)1981 5112 y Fh(X)1976 5297 y Fm(k)r Fl(=0)2122 5099 y Fh(\020)2171 5195 y Fq(4)17 b Fo(g)2288 5144 y Fl(\(1\))2284 5220 y Fm(\013;\014)2395 5099 y Fh(\021)2445 5122 y Fm(k)2548 5195 y Fn(\024)61 b Fq(4)2780 5087 y Fi(1)2755 5112 y Fh(X)2751 5297 y Fm(k)r Fl(=0)2896 5074 y Fh(\022)2967 5128 y Fq(2)p 2967 5172 49 4 v 2967 5264 a(3)3026 5074 y Fh(\023)3087 5097 y Fm(k)3190 5195 y Fq(=)f(12)22 b Fo(:)3222 5388 y Fq(\(I)s(I.80\))p eop %%Page: 19 20 19 19 bop 257 266 a Fr(BFP-GrSt,)32 b(04-F)-8 b(eb-2005)2130 b Fq(19)257 573 y(Com)m(bining)31 b(Lemma)g(I)s(I.11,)i(Eq.)g(\(I)s (I.77\),)f(and)h(Eq.)h(\(I)s(I.80\),)e(w)m(e)h(obtain)f(the)h(b)s(ound) 1048 863 y Fn(k)1121 831 y Fh(e)1098 863 y Fo(P)1161 878 y Fm(n)1229 863 y Fn(\000)23 b Fo(P)1392 878 y Fm(n)p Fl(+1)1529 863 y Fn(k)60 b(\024)1793 714 y(p)p 1876 714 98 4 v 82 x Fq(12)p 1787 840 193 4 v 1787 951 a Fo(\033)1846 900 y Fl(1)p Fm(=)p Fl(2)1842 972 y Fm(n)p Fl(+1)2011 739 y Fh(\015)2011 788 y(\015)2011 838 y(\015)2011 888 y(\015)2080 831 y(e)2057 863 y Fo(R)2132 822 y Fl(1)p Fm(=)p Fl(2)2131 888 y Fm(n)2259 863 y Fo(W)2365 822 y Fm(n)2351 888 y(n)p Fl(+1)2518 837 y Fq(~)2505 863 y Fo(\036)2563 878 y Fm(n)2610 739 y Fh(\015)2610 788 y(\015)2610 838 y(\015)2610 888 y(\015)2678 863 y Fo(:)517 b Fq(\(I)s(I.81\))257 1180 y(App)s(ealing)26 b(to)h(\(I)s(I.28\),)h(w)m (e)h(write)1566 1147 y Fh(e)1544 1180 y Fo(R)1619 1143 y Fl(1)p Fm(=)p Fl(2)1618 1204 y Fm(n)1745 1180 y Fo(W)1851 1143 y Fm(n)1837 1204 y(n)p Fl(+1)1987 1153 y Fq(~)1975 1180 y Fo(\036)2033 1195 y Fm(n)2107 1180 y Fq(=)f Fo( )2278 1143 y Fl(\(1\))2383 1180 y Fq(+)12 b Fo( )2538 1143 y Fl(\(2\))2644 1180 y Fq(+)g(2)p Fo( )2848 1143 y Fl(\(3\))2953 1180 y Fq(+)g(2)p Fo( )3157 1143 y Fl(\(4\))3250 1180 y Fq(,)29 b(with)1047 1435 y Fo( )1114 1394 y Fl(\(1\))1292 1435 y Fq(:=)83 b Fo(\033)1537 1384 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)1533 1457 y Fm(n)p Fl(+1)1724 1336 y Fh(\015)1724 1385 y(\015)1724 1435 y(\015)1784 1410 y Fo(~)1770 1435 y(G)1847 1394 y Fm(n)1847 1460 y(n)p Fl(+1)1984 1336 y Fh(\015)1984 1385 y(\015)1984 1435 y(\015)2030 1362 y Fl(2)2099 1409 y Fq(~)2086 1435 y Fo(\036)2144 1450 y Fm(n)2213 1435 y Fo(;)982 b Fq(\(I)s(I.82\))1047 1614 y Fo( )1114 1573 y Fl(\(2\))1292 1614 y Fq(:=)1500 1582 y Fh(e)1478 1614 y Fo(R)1553 1573 y Fl(1)p Fm(=)p Fl(2)1552 1639 y Fm(n)1685 1614 y Fo(a)1736 1573 y Fi(\003)1776 1614 y Fq(\()1829 1589 y Fo(~)1814 1614 y(G)1891 1573 y Fm(n)1891 1639 y(n)p Fl(+1)2028 1614 y Fq(\))22 b Fn(\001)f Fo(a)2188 1573 y Fi(\003)2228 1614 y Fq(\()2281 1589 y Fo(~)2266 1614 y(G)2343 1573 y Fm(n)2343 1639 y(n)p Fl(+1)2480 1614 y Fq(\))2547 1588 y(~)2535 1614 y Fo(\036)2593 1629 y Fm(n)2662 1614 y Fo(;)533 b Fq(\(I)s(I.83\))1047 1775 y Fo( )1114 1734 y Fl(\(3\))1292 1775 y Fq(:=)1500 1743 y Fh(e)1478 1775 y Fo(R)1553 1734 y Fl(1)p Fm(=)p Fl(2)1552 1800 y Fm(n)1685 1775 y Fo(a)1736 1734 y Fi(\003)1776 1775 y Fq(\(\001)1910 1750 y Fo(~)1895 1775 y(G)1972 1734 y Fm(n)1972 1800 y(n)p Fl(+1)2109 1775 y Fq(\))22 b Fn(\001)c Fo(~)-45 b(v)2266 1790 y Fm(n)2342 1749 y Fq(~)2329 1775 y Fo(\036)2387 1790 y Fm(n)3222 1775 y Fq(\(I)s(I.84\))1305 1936 y(=)1500 1904 y Fh(e)1478 1936 y Fo(R)1553 1895 y Fl(1)p Fm(=)p Fl(2)1552 1961 y Fm(n)1681 1936 y Fo(~)g(v)1732 1951 y Fm(n)1801 1936 y Fn(\001)22 b Fo(a)1902 1895 y Fi(\003)1942 1936 y Fq(\(\001)2076 1911 y Fo(~)2061 1936 y(G)2138 1895 y Fm(n)2138 1961 y(n)p Fl(+1)2275 1936 y Fq(\))2342 1910 y(~)2330 1936 y Fo(\036)2388 1951 y Fm(n)2456 1936 y Fo(;)739 b Fq(\(I)s(I.85\))1047 2097 y Fo( )1114 2056 y Fl(\(4\))1292 2097 y Fq(:=)1500 2065 y Fh(e)1478 2097 y Fo(R)1553 2056 y Fl(1)p Fm(=)p Fl(2)1552 2122 y Fm(n)1685 2097 y Fo(a)1736 2056 y Fi(\003)1776 2097 y Fq(\()1829 2072 y Fo(~)1814 2097 y(G)1891 2056 y Fm(n)1886 2117 y(~)1891 2131 y Fl(0)p Fm(;n)p Fl(+1)2082 2097 y Fq(\))23 b Fn(\001)17 b Fo(~)-45 b(v)2239 2112 y Fm(n)2316 2071 y Fq(~)2303 2097 y Fo(\036)2361 2112 y Fm(n)2430 2097 y Fo(;)765 b Fq(\(I)s(I.86\))257 2348 y(where)34 b(\001)635 2323 y Fo(~)620 2348 y(G)697 2312 y Fm(n)697 2373 y(n)p Fl(+1)862 2348 y Fq(:=)1007 2323 y Fo(~)993 2348 y(G)1070 2312 y Fm(n)1070 2373 y(n)p Fl(+1)1229 2348 y Fn(\000)1343 2323 y Fo(~)1329 2348 y(G)1406 2312 y Fm(n)1401 2373 y(~)1406 2387 y Fl(0)p Fm(;n)p Fl(+1)1630 2348 y Fq(and)508 2651 y Fo(~)493 2676 y(G)570 2635 y Fm(n)565 2696 y(~)570 2710 y Fl(0)p Fm(;n)p Fl(+1)762 2676 y Fq(\()p Fo(k)s Fq(\))60 b(:=)1130 2651 y Fo(~)1115 2676 y(G)1192 2635 y Fm(n)1192 2701 y(n)p Fl(+1)1329 2676 y Fq(\()p Fo(k)s Fq(\))1459 2577 y Fh(\014)1459 2626 y(\014)1459 2676 y(\014)1486 2730 y Fm(~)-34 b(x)p Fl(=)1577 2716 y Fm(~)1582 2730 y Fl(0)1681 2676 y Fq(=)60 b Fn(\000)p Fo(\013)1957 2635 y Fl(3)p Fm(=)p Fl(2)2095 2515 y Fh(s)p 2178 2515 79 4 v 2193 2609 a Fq(2)p 2188 2653 59 4 v 2188 2745 a Fo(\031)2295 2609 y Fc(1)p Fq(\()p Fn(j)p Fo(k)s Fn(j)27 b Fo(<)g(\024)p Fq(\))p 2295 2653 429 4 v 2357 2745 a Fo(!)t Fq(\()p Fo(k)s Fq(\))2552 2716 y Fl(1)p Fm(=)p Fl(2)2760 2676 y Fo(~)-48 b(")p Fq(\()p Fo(k)s Fq(\))22 b Fo(:)236 b Fq(\(I)s(I.87\))257 2952 y(Eq.)34 b(\(I)s(I.85\))e(deriv)m(es)i(from)d (the)i(transv)m(ersalit)m(y)g(of)f(the)h(photon)g(p)s(olarization.)404 3072 y(The)g(con)m(tributions)f Fo( )1265 3036 y Fl(\(1\))1392 3072 y Fq(and)h Fo( )1649 