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(tw)o(o)f(appendices.)61 b(In)35 b(Appendix)f(A)h(we)h(construct)e(a) 257 1175 y(Bogoliubo)o(v)g(transformation)g(that)g(eliminates)g(a)i (single,)g(arbitrary)g(matrix)e(element)h(in)257 1295 y(the)27 b(interaction,)h(and)f(in)g(Appendix)f(B)i(we)f(sho)n(w)g (that)f(bound)h(or)g(ne)o(gati)n(v)o(e)e(ener)n(gy)j(states)257 1416 y(do)d(not)f(contrib)n(ute)g(to)g(the)h(transported)f(char)n(ge.) 257 1607 y Fk(Ackno)o(wledgement:)37 b Fv(W)-8 b(e)28 b(thank)e(S.)h(De)g(Bi)6 b(\036)-39 b(evre,)28 b(J.)f(Fr)8 b(\250)-41 b(ohlich,)27 b(Ch.)g(G)6 b(\264)-39 b(erard,)28 b(M.)e(Griese-)257 1727 y(mer)l(,)20 b(V)-13 b(.)19 b(Jak)s(\020)-36 b(si)6 b(\264)-39 b(c,)20 b(V)-13 b(.)19 b(K)m(ostrykin,)f(T)-7 b(.)18 b(P)o(aul,)i(B.)f(Schlein,)h(R.)f(Schrader)l(,)i(I.)e(M.)g (Sigal,)g(H.)g(Spohn,)257 1848 y(and)25 b(S.)g(T)-7 b(eufel)25 b(for)g(helpful)g(discussions)d(and)j(remarks.)404 1968 y(W)-8 b(e)34 b(gratefully)g(ackno)n(wledge)f(the)h(support)f(of)h(our) f(project)h(by)g(the)g(TMR)g(netw)o(ork)257 2088 y(ERBFMRXCT960001)26 b(of)e(the)h(European)g(Union)f(and)h(the)f(Gerhard)h(Hess-grant)257 2209 y(no.)g(Ba)g(1477/3-1)f(of)h(the)g(german)f(science)h(foundation)f (DFG.)p eop %%Page: 4 5 4 4 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2409 b Fv(4)257 573 y Fj(I.1)119 b(Mathematical)29 b(Model)i(f)m(or)e(the)i (Photoelectric)f(Effect)257 760 y Fv(Our)25 b(goal)g(is)f(to)g(analyze) h(the)g(photoelectric)f(ef)n(fect)h(on)f(the)h(e)o(xample)f(of)h(an)g (atom)f(coupled)257 881 y(to)30 b(the)g(quantized)g(radiation)f (\002eld.)47 b(The)30 b(mathematical)f(model)h(for)g(our)g(analysis)f (of)h(the)257 1001 y(photoelectric)e(ef)n(fect)g(is)g(a)h(v)n(ariant)e (of)i(the)f Fq(standar)l(d)f(model)g(of)h(nonr)l(elativistic)f(quantum) 257 1122 y(electr)l(odynamics)f Fv(as)h(gi)n(v)o(en)f(in)g([2])h(which) g(we)g(brie\003y)g(recall)h(and)f(adapt)g(to)f(our)h(problem)257 1242 y(at)e(hand.)257 1505 y Fk(I.1.1