3036 y Fl(\(2\))1776 3072 y Fq(can)f(easily)g(b)s(e)h(estimated)f(b)m(y)499 3331 y Fn(k)p Fo( )616 3290 y Fl(\(1\))710 3331 y Fn(k)83 b(\024)g Fo(\033)1062 3280 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)1058 3353 y Fm(n)p Fl(+1)1249 3232 y Fh(\015)1249 3282 y(\015)1249 3331 y(\015)1310 3306 y Fo(~)1295 3331 y(G)1372 3290 y Fm(n)1372 3356 y(n)p Fl(+1)1509 3232 y Fh(\015)1509 3282 y(\015)1509 3331 y(\015)1556 3258 y Fl(2)1655 3331 y Fn(\024)1803 3264 y Fq(1)p 1803 3308 49 4 v 1803 3400 a(5)1878 3331 y Fo(\013)1941 3290 y Fl(3)1997 3331 y Fo(\033)2056 3290 y Fl(2)2052 3356 y Fm(n)2116 3331 y Fo(\033)2175 3280 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)2171 3353 y Fm(n)p Fl(+1)2362 3331 y Fo(;)833 b Fq(\(I)s(I.88\))499 3549 y Fn(k)p Fo( )616 3508 y Fl(\(2\))710 3549 y Fn(k)83 b Fq(=)h Fo(\033)1062 3498 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)1058 3571 y Fm(n)p Fl(+1)1249 3449 y Fh(\015)1249 3499 y(\015)1249 3549 y(\015)p Fo(a)1346 3508 y Fi(\003)1386 3549 y Fq(\()1439 3524 y Fo(~)1424 3549 y(G)1501 3508 y Fm(n)1501 3574 y(n)p Fl(+1)1638 3549 y Fq(\))22 b Fn(\001)g Fo(a)1799 3508 y Fi(\003)1839 3549 y Fq(\()1892 3524 y Fo(~)1877 3549 y(G)1954 3508 y Fm(n)1954 3574 y(n)p Fl(+1)2091 3549 y Fq(\))17 b(\012)2216 3508 y Fm(n)2216 3574 y(n)p Fl(+1)2353 3449 y Fh(\015)2353 3499 y(\015)2353 3549 y(\015)60 b Fn(\024)h Fq(2)17 b Fo(\033)2722 3498 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)2718 3571 y Fm(n)p Fl(+1)2908 3449 y Fh(\015)2908 3499 y(\015)2908 3549 y(\015)2969 3524 y Fo(~)2954 3549 y(G)3031 3508 y Fm(n)3031 3574 y(n)p Fl(+1)3168 3449 y Fh(\015)3168 3499 y(\015)3168 3549 y(\015)3214 3476 y Fl(2)843 3764 y Fn(\024)1013 3696 y Fq(2)p 1013 3740 V 1013 3832 a(5)1088 3764 y Fo(\013)1151 3723 y Fl(3)1207 3764 y Fo(\033)1266 3723 y Fl(2)1262 3788 y Fm(n)1326 3764 y Fo(\033)1385 3713 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)1381 3785 y Fm(n)p Fl(+1)1572 3764 y Fo(:)1623 b Fq(\(I)s(I.89\))257 4026 y(T)-8 b(o)34 b(estimate)f Fo( )861 3990 y Fl(\(3\))955 4026 y Fq(,)i(w)m(e)f(recall)f(from)f(\(I)s(I.12\))i(the)g (de\014nition)f(of)g Fn(K)2755 3990 y Fm(n)2754 4051 y(n)p Fl(+1)2921 4026 y Fq(:=)d Fn(f)p Fo(k)i Fn(2)e Fk(R)3349 3990 y Fl(3)3418 4026 y Fn(\002)257 4146 y Fk(Z)326 4161 y Fl(2)363 4146 y Fn(j)p Fo(\033)446 4161 y Fm(n)p Fl(+1)611 4146 y Fn(\024)e Fo(!)t Fq(\()p Fo(k)s Fq(\))f Fo(<)h(\033)1097 4161 y Fm(n)1144 4146 y Fn(g)p Fq(.)43 b(W)-8 b(e)33 b(observ)m(e)i(that,)d(for)g Fo(k)f Fn(2)d(K)2420 4110 y Fm(n)2419 4171 y(n)p Fl(+1)2589 4146 y Fq(and)k(an)m(y)i Fo( )d Fn(2)d(H)3235 4161 y Fm(n)3283 4146 y Fq(,)722 4346 y Fh(e)699 4378 y Fo(R)774 4337 y Fl(1)p Fm(=)p Fl(2)773 4403 y Fm(n)906 4378 y Fo( )f Fn(\012)22 b Fo(a)1146 4337 y Fi(\003)1186 4378 y Fq(\()p Fo(k)s Fq(\)\012)1386 4337 y Fm(n)1386 4403 y(n)p Fl(+1)782 4566 y Fq(=)941 4470 y Fh(h)980 4566 y Fo(H)1069 4525 y Fl(+)1061 4591 y Fm(n)1150 4566 y Fn(\012)h Fc(1)f Fq(+)g Fc(1)g Fn(\012)h Fq(\()1667 4541 y(\024)1642 4566 y Fo(H)1731 4525 y Fm(n)1723 4591 y(n)p Fl(+1)1882 4566 y Fq(+)f Fo(\033)2035 4581 y Fm(n)p Fl(+1)2172 4566 y Fq(\))2210 4470 y Fh(i)2249 4493 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)2436 4566 y Fo( )27 b Fn(\012)22 b Fo(a)2676 4525 y Fi(\003)2716 4566 y Fq(\()p Fo(k)s Fq(\)\012)2916 4525 y Fm(n)2916 4591 y(n)p Fl(+1)782 4772 y Fq(=)941 4676 y Fh(\020)991 4772 y Fo(H)1080 4731 y Fl(+)1072 4797 y Fm(n)1160 4772 y Fq(+)g Fo(!)t Fq(\()p Fo(k)s Fq(\))g(+)g Fo(\033)1628 4787 y Fm(n)p Fl(+1)1765 4676 y Fh(\021)1815 4699 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)1980 4772 y Fo( )k Fn(\012)d Fo(a)2220 4731 y Fi(\003)2259 4772 y Fq(\()p Fo(k)s Fq(\))f(\012)2481 4731 y Fm(n)2481 4797 y(n)p Fl(+1)2641 4772 y Fo(:)554 b Fq(\(I)s(I.90\))257 5016 y(W)-8 b(riting)28 b Fo( )679 4980 y Fl(\(3\))802 5016 y Fq(as)h(an)g Fn(F)1122 5031 y Fm(n)p Fl(+1)1259 5016 y Fq(-v)-5 b(alued)28 b(function)h(of)g(the)g(electron)g(co)s (ordinate)f(and)h(using)p eop %%Page: 20 21 20 20 bop 257 266 a Fr(BFP-GrSt,)32 b(04-F)-8 b(eb-2005)2130 b Fq(20)257 573 y(\(I)s(I.90\),)33 b(w)m(e)g(hence)h(ha)m(v)m(e)563 747 y Fh(\015)563 797 y(\015)563 846 y(\015)p Fo( )676 805 y Fl(\(3\))771 846 y Fq(\()p Fo(x)p Fq(\))902 747 y Fh(\015)902 797 y(\015)902 846 y(\015)948 773 y Fl(2)948 900 y Fi(F)998 909 y Fa(n)p Fe(+1)1182 846 y Fq(=)1318 697 y Fh(\015)1318 747 y(\015)1318 797 y(\015)1318 846 y(\015)1318 896 y(\015)1381 729 y(Z)1481 846 y Fo(dk)1636 814 y Fh(e)1613 846 y Fo(R)1688 805 y Fl(1)p Fm(=)p Fl(2)1687 871 y Fm(n)1820 846 y Fq([)l Fo(~)-45 b(v)1894 861 y Fm(n)1964 846 y Fn(\001)22 b Fq(\001)2110 821 y Fo(~)2095 846 y(G)2172 805 y Fm(n)2172 871 y(n)p Fl(+1)2309 846 y Fq(\()p Fo(k)s Fq(\))17 b Fo(\036)2514 861 y Fm(n)2560 846 y Fq(]\()p Fo(x)p Fq(\))23 b Fn(\012)f Fo(a)2891 805 y Fi(\003)2931 846 y Fq(\()p Fo(k)s Fq(\)\012)3131 805 y Fm(n)3131 871 y(n)p Fl(+1)3269 697 y Fh(\015)3269 747 y(\015)3269 797 y(\015)3269 846 y(\015)3269 896 y(\015)3315 723 y Fl(2)3315 950 y Fi(F)3365 959 y Fa(n)p Fe(+1)647 1141 y Fq(=)807 1023 y Fh(Z)906 1141 y Fo(dk)1039 1016 y Fh(\015)1039 1066 y(\015)1039 1116 y(\015)1039 1165 y(\015)1085 1044 y(h\020)1174 1141 y Fo(H)1263 1099 y Fl(+)1255 1165 y Fm(n)1343 1141 y Fq(+)g Fo(!)t Fq(\()p Fo(k)s Fq(\))g(+)g Fo(\033)1811 1156 y Fm(n)p Fl(+1)1948 1044 y Fh(\021)1998 1067 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)2159 1141 y Fo(~)-45 b(v)2210 1156 y Fm(n)2279 1141 y Fn(\001)22 b Fq(\001)2425 1115 y Fo(~)2410 1141 y(G)2487 1099 y Fm(n)2487 1165 y(n)p Fl(+1)2624 1141 y Fq(\()p Fo(k)s Fq(\))17 b Fo(\036)2829 1156 y Fm(n)2876 1044 y Fh(i)2915 1141 y Fq(\()p Fo(x)p Fq(\))3046 1016 y Fh(\015)3046 1066 y(\015)3046 1116 y(\015)3046 1165 y(\015)3092 1042 y Fl(2)3092 1219 y Fi(F)3142 1227 y Fa(n)646 1402 y Fn(\024)84 b Fo(C)877 1417 y Fl(1)949 1285 y Fh(Z)1104 1335 y Fo(dk)p 1059 1379 195 4 v 1059 1470 a(!)t Fq(\()p Fo(k)s Fq(\))1291 1277 y Fh(\015)1291 1327 y(\015)1291 1377 y(\015)1291 1427 y(\015)1337 1306 y(h)1376 1402 y Fq(\001)1472 1377 y Fo(~)1457 1402 y(G)1534 1361 y Fm(n)1534 1427 y(n)p Fl(+1)1671 1402 y Fq(\()p Fo(k)s Fq(\))17 b Fo(\036)1876 1417 y Fm(n)1923 1306 y Fh(i)1962 1402 y Fq(\()p Fo(x)p Fq(\))2093 1277 y Fh(\015)2093 1327 y(\015)2093 1377 y(\015)2093 1427 y(\015)2139 1304 y Fl(2)2139 1481 y Fi(F)2189 1489 y Fa(n)2258 1402 y Fo(;)937 b Fq(\(I)s(I.91\))257 1653 y(where)25 b(the)f(last)e(inequalit)m(y)g(follo)m(ws)g(from)g(\(I) s(I.42\).)40 b(Next,)27 b(w)m(e)d(use)g(that,)h(as)f(a)f(function)257 1773 y(of)32 b(the)h(t)m(w)m(o)h(v)-5 b(ariables)29 b Fo(~)-47 b(x)28 b Fn(2)g Fk(R)1368 1737 y Fl(3)1446 1773 y Fq(and)33 b Fo(k)e Fn(2)d Fk(R)1878 1737 y Fl(3)1945 1773 y Fn(\002)23 b Fk(Z)2114 1788 y Fl(2)2151 1773 y Fq(,)257 1960 y Fh(\014)257 2009 y(\014)257 2059 y(\014)p Fq(\001)381 2034 y Fo(~)366 2059 y(G)443 2018 y Fm(n)443 2084 y(n)p Fl(+1)580 2059 y Fq(\()p Fo(k)s Fq(\))710 1960 y Fh(\014)710 2009 y(\014)710 2059 y(\014)55 b Fq(=)923 1882 y Fh(v)923 1928 y(u)923 1978 y(u)923 2028 y(t)p 1011 1882 290 4 v 1072 1992 a Fq(2)17 b Fo(\013)1201 1963 y Fl(3)p 1021 2036 270 4 v 1021 2128 a Fo(\031)j(!)t Fq(\()p Fo(k)s Fq(\))1317 1960 y Fh(\014)1317 2009 y(\014)1317 2059 y(\014)p Fq(1)11 b Fn(\000)g Fo(e)1538 2018 y Fi(\000)p Fm(i)1614 2001 y(~)1617 2018 y(k)q Fi(\001)o Fm(~)-34 b(x)1718 1960 y Fh(\014)1718 2009 y(\014)1718 2059 y(\014)54 b Fn(\024)1932 1893 y Fh(s)p 2015 1893 399 4 v 2025 1992 a Fq(2)17 b Fo(\013)2154 1963 y Fl(3)2209 1992 y Fo(!)t Fq(\()p Fo(k)s Fq(\))p 2025 2036 379 4 v 2185 2128 a Fo(\031)2424 2059 y Fn(j)n Fo(~)-47 b(x)p Fn(j)55 b Fq(=)f Fo(!)t Fq(\()p Fo(k)s Fq(\))2932 1960 y Fh(\014)2931 2009 y(\014)2931 2059 y(\014)2973 2034 y Fo(~)2958 2059 y(G)3035 2018 y Fm(n)3035 2084 y(n)p Fl(+1)3173 2059 y Fq(\()p Fo(k)s Fq(\))3303 1960 y Fh(\014)3303 2009 y(\014)3303 2059 y(\014)16 b Fn(j)n Fo(~)-47 b(x)p Fn(j)11 b Fo(:)3222 2260 y Fq(\(I)s(I.92\))257 2381 y(Therefore,)424 2515 y Fh(\015)424 2564 y(\015)424 2614 y(\015)p Fo( )537 2573 y Fl(\(3\))632 2515 y Fh(\015)632 2564 y(\015)632 2614 y(\015)678 2668 y Fi(H)738 2677 y Fa(n)p Fe(+1)922 2614 y Fn(\024)60 b Fo(C)1136 2563 y Fl(1)p Fm(=)p Fl(2)1129 2636 y(1)1263 2490 y Fh(\015)1263 2539 y(\015)1263 2589 y(\015)1263 2639 y(\015)1309 2614 y Fo(!)1374 2573 y Fl(1)p Fm(=)p Fl(2)1498 2589 y Fo(~)1483 2614 y(G)1560 2573 y Fm(n)1560 2639 y(n)p Fl(+1)1697 2490 y Fh(\015)1697 2539 y(\015)1697 2589 y(\015)1697 2639 y(\015)1766 2515 y(\015)1766 2564 y(\015)1766 2614 y(\015)n Fo(~)-47 b(x)17 b(\036)1942 2629 y Fm(n)1989 2515 y Fh(\015)1989 2564 y(\015)1989 2614 y(\015)60 b Fn(\024)2243 2547 y Fq(1)p 2243 2591 49 4 v 2243 2683 a(2)2318 2614 y Fo(C)2395 2563 y Fl(1)p Fm(=)p Fl(2)2388 2636 y(1)2521 2614 y Fo(C)2591 2629 y Fl(2)2647 2614 y Fo(\013)2710 2573 y Fl(3)p Fm(=)p Fl(2)2837 2614 y Fo(\033)2896 2573 y Fl(3)p Fm(=)p Fl(2)2892 2639 y Fm(n)3028 2614 y Fo(;)167 b Fq(\(I)s(I.93\))257 2846 y(where)34 b Fo(C)609 2861 y Fl(2)681 2846 y Fq(is)e(the)h(univ)m (ersal)f(constan)m(t)i(in)e(\(I)s(I.72\).)404 2976 y(It)i(remains)f(to) g(estimate)g Fo( )1455 2940 y Fl(\(4\))1550 2976 y Fq(,)h(our)g(k)m(ey) h(estimate.)47 b(W)-8 b(e)34 b(use)h(that)2962 2951 y Fo(~)2948 2976 y(G)3025 2940 y Fm(n)3020 3001 y(~)3025 3015 y Fl(0)p Fm(;n)p Fl(+1)3217 2976 y Fq(\()p Fo(k)s Fq(\),)f(as)257 3105 y(an)j(op)s(erator)f(on)h Fn(H)1018 3120 y Fm(el)1077 3105 y Fq(,)h(is)f(a)f(m)m(ultiple)f(of)h(the)h(iden) m(tit)m(y)-8 b(,)38 b(for)f(eac)m(h)g Fo(k)i Fn(2)c(K)3034 3069 y Fm(n)3033 3130 y(n)p Fl(+1)3170 3105 y Fq(.)57 b(Th)m(us,)257 3225 y(b)m(y)34 b(\(I)s(I.90\),)e(w)m(e)i(ha)m(v)m(e)307 3456 y Fo( )374 3415 y Fl(\(4\))528 3456 y Fq(=)663 3339 y Fh(Z)762 3456 y Fo(dk)909 3431 y(~)895 3456 y(G)972 3415 y Fm(n)967 3476 y(~)972 3490 y Fl(0)p Fm(;n)p Fl(+1)1164 3456 y Fq(\()p Fo(k)s Fq(\))19 b Fn(\001)1361 3360 y Fh(\020)1411 3456 y Fo(H)1500 3415 y Fl(+)1492 3481 y Fm(n)1578 3456 y Fq(+)h Fo(!)t Fq(\()p Fo(k)s Fq(\))f(+)h Fo(\033)2039 3471 y Fm(n)p Fl(+1)2176 3360 y Fh(\021)2226 3383 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)2387 3456 y Fo(~)-45 b(v)2438 3471 y Fm(n)2502 3456 y Fo(\036)2560 3471 y Fm(n)2626 3456 y Fn(\012)21 b Fo(a)2775 3415 y Fi(\003)2814 3456 y Fq(\()p Fo(k)s Fq(\))f(\012)3034 3415 y Fm(n)3034 3481 y(n)p Fl(+1)3222 3456 y Fq(\(I)s(I.94\))257 3675 y(and)33 b(then)565 3799 y Fh(\015)565 3849 y(\015)565 3899 y(\015)p Fo( )678 3858 y Fl(\(4\))772 3799 y Fh(\015)772 3849 y(\015)772 3899 y(\015)818 3826 y Fl(2)818 3953 y Fi(H)878 3962 y Fa(n)p Fe(+1)1062 3899 y Fn(\024)1200 3782 y Fh(Z)1299 3899 y Fo(dk)1421 3799 y Fh(\015)1421 3849 y(\015)1421 3899 y(\015)1481 3874 y Fo(~)1467 3899 y(G)1544 3858 y Fm(n)1544 3924 y(n)p Fl(+1)1681 3899 y Fq(\()p Fo(k)s Fq(\))1811 3799 y Fh(\015)1811 3849 y(\015)1811 3899 y(\015)1857 3826 y Fl(2)1919 3774 y Fh(\015)1919 3824 y(\015)1919 3874 y(\015)1919 3924 y(\015)1965 3803 y(\020)2014 3899 y Fo(H)2103 3858 y Fl(+)2095 3924 y Fm(n)2184 3899 y Fq(+)22 b Fo(!)t Fq(\()p Fo(k)s Fq(\))g(+)g Fo(\033)2652 3914 y Fm(n)p Fl(+1)2789 3803 y Fh(\021)2839 3826 y Fi(\000)p Fl(1)p Fm(=)p Fl(2)3000 3899 y Fo(~)-45 b(v)3051 3914 y Fm(n)3114 3899 y Fo(\036)3172 3914 y Fm(n)3219 3774 y Fh(\015)3219 3824 y(\015)3219 3874 y(\015)3219 3924 y(\015)3265 3801 y Fl(2)3265 3978 y Fi(H)3325 3986 y Fa(n)562 4144 y Fn(\024)731 4027 y Fh(Z)830 4144 y Fo(dk)963 4044 y Fh(\015)963 4094 y(\015)963 4144 y(\015)1023 4119 y Fo(~)1009 4144 y(G)1086 4103 y Fm(n)1086 4169 y(n)p Fl(+1)1223 4144 y Fq(\()p Fo(k)s Fq(\))1353 4044 y Fh(\015)1353 4094 y(\015)1353 4144 y(\015)1399 4071 y Fl(2)1461 4044 y Fh(\015)1461 4094 y(\015)1461 4144 y(\015)l Fo(~)g(v)1554 4159 y Fm(n)1617 4144 y Fo(\036)1675 4159 y Fm(n)1722 4044 y Fh(\015)1722 4094 y(\015)1722 4144 y(\015)1790 4019 y(\015)1790 4069 y(\015)1790 4119 y(\015)1790 4169 y(\015)1837 4048 y(\020)1886 4144 y Fo(H)1975 4103 y Fl(+)1967 4169 y Fm(n)2056 4144 y Fq(+)22 b Fo(!)t Fq(\()p Fo(k)s Fq(\))f(+)h Fo(\033)2523 4159 y Fm(n)p Fl(+1)2661 4048 y Fh(\021)2710 4071 y Fi(\000)p Fl(1)2801 4144 y Fo(~)-45 b(v)2852 4159 y Fm(n)2916 4144 y Fo(\036)2974 4159 y Fm(n)3020 4019 y Fh(\015)3020 4069 y(\015)3020 4119 y(\015)3020 4169 y(\015)3066 4144 y Fo(;)129 b Fq(\(I)s(I.95\))257 4370 y(where)39 b(the)e(last)g(estimate) f(follo)m(ws)g(from)g(Sc)m(h)m(w)m(arz')j(inequalit)m(y)-8 b(.)56 b(W)-8 b(e)37 b(observ)m(e)i(that,)257 4491 y(b)m(y)34 b(\(I)s(I.71\),)307 4618 y Fh(\015)307 4667 y(\015)307 4717 y(\015)307 4767 y(\015)353 4646 y(\020)403 4742 y Fo(H)492 4701 y Fl(+)484 4767 y Fm(n)551 4742 y Fq(+)q Fo(!)t Fq(\()p Fo(k)s Fq(\))q(+)q Fo(\033)956 4757 y Fm(n)p Fl(+1)1092 4646 y Fh(\021)1142 4669 y Fi(\000)p Fl(1)1232 4742 y Fo(~)-45 b(v)1283 4757 y Fm(n)1347 4742 y Fo(\036)1405 4757 y Fm(n)1451 4618 y Fh(\015)1451 4667 y(\015)1451 4717 y(\015)1451 4767 y(\015)1547 4742 y Fn(\024)1674 4618 y Fh(\015)1674 4667 y(\015)1674 4717 y(\015)1674 4767 y(\015)2044 4675 y Fo(H)2133 4639 y Fl(+)2125 4699 y Fm(n)p 1730 4719 775 4 v 1730 4810 a Fo(H)1819 4782 y Fl(+)1811 4835 y Fm(n)1900 4810 y Fq(+)22 b Fo(!)t Fq(\()p Fo(k)s Fq(\))g(+)g Fo(\033)2368 4825 y Fm(n)p Fl(+1)2515 4618 y Fh(\015)2515 4667 y(\015)2515 4717 y(\015)2515 4767 y(\015)2589 4642 y(\015)2589 4692 y(\015)2589 4742 y(\015)n Fo(~)-47 b(x)17 b(\036)2765 4757 y Fm(n)2812 4642 y Fh(\015)2812 4692 y(\015)2812 4742 y(\015)50 b Fn(\024)g Fo(C)3105 4757 y Fl(2)3145 4742 y Fo(;)g Fq(\(I)s(I.96\))257 4993 y(with)39 b Fo(C)556 5008 y Fl(2)635 4993 y Fq(b)s(eing)g(the)h(univ)m(ersal)f(constan)m(t)h (in)f(\(I)s(I.72\).)63 b(Inserting)40 b(Estimate)f(\(I)s(I.96\))257 5127 y(and)33 b(using)f(that)h Fn(k)l Fo(~)-45 b(v)1011 5142 y Fm(n)1074 5127 y Fo(\036)1132 5142 y Fm(n)1179 5127 y Fn(k)27 b(\024)i Fo(\033)1421 5076 y Fl(1)p Fm(=)p Fl(2)1417 5148 y Fm(n)p Fl(+1)1554 5127 y Fn(k)l Fo(~)-45 b(v)1651 5142 y Fm(n)1737 5094 y Fh(e)1715 5127 y Fo(R)1790 5091 y Fl(1)p Fm(=)p Fl(2)1789 5151 y Fm(n)1900 5127 y Fn(k)27 b(\024)2082 5050 y(p)p 2165 5050 110 4 v 77 x Fo(C)2235 5142 y Fl(1)2275 5127 y Fq(,)32 b(w)m(e)i(arriv)m(e)e(at) 563 5270 y Fh(\015)563 5320 y(\015)563 5370 y(\015)p Fo( )676 5328 y Fl(\(4\))770 5270 y Fh(\015)770 5320 y(\015)770 5370 y(\015)60 b Fn(\024)h Fo(C)1091 5319 y Fl(1)p Fm(=)p Fl(4)1084 5391 y(1)1218 5370 y Fo(C)1295 5319 y Fl(1)p Fm(=)p Fl(2)1288 5391 y(2)1427 5270 y Fh(\015)1427 5320 y(\015)1427 5370 y(\015)1487 5344 y Fo(~)1473 5370 y(G)1550 5328 y Fm(n)1550 5394 y(n)p Fl(+1)1687 5370 y Fq(\()p Fo(k)s Fq(\))1817 5270 y Fh(\015)1817 5320 y(\015)1817 5370 y(\015)82 b Fn(\024)2093 5302 y Fq(1)p 2093 5346 49 4 v 2093 5438 a(2)2168 5370 y Fo(C)2245 5319 y Fl(1)p Fm(=)p Fl(4)2238 5391 y(1)2372 5370 y Fo(C)2449 5319 y Fl(1)p Fm(=)p Fl(2)2442 5391 y(2)2575 5370 y Fo(\013)2638 5328 y Fl(3)p Fm(=)p Fl(2)2765 5370 y Fo(\033)2820 5385 y Fm(n)2889 5370 y Fo(:)306 b Fq(\(I)s(I.97\))p eop %%Page: 21 22 21 21 bop 257 266 a Fr(BFP-GrSt,)32 b(04-F)-8 b(eb-2005)2130 b Fq(21)257 573 y(Adding)32 b(up)h(the)g(estimates)g(\(I)s(I.88\),)f (\(I)s(I.89\),)g(\(I)s(I.93\),)g(and)h(\(I)s(I.97\),)f(w)m(e)i(obtain) 350 831 y Fn(k)423 799 y Fh(e)400 831 y Fo(P)463 846 y Fm(n)532 831 y Fn(\000)22 b Fo(P)694 846 y Fm(n)p Fl(+1)831 831 y Fn(k)83 b(\024)1141 682 y(p)p 1224 682 98 4 v 82 x Fq(12)p 1135 808 193 4 v 1135 919 a Fo(\033)1194 868 y Fl(1)p Fm(=)p Fl(2)1190 940 y Fm(n)p Fl(+1)1337 710 y Fh(\024)1391 764 y Fq(3)p 1391 808 49 4 v 1391 899 a(5)1476 764 y Fo(\013)1539 728 y Fl(3)1595 764 y Fo(\033)1654 728 y Fl(2)1650 788 y Fm(n)p 1476 808 222 4 v 1491 919 a Fo(\033)1550 868 y Fl(1)p Fm(=)p Fl(2)1546 940 y Fm(n)p Fl(+1)1730 831 y Fq(+)1838 764 y Fo(C)1915 713 y Fl(1)p Fm(=)p Fl(2)1908 785 y(1)2025 764 y Fo(C)2095 779 y Fl(2)p 1838 808 297 4 v 1961 899 a Fq(2)2160 831 y Fo(\013)2223 790 y Fl(3)p Fm(=)p Fl(2)2350 831 y Fo(\033)2409 790 y Fl(3)p Fm(=)p Fl(2)2405 856 y Fm(n)2541 831 y Fq(+)2649 764 y Fo(C)2726 713 y Fl(1)p Fm(=)p Fl(4)2719 785 y(1)2836 764 y Fo(C)2913 713 y Fl(1)p Fm(=)p Fl(2)2906 785 y(2)p 2649 808 374 4 v 2811 899 a Fq(2)3049 831 y Fo(\013)3112 790 y Fl(3)p Fm(=)p Fl(2)3239 831 y Fo(\033)3294 846 y Fm(n)3341 710 y Fh(\025)964 1149 y Fn(\024)1135 1081 y Fq(3)1201 999 y Fn(p)p 1283 999 98 4 v 1283 1081 a Fq(12)p 1135 1125 246 4 v 1233 1217 a(5)1407 1027 y Fh(\024)1461 1081 y Fo(\013)1524 1045 y Fl(3)1580 1081 y Fo(\033)1635 1096 y Fm(n)p 1461 1125 222 4 v 1541 1217 a Fo(\014)1714 1149 y Fq(+)1822 1081 y Fo(C)1899 1030 y Fl(1)p Fm(=)p Fl(2)1892 1103 y(1)2026 1081 y Fo(C)2096 1096 y Fl(2)2152 1081 y Fo(\013)2215 1045 y Fl(3)p Fm(=)p Fl(2)2341 1081 y Fo(\033)2396 1096 y Fm(n)p 1822 1125 622 4 v 2047 1217 a Fo(\014)2108 1188 y Fl(1)p Fm(=)p Fl(2)2475 1149 y Fq(+)2583 1081 y Fo(C)2660 1030 y Fl(1)p Fm(=)p Fl(4)2653 1103 y(1)2787 1081 y Fo(C)2864 1030 y Fl(1)p Fm(=)p Fl(2)2857 1103 y(2)2991 1081 y Fo(\013)3054 1045 y Fl(3)p Fm(=)p Fl(2)3180 1081 y Fo(\033)3239 1045 y Fl(1)p Fm(=)p Fl(2)3235 1106 y Fm(n)p 2583 1125 766 4 v 2881 1217 a Fo(\014)2942 1188 y Fl(1)p Fm(=)p Fl(2)3359 1027 y Fh(\025)964 1427 y Fn(\024)1135 1359 y Fq(3)1201 1277 y Fn(p)p 1283 1277 98 4 v 1283 1359 a Fq(12)p 1135 1404 246 4 v 1233 1495 a(5)1417 1359 y Fo(\013)1480 1323 y Fl(3)p Fm(=)p Fl(2)1607 1359 y Fo(\033)1666 1323 y Fl(1)p Fm(=)p Fl(2)1662 1384 y Fm(n)p 1417 1404 359 4 v 1511 1495 a Fo(\014)1572 1466 y Fl(1)p Fm(=)p Fl(2)1802 1331 y Fh(\020)1852 1427 y Fq(1)22 b(+)g Fo(C)2098 1376 y Fl(1)p Fm(=)p Fl(4)2091 1449 y(1)2208 1427 y Fo(C)2285 1376 y Fl(1)p Fm(=)p Fl(2)2278 1449 y(2)2417 1427 y Fq(+)g Fo(C)2592 1376 y Fl(1)p Fm(=)p Fl(2)2585 1449 y(1)2702 1427 y Fo(C)2772 1442 y Fl(2)2811 1331 y Fh(\021)964 1701 y Fn(\024)84 b Fq(3)17 b(\()p Fo(C)1299 1716 y Fl(2)1359 1701 y Fq(+)22 b(1\))1571 1634 y Fo(C)1641 1649 y Fl(1)1697 1634 y Fo(\013)1760 1598 y Fl(3)p Fm(=)p Fl(2)p 1571 1678 299 4 v 1635 1769 a Fo(\014)1696 1741 y Fl(1)p Fm(=)p Fl(2)1896 1701 y Fo(\014)1957 1660 y Fm(n=)p Fl(2)2096 1701 y Fo(;)1099 b Fq(\(I)s(I.98\))257 1923 y(where)35 b(w)m(e)f(mak)m(e)f(use)h(of)f Fo(C)1290 1938 y Fl(1)1358 1923 y Fn(\025)c Fq(8)p Fo(\024)f Fn(\025)h Fq(8)k(to)g(deriv)m(e)g(the)h(last)e(inequalit)m(y)-8 b(.)44 b(This)34 b(yields)257 2044 y(\(I)s(I.75\).)p 3425 2044 4 66 v 3429 1981 59 4 v 3429 2044 V 3487 2044 4 66 v 257 2284 a Fc(Ac)m(kno)m(wledgemen)m(t:)50 b Fq(W)-8 b(e)37 b(thank)g(M.)g(Shoufan)g(and)g(M.)g(K\177)-49 b(onen)m(b)s(erg)37 b(for)f(careful)257 2405 y(pro)s(ofreading)23 b(of)i(the)g(man)m(uscript.)40 b(Supp)s(ort)25 b(b)m(y)h(the)f(EU)g (net)m(w)m(ork)h(R)-8 b(TN)25 b(2-2001-065)257 2525 y(is)32 b(gratefully)f(ac)m(kno)m(wledged.)257 2852 y Fp(A)162 b(Pro)t(ofs)53 b(of)h(Lemma)f(I)t(I.3)h(and)g(Theorem)e(I)t(I.10)257 3071 y Fc(Pro)s(of)38 b(of)g(Lemma)g(I)s(I.3.)44 b Fq(W)-8 b(e)34 b(\014rst)f(note)g(that)g(\(I)s(I.43\))g(follo)m(ws)e(from)h (\(I)s(I.42\))h(with)257 3191 y Fo(\032)28 b Fq(:=)g Fo(\033)521 3206 y Fm(n)p Fl(+1)659 3191 y Fq(.)39 b(T)-8 b(o)22 b(pro)m(v)m(e)i(\(I)s(I.42\),)f(w)m(e)g(observ)m(e)h(that)d (Lemma)g(I)s(I.2)h(implies)d(the)j(quadratic)257 3311 y(form)32 b(estimate)274 3392 y Fh(\020)324 3489 y Fo(a)375 3448 y Fi(\003)415 3489 y Fq(\()468 3464 y Fo(~)453 3489 y(G)530 3504 y Fm(n)577 3489 y Fq(\))22 b(+)g Fo(a)p Fq(\()839 3464 y Fo(~)824 3489 y(G)901 3504 y Fm(n)948 3489 y Fq(\))986 3392 y Fh(\021)1035 3415 y Fl(2)1135 3489 y Fn(\024)61 b Fq(4)1339 3389 y Fh(\015)1338 3439 y(\015)1338 3489 y(\015)p Fq(\()p Fo(\024)1478 3448 y Fi(\000)p Fl(1)1595 3489 y Fq(+)22 b Fo(!)1758 3448 y Fi(\000)p Fl(1)1851 3489 y Fq(\))1889 3448 y Fl(1)p Fm(=)p Fl(2)2014 3464 y Fo(~)1999 3489 y(G)2076 3504 y Fm(n)2123 3389 y Fh(\015)2123 3439 y(\015)2123 3489 y(\015)2169 3415 y Fl(2)2231 3489 y Fq(\()2294 3464 y(\024)2269 3489 y Fo(H)2350 3504 y Fm(n)2419 3489 y Fq(+)g Fo(\024)p Fq(\))60 b Fn(\024)h Fq(8)p Fo(\024)17 b(\013)2994 3448 y Fl(3)3049 3489 y Fq(\()3112 3464 y(\024)3087 3489 y Fo(H)3168 3504 y Fm(n)3237 3489 y Fq(+)22 b Fo(\024)p Fq(\))g Fo(:)3270 3622 y Fq(\(A.1\))257 3742 y(This)33 b(implies)d(that)307 3943 y Fn(\000)p Fq(\001)59 b Fn(\024)g Fq(2)17 b(\()l Fo(~)-45 b(v)811 3958 y Fm(n)857 3943 y Fq(\))895 3902 y Fl(2)952 3943 y Fq(+)18 b(2)1112 3846 y Fh(\020)1161 3943 y Fo(a)1212 3902 y Fi(\003)1251 3943 y Fq(\()1304 3917 y Fo(~)1289 3943 y(G)1366 3958 y Fm(n)1413 3943 y Fq(\))g(+)f Fo(a)p Fq(\()1666 3917 y Fo(~)1651 3943 y(G)1728 3958 y Fm(n)1776 3943 y Fq(\))1814 3846 y Fh(\021)1863 3869 y Fl(2)1961 3943 y Fn(\024)58 b Fq(2)17 b(\()l Fo(~)-45 b(v)2247 3958 y Fm(n)2294 3943 y Fq(\))2332 3902 y Fl(2)2405 3943 y Fq(+)34 b(16)p Fo(\024)17 b(\013)2749 3902 y Fl(3)2805 3943 y Fq(\()2868 3917 y(\024)2843 3943 y Fo(H)2924 3958 y Fm(n)2988 3943 y Fq(+)h Fo(\024)p Fq(\))f Fo(:)50 b Fq(\(A.2\))257 4120 y(Since)33 b Fo(V)54 b Fq(is)32 b(a)h(small)d(p)s(erturbation)i(relativ)m(e)f(to)i Fn(\000)p Fq(\001,)g(w)m(e)h(ha)m(v)m(e)g(that)1020 4297 y Fo(V)82 b Fn(\024)60 b Fq(2)p Fo(")p Fq(\()l Fo(~)-45 b(v)1476 4312 y Fm(n)1523 4297 y Fq(\))1561 4256 y Fl(2)1639 4297 y Fq(+)39 b(16)p Fo(\024"\013)2017 4256 y Fl(3)2072 4297 y Fq(\()2135 4272 y(\024)2110 4297 y Fo(H)2191 4312 y Fm(n)2260 4297 y Fq(+)22 b Fo(\024)p Fq(\))38 b(+)h Fo(b)2646 4312 y Fm(")2705 4297 y Fo(;)538 b Fq(\(A.3\))257 4475 y(for)32 b(an)m(y)i Fo(")27 b(>)h Fq(0.)43 b(As)33 b(a)f(consequence,)k(w)m(e)e(ha)m(v)m(e,)g(for)e(an)m(y)h Fo(";)17 b(\026)27 b(>)g Fq(0,)486 4652 y(\()l Fo(~)-45 b(v)571 4667 y Fm(n)618 4652 y Fq(\))656 4611 y Fl(2)755 4652 y Fn(\024)61 b Fq(\(1)22 b(+)g(2)p Fo("\026)p Fq(\)\()l Fo(~)-45 b(v)1377 4667 y Fm(n)1423 4652 y Fq(\))1461 4611 y Fl(2)1539 4652 y Fn(\000)39 b Fo(\026V)60 b Fq(+)39 b(16)p Fo(\024"\026\013)2268 4611 y Fl(3)2323 4652 y Fq(\()2386 4627 y(\024)2361 4652 y Fo(H)2442 4667 y Fm(n)2510 4652 y Fq(+)23 b Fo(\024)p Fq(\))38 b(+)h Fo(\026b)2956 4667 y Fm(")3015 4652 y Fo(:)228 b Fq(\(A.4\))257 4829 y(W)-8 b(e)41 b(c)m(ho)s(ose)h Fo(")f Fq(:=)g(1)p Fo(=)p Fq(8,)h Fo(\026)f Fq(:=)g(2)f(and)h(observ)m(e)h(that)f(then)g(16)p Fo(\024"\026\013)2856 4793 y Fl(3)2935 4829 y Fq(=)g(4)p Fo(\024\013)3220 4793 y Fl(3)3301 4829 y Fn(\024)g Fq(4,)257 4949 y(since)33 b Fo(\013)c Fn(\024)f Fo(\024)748 4913 y Fi(\000)p Fl(1)p Fm(=)p Fl(3)913 4949 y Fq(,)33 b(whic)m(h)g(yields) 595 5127 y(\()l Fo(~)-45 b(v)680 5142 y Fm(n)727 5127 y Fq(\))765 5086 y Fl(2)888 5127 y Fn(\024)83 b Fq(4)1097 5030 y Fh(h)1136 5127 y Fq(\()l Fo(~)-45 b(v)1221 5142 y Fm(n)1268 5127 y Fq(\))1306 5086 y Fl(2)1368 5127 y Fn(\000)22 b Fo(V)44 b Fq(+)1691 5102 y(\024)1666 5127 y Fo(H)1747 5142 y Fm(n)1816 5127 y Fq(+)22 b Fo(\024)g Fq(+)g Fo(b)2131 5142 y Fl(1)p Fm(=)p Fl(8)2242 5030 y Fh(i)2341 5127 y Fn(\024)61 b Fq(4[)p Fo(H)2636 5142 y Fm(n)2705 5127 y Fq(+)22 b Fo(\024)g Fq(+)g Fo(b)3020 5142 y Fl(1)p Fm(=)p Fl(8)3130 5127 y Fq(])888 5350 y Fn(\024)1058 5283 y Fo(C)1135 5247 y Fi(0)1128 5307 y Fl(1)p 1058 5327 110 4 v 1088 5419 a Fo(\032)1177 5350 y Fq(\()p Fo(H)1296 5365 y Fm(n)1365 5350 y Fn(\000)h Fo(E)1537 5365 y Fm(n)1606 5350 y Fq(+)f Fo(\032)p Fq(\))g Fo(;)1429 b Fq(\(A.5\))p eop %%Page: 22 23 22 22 bop 257 266 a Fr(BFP-GrSt,)32 b(04-F)-8 b(eb-2005)2130 b Fq(22)257 573 y(where)34 b(the)f(constan)m(t)h Fo(C)1178 537 y Fi(0)1171 597 y Fl(1)1242 573 y Fq(is)e(giv)m(en)h(b)m(y)321 843 y Fo(C)398 802 y Fi(0)391 868 y Fl(1)490 843 y Fq(:=)60 b(4)17 b Fo(\032)39 b Fq(sup)819 921 y Fm(r)r Fi(\025)p Fl(0)971 722 y Fh(\032)1043 775 y Fo(r)25 b Fq(+)d Fo(E)1282 790 y Fm(n)1351 775 y Fq(+)g Fo(\024)h Fq(+)f Fo(b)1667 790 y Fl(1)p Fm(=)p Fl(8)p 1043 820 734 4 v 1302 912 a Fo(r)j Fq(+)d Fo(\032)1787 722 y Fh(\033)1910 843 y Fq(=)60 b(4)33 b(max)2326 747 y Fh(n)2381 843 y Fo(E)2453 858 y Fm(n)2523 843 y Fq(+)22 b Fo(\024)g Fq(+)g Fo(b)2838 859 y Fl(1)p Fm(=)p Fl(8)2976 843 y Fo(;)49 b(\032)3102 747 y Fh(o)3180 843 y Fo(:)63 b Fq(\(A.6\))257 1117 y(Since)48 b(the)g(groundstate)g(energy)h Fo(e)1634 1132 y Fm(el)1740 1117 y Fq(of)e Fo(H)1947 1132 y Fm(el)2054 1117 y Fq(is)g(strictly)g (negativ)m(e,)k Fo(e)2991 1132 y Fm(el)3103 1117 y Fo(<)j Fq(0,)d(the)257 1238 y(v)-5 b(ariational)26 b(principle)h(with)i(trial) d(v)m(ector)k Fo(')1915 1253 y Fm(el)1989 1238 y Fn(\012)15 b Fq(\012)2151 1253 y Fm(n)2198 1238 y Fq(,)30 b(where)g Fo(')2597 1253 y Fm(el)2683 1238 y Fn(2)e(H)2861 1253 y Fm(el)2949 1238 y Fq(is)h(the)g(atomic)257 1358 y(groundstate,)34 b Fo(H)903 1373 y Fm(el)961 1358 y Fo(')1025 1373 y Fm(el)1111 1358 y Fq(=)28 b Fo(e)1260 1373 y Fm(el)1319 1358 y Fo(')1383 1373 y Fm(el)1442 1358 y Fq(,)k(yields)307 1617 y Fo(E)379 1632 y Fm(n)480 1617 y Fn(\024)55 b(h)p Fo(')715 1632 y Fm(el)783 1617 y Fn(\012)9 b Fq(\012)939 1632 y Fm(n)997 1617 y Fn(j)26 b Fo(H)1132 1632 y Fm(n)1195 1617 y Fq(\()p Fo(')1297 1632 y Fm(el)1365 1617 y Fn(\012)9 b Fq(\012)1521 1632 y Fm(n)1569 1617 y Fq(\))p Fn(i)54 b Fq(=)g Fo(e)1875 1632 y Fm(el)1943 1617 y Fq(+)9 b Fn(k)2093 1592 y Fo(~)2078 1617 y(G)2155 1632 y Fm(n)2202 1617 y Fn(k)2252 1576 y Fl(2)2346 1617 y Fn(\024)54 b Fo(e)2522 1632 y Fm(el)2591 1617 y Fq(+)2686 1550 y(1)p 2686 1594 49 4 v 2686 1685 a(5)2745 1617 y Fo(\024)2801 1576 y Fl(2)2840 1617 y Fo(\013)2903 1576 y Fl(3)2996 1617 y Fn(\024)h Fo(\024)9 b(:)50 b Fq(\(A.7\))257 1868 y(Moreo)m(v)m(er,)35 b Fo(\032)28 b Fn(\024)g Fo(\033)951 1883 y Fm(n)1026 1868 y Fn(\024)g Fo(\024)p Fq(,)33 b(and)f(th)m(us)i Fo(C)1728 1832 y Fi(0)1721 1893 y Fl(1)1788 1868 y Fn(\024)28 b Fo(C)1963 1883 y Fl(1)2030 1868 y Fq(=)g(8\()p Fo(\024)22 b Fq(+)g Fo(b)2438 1884 y Fl(1)p Fm(=)p Fl(8)2548 1868 y Fq(\).)p 3425 1868 4 66 v 3429 1806 59 4 v 3429 1868 V 3487 1868 4 66 v 257 2227 a Fc(Pro)s(of)51 b(of)h(Theorem)f(I)s(I.10.)79 b Fq(Fix)44 b Fo(n)k Fn(2)h Fk(N)2025 2242 y Fl(0)2070 2227 y Fq(.)80 b(W)-8 b(e)44 b(\014rst)i(observ)m(e)g(that,)h(for)d(an) m(y)257 2347 y Fo(f)39 b Fn(2)28 b Fo(C)515 2311 y Fi(1)508 2372 y Fl(0)590 2347 y Fq(\()p Fk(R)693 2311 y Fl(3)739 2347 y Fq(;)17 b Fk(R)t Fq(\),)39 b(the)33 b(v)m(ector)h Fo(f)11 b Fq(\()n Fo(~)-47 b(x)p Fq(\))p Fo(\036)1661 2362 y Fm(n)1740 2347 y Fq(lies)31 b(in)h(the)h(domain)e(of)h Fo(H)2729 2362 y Fm(n)2776 2347 y Fq(,)h(and)f(w)m(e)i(ha)m(v)m(e)392 2580 y Fo(H)481 2539 y Fl(+)473 2604 y Fm(n)556 2580 y Fo(f)11 b Fq(\()n Fo(~)-47 b(x)p Fq(\))17 b Fo(\036)821 2595 y Fm(n)928 2580 y Fq(=)60 b([)p Fo(H)1172 2595 y Fm(n)1235 2580 y Fo(;)33 b(f)11 b Fq(\()n Fo(~)-47 b(x)q Fq(\)])17 b Fo(\036)1588 2595 y Fm(n)1694 2580 y Fq(=)1830 2484 y Fh(\020)1902 2580 y Fn(\000)22 b Fq(\001)p Fo(f)11 b Fq(\()n Fo(~)-47 b(x)q Fq(\))22 b Fn(\000)h Fq(2)p Fo(i)2486 2555 y(~)2477 2580 y Fn(r)p Fo(f)11 b Fq(\()n Fo(~)-47 b(x)p Fq(\))22 b Fn(\001)c Fo(~)-45 b(v)2869 2595 y Fm(n)2916 2484 y Fh(\021)2982 2580 y Fo(\036)3040 2595 y Fm(n)3109 2580 y Fo(:)134 b Fq(\(A.8\))257 2812 y(By)33 b(Lemma)f(I)s(I.3,)g Fn(k)l Fo(~)-45 b(v)1064 2827 y Fm(n)1128 2812 y Fo(\036)1186 2827 y Fm(n)1232 2812 y Fn(k)33 b Fq(is)f(b)s(ounded)h(b)m(y)h Fo(C)2017 2827 y Fl(1)2056 2812 y Fq(,)f(uniformly)d(in)i Fo(n)c Fn(2)g Fk(N)9 b Fq(.)49 b(Th)m(us,)834 2945 y Fh(\015)834 2995 y(\015)834 3045 y(\015)p Fo(H)969 3004 y Fl(+)961 3069 y Fm(n)1045 3045 y Fo(f)11 b Fq(\()n Fo(~)-47 b(x)p Fq(\))17 b Fo(\036)1310 3060 y Fm(n)1356 2945 y Fh(\015)1356 2995 y(\015)1356 3045 y(\015)60 b Fn(\024)1600 2948 y Fh(\020)1649 3045 y Fn(k)p Fq(\001)p Fo(f)11 b Fn(k)1889 3060 y Fi(1)1986 3045 y Fq(+)22 b Fn(k)2143 3020 y Fo(~)2134 3045 y Fn(r)p Fo(f)11 b Fn(k)2326 3060 y Fi(1)2400 2948 y Fh(\021)2466 3045 y Fq(\(2)p Fo(C)2623 3060 y Fl(1)2684 3045 y Fq(+)22 b(1\))g Fo(:)352 b Fq(\(A.9\))257 3281 y(W)-8 b(e)42 b(c)m(ho)s(ose)g(a)f(smo)s(oth)f(c)m(haracteristic)h (function)g Fo(\037)h Fn(2)h Fo(C)2483 3245 y Fi(1)2476 3306 y Fl(0)2558 3281 y Fq(\()p Fk(R)2662 3245 y Fl(3)2707 3281 y Fq(;)17 b([0)p Fo(;)g Fq(1]\))40 b(of)h(the)h(unit)257 3402 y(ball)31 b(in)h Fk(R)627 3366 y Fl(3)672 3402 y Fq(,)h(i.e.,)f Fo(\037)p Fq(\()n Fo(~)-47 b(x)q Fq(\))27 b(=)h(1,)k(for)g Fn(j)n Fo(~)-47 b(x)p Fn(j)28 b(\024)g Fq(1,)k(and)h Fo(\037)p Fq(\()n Fo(~)-47 b(x)p Fq(\))28 b(=)g(0,)k(for)g Fn(j)n Fo(~)-47 b(x)p Fn(j)27 b(\025)i Fq(2,)j(and)h(de\014ne)1162 3648 y Fo(~)1147 3673 y(F)1210 3688 y Fm(r)n(;R)1318 3673 y Fq(\()n Fo(~)-47 b(x)p Fq(\))60 b(:=)f Fo(~)-48 b(x)1744 3552 y Fh(\024)1788 3673 y Fq(1)22 b Fn(\000)h Fo(\037)2020 3552 y Fh(\022)2089 3606 y Fo(~)-47 b(x)p 2091 3650 56 4 v 2095 3741 a(r)2156 3552 y Fh(\023\025)2278 3673 y Fo(\037)2339 3552 y Fh(\022)2418 3606 y Fo(~)g(x)p 2410 3650 76 4 v 2410 3741 a(R)2495 3552 y Fh(\023)2578 3673 y Fo(;)617 b Fq(\(A.10\))257 3930 y(where)35 b Fo(r)e Fn(\025)d Fq(1)j(is)g(c)m(hosen)j(su\016cien)m(tly)e(large)f(suc)m(h)i (that)e(sup)2528 3954 y Fi(j)o Fm(~)-34 b(x)p Fi(j\025)p Fm(r)r(=)p Fl(2)2787 3930 y Fn(j)p Fo(V)21 b Fq(\()n Fo(~)-47 b(x)q Fq(\))p Fn(j)29 b(\024)h(\000)p Fo(e)3311 3945 y Fm(el)3371 3930 y Fo(=)p Fq(3,)257 4051 y(and)j Fo(R)c(>)e(r)s Fq(.)43 b(One)33 b(easily)f(c)m(hec)m(ks)k(that)944 4337 y(sup)901 4416 y Fm(R>r)r(>)p Fl(1)1194 4229 y(3)1152 4254 y Fh(X)1151 4436 y Fm(j)t Fl(=1)1306 4241 y Fh(n)1362 4337 y Fn(k)p Fq(\001)p Fo(F)1556 4352 y Fm(r)n(;R)p Fl(;)p Fm(j)1715 4337 y Fn(k)1765 4352 y Fi(1)1862 4337 y Fq(+)22 b Fn(k)2019 4312 y Fo(~)2010 4337 y Fn(r)p Fo(F)2156 4352 y Fm(r)n(;R)p Fl(;)p Fm(j)2315 4337 y Fn(k)2365 4352 y Fi(1)2439 4241 y Fh(o)2555 4337 y Fn(\024)61 b Fo(C)2770 4296 y Fi(0)2763 4362 y Fl(2)2824 4337 y Fo(;)371 b Fq(\(A.11\))257 4639 y(where)29 b Fo(C)611 4602 y Fi(0)604 4663 y Fl(2)671 4639 y Fo(<)f Fn(1)f Fq(dep)s(ends)i(only)f(on)f Fo(\037)h Fq(and)g(its)f(deriv)-5 b(ativ)m(es)28 b(up)g(to)f(second)i(order,)g(and)272 4743 y Fo(~)257 4768 y(F)320 4783 y Fm(r)n(;R)455 4768 y Fq(=:)f(\()p Fo(F)687 4783 y Fm(r)n(;R)p Fl(;1)849 4768 y Fo(;)17 b(F)956 4783 y Fm(r)n(;R)p Fl(;2)1118 4768 y Fo(;)g(F)1225 4783 y Fm(r)n(;R)p Fl(;3)1387 4768 y Fq(\).)43 b(Note)33 b(that)1957 4743 y Fo(~)1943 4768 y(F)2006 4783 y Fm(r)n(;R)2141 4768 y Fn(6)p Fq(=)27 b(0)32 b(implies)f Fn(j)n Fo(~)-47 b(x)p Fn(j)27 b(\025)h Fo(r)s(=)p Fq(2.)43 b(Hence)501 4907 y Fh(D)566 4978 y Fo(~)552 5003 y(F)615 5018 y Fm(r)n(;R)722 5003 y Fo(\036)780 5018 y Fm(n)827 4904 y Fh(\014)827 4953 y(\014)827 5003 y(\014)22 b Fo(H)966 4962 y Fl(+)958 5028 y Fm(n)1055 4978 y Fo(~)1041 5003 y(F)1104 5018 y Fm(r)n(;R)1211 5003 y Fo(\036)1269 5018 y Fm(n)1316 4907 y Fh(E)1450 5003 y Fn(\025)1610 4907 y Fh(D)1675 4978 y Fo(~)1661 5003 y(F)1724 5018 y Fm(r)n(;R)1831 5003 y Fo(\036)1889 5018 y Fm(n)1936 4904 y Fh(\014)1936 4953 y(\014)1936 5003 y(\014)g Fq([)p Fo(V)f Fq(\()n Fo(~)-47 b(x)q Fq(\))22 b Fn(\000)h Fo(e)2390 5018 y Fm(el)2471 5003 y Fn(\000)f Fq(2)p Fn(k)2684 4978 y Fo(~)2669 5003 y(G)2746 5018 y Fm(n)2793 5003 y Fn(k)2843 4962 y Fl(2)2882 5003 y Fq(])2940 4978 y Fo(~)2926 5003 y(F)2989 5018 y Fm(r)n(;R)3096 5003 y Fo(\036)3154 5018 y Fm(n)3201 4907 y Fh(E)1450 5228 y Fn(\025)1620 5161 y(j)p Fo(e)1693 5176 y Fm(el)1752 5161 y Fn(j)p 1620 5205 160 4 v 1675 5297 a Fq(3)1789 5129 y Fh(\015)1789 5178 y(\015)1789 5228 y(\015)1850 5203 y Fo(~)1836 5228 y(F)1899 5243 y Fm(r)n(;R)2006 5228 y Fo(\036)2064 5243 y Fm(n)2111 5129 y Fh(\015)2111 5178 y(\015)2111 5228 y(\015)2157 5155 y Fl(2)2218 5228 y Fo(;)977 b Fq(\(A.12\))p eop %%Page: 23 24 23 23 bop 257 266 a Fr(BFP-GrSt,)32 b(04-F)-8 b(eb-2005)2130 b Fq(23)257 573 y(where)548 548 y Fo(~)534 573 y(F)597 588 y Fm(r)n(;R)704 573 y Fo(\036)762 588 y Fm(n)837 573 y Fq(:=)982 548 y Fo(~)967 573 y(F)1030 588 y Fm(r)n(;R)1138 573 y Fq(\()n Fo(~)-47 b(x)p Fq(\))p Fo(\036)1327 588 y Fm(n)1374 573 y Fq(,)28 b(using)f(that)g Fn(k)1950 548 y Fo(~)1935 573 y(G)2012 588 y Fm(n)2059 573 y Fn(k)2109 537 y Fl(2)2176 573 y Fn(\024)h(j)p Fo(e)2354 588 y Fm(el)2412 573 y Fn(j)p Fo(=)p Fq(6.)41 b(The)29 b(Cauc)m(h)m(y-Sc)m(h)m(w)m(arz) 257 693 y(inequalit)m(y)-8 b(,)32 b(\(A.9\),)g(and)h(\(A.11\))f(th)m (us)i(yield)507 945 y Fn(k)571 920 y Fo(~)557 945 y(F)620 960 y Fm(r)n(;R)727 945 y Fo(\036)785 960 y Fm(n)832 945 y Fn(k)60 b(\024)1145 878 y Fq(3)p 1089 922 160 4 v 1089 1013 a Fn(j)p Fo(e)1162 1028 y Fm(el)1221 1013 y Fn(j)1275 945 y(k)p Fq(\()p Fo(H)1444 960 y Fm(n)1513 945 y Fn(\000)23 b Fo(E)1685 960 y Fm(n)1732 945 y Fq(\))1801 920 y Fo(~)1787 945 y(F)1850 960 y Fm(r)n(;R)1957 945 y Fo(\036)2015 960 y Fm(n)2061 945 y Fn(k)60 b(\024)2319 878 y Fq(3)17 b(\(2)p Fo(C)2542 893 y Fl(1)2602 878 y Fq(+)22 b(1\))17 b Fo(C)2881 841 y Fi(0)2874 902 y Fl(2)p 2319 922 595 4 v 2536 1013 a Fn(j)p Fo(e)2609 1028 y Fm(el)2668 1013 y Fn(j)2945 945 y Fo(:)250 b Fq(\(A.13\))257 1219 y(W)-8 b(e)35 b(ha)m(v)m(e)g Fo(~)-48 b(x)q(\036)768 1234 y Fm(n)814 1219 y Fq(\()n Fo(~)h(x)q Fq(\))65 b(=)f Fo(~)-47 b(x\037)p Fq(\()n Fo(~)g(x)q(=r)s Fq(\))p Fo(\036)1555 1234 y Fm(n)1601 1219 y Fq(\()n Fo(~)g(x)p Fq(\))24 b(+)f(lim)1991 1234 y Fm(R)p Fi(!1)2221 1194 y Fo(~)2206 1219 y(F)2269 1234 y Fm(r)n(;R)2377 1219 y Fq(\()n Fo(~)-47 b(x)p Fq(\))p Fo(\036)2566 1234 y Fm(n)2613 1219 y Fq(\()n Fo(~)g(x)p Fq(\),)35 b(for)f(a.)h(e.)e Fo(~)-47 b(x)31 b Fn(2)h Fk(R)3423 1183 y Fl(3)3468 1219 y Fq(,)257 1339 y(b)m(y)i(Leb)s (esgue's)g(monotone)e(con)m(v)m(ergence)j(theorem,)e(and)f(w)m(e)i (obtain)e(the)h(b)s(ound)257 1536 y Fn(k)n Fo(~)-47 b(x)17 b(\036)437 1551 y Fm(n)484 1536 y Fn(k)55 b(\024)h(k)p Fo(\037)p Fq(\()n Fo(~)-47 b(x=r)s Fq(\))15 b Fo(~)-47 b(x)16 b(\036)1206 1551 y Fm(n)1253 1536 y Fn(k)c Fq(+)g Fn(k)p Fq([1)g Fn(\000)g Fo(\037)p Fq(\()n Fo(~)-47 b(x=r)s Fq(\)])15 b Fo(~)-47 b(x)16 b(\036)2091 1551 y Fm(n)2138 1536 y Fn(k)55 b(\024)h Fq(2)17 b Fo(r)26 b Fq(+)e(3)17 b(\(2)p Fo(C)2835 1551 y Fl(1)2885 1536 y Fq(+)12 b(1\))17 b Fo(C)3154 1495 y Fi(0)3147 1561 y Fl(2)3202 1536 y Fn(j)p Fo(e)3275 1551 y Fm(el)3334 1536 y Fn(j)3362 1495 y Fi(\000)p Fl(1)3468 1536 y Fo(:)3222 1657 y Fq(\(A.14\))257 1777 y(Moreo)m(v)m(er,)29 b(the)e(closedness)g(of)e Fo(H)1509 1792 y Fm(n)1582 1777 y Fq(in)g(com)m(bination)f(with)h(the)i(limit)22 b Fo(R)29 b Fn(!)e(1)f Fq(implies)257 1897 y(that)31 b Fo(~)-47 b(x)17 b(\036)599 1912 y Fm(n)678 1897 y Fq(b)s(elongs)32 b(to)g(its)g(domain.)p 3425 1897 4 66 v 3429 1835 59 4 v 3429 1897 V 3487 1897 4 66 v 257 2347 a Fp(References)306 2566 y Fq([1])49 b(V.)e(Bac)m(h,)k(J.)c(F)-8 b(r\177)-49 b(ohlic)m(h,)49 b(and)d(I.)h(M.)h(Sigal.)83 b(Quan)m(tum)47 b(electro)s(dynamics)f(of)458 2687 y(con\014ned)34 b(non-relativistic)c (particles.)42 b Fj(A)-5 b(dv.)35 b(in)f(Math.)i Fq(,)c(137:299{395,)f (1998.)306 2883 y([2])49 b(V.)28 b(Bac)m(h,)i(J.)f(F)-8 b(r\177)-49 b(ohlic)m(h,)28 b(and)g(I.)h(M.)g(Sigal.)34 b(Renormalization)25 b(group)j(analysis)g(of)458 3004 y(sp)s(ectral)f(problems)g(in)g(quan)m(tum)g(\014eld)h(theory)-8 b(.)35 b Fj(A)-5 b(dv.)30 b(in)g(Math.)h Fq(,)d(137:205{298,)458 3124 y(1998.)306 3321 y([3])49 b(V.)37 b(Bac)m(h,)j(J.)d(F)-8 b(r\177)-49 b(ohlic)m(h,)37 b(and)h(I.)f(M.)h(Sigal.)56 b(Sp)s(ectral)36 b(analysis)h(for)g(systems)i(of)458 3441 y(atoms)44 b(and)g(molecules)f(coupled)i(to)f(the)h(quan)m(tized)g (radiation)e(\014eld.)78 b Fj(Com-)458 3562 y(mun.)34 b(Math.)i(Phys.)p Fq(,)c(207\(2\):249{290,)e(1999.)306 3758 y([4])49 b(M.)60 b(Griesemer,)65 b(E.)60 b(Lieb,)66 b(and)60 b(M.)g(Loss.)124 b(Ground)59 b(states)i(in)d(nonrela-)458 3879 y(tivistic)51 b(quan)m(tum)j(electro)s(dynamics.)104 b Fj(Invent.)52 b(math.)p Fq(,)58 b(June)c(2001.)104 b(DOI)458 3999 y(10.1007/s002220100159.)306 4196 y([5])49 b(F.)40 b(Hiroshima.)66 b(F)-8 b(unctional)39 b(in)m(tegral)g(represen) m(tation)j(of)e(a)h(mo)s(del)e(in)h(QED.)458 4316 y Fj(R)-5 b(ev.)34 b(Math.)i(Phys.)p Fq(,)c(9\(4\):489{530,)f(1997.)306 4513 y([6])49 b(T.)54 b(Kato.)106 b Fj(Perturb)-5 b(ation)55 b(The)-5 b(ory)54 b(of)g(Line)-5 b(ar)54 b(Op)-5 b(er)g(ators)p Fq(,)58 b(v)m(olume)c(132)f(of)458 4633 y Fj(Grund)5 b(lehr)-5 b(en)46 b(der)g(mathematischen)f(Wissenschaften)p Fq(.)80 b(Springer-V)-8 b(erlag,)46 b(2)458 4754 y(edition,)31 b(1976.)306 4951 y([7])49 b(E.)23 b(Lieb)e(and)i(M.)f(Loss.)28 b(Existence)c(of)d(atoms)h(and)g(molecules)f(in)h(non-relativistic)458 5071 y(quan)m(tum)h(electro)s(dynamics.)j Fj(A)-5 b(dv.)25 b(The)-5 b(or.)25 b(Math.)h(Phys.)p Fq(,)f(7\(4\):667{710,)d(2003.)306 5268 y([8])49 b(A.)43 b(Pizzo.)75 b(One-particle)42 b(\(improp)s(er\))f (states)j(in)f(Nelson's)g(massless)h(mo)s(del.)458 5388 y Fj(A)n(nn.)34 b(H.)h(Poinc)-5 b(ar)n(\023)-47 b(e)p Fq(,)31 b(4\(3\):439{486,)g(2003.)p eop %%Page: 24 25 24 24 bop 257 266 a Fr(BFP-GrSt,)32 b(04-F)-8 b(eb-2005)2130 b Fq(24)306 573 y([9])49 b(M.)23 b(Reed)g(and)f(B.)h(Simon.)i Fj(Metho)-5 b(ds)26 b(of)f(Mo)-5 b(dern)25 b(Mathematic)-5 b(al)25 b(Physics:)40 b(A)n(nal-)458 693 y(ysis)28 b(of)g(Op)-5 b(er)g(ators)p Fq(,)26 b(v)m(olume)f(4.)32 b(Academic)25 b(Press,)j(San)e(Diego,)f(1)g(edition,)h(1978.)257 897 y([10])49 b(M.)35 b(Reed)h(and)f(B.)g(Simon.)49 b Fj(Metho)-5 b(ds)37 b(of)g(Mo)-5 b(dern)37 b(Mathematic)-5 b(al)36 b(Physics:)49 b(II.)458 1017 y(Fourier)36 b(A)n(nalysis)f(and)h(Self-A) -5 b(djointness)p Fq(,)32 b(v)m(olume)i(2.)47 b(Academic)33 b(Press,)j(San)458 1137 y(Diego,)31 b(2)h(edition,)g(1980.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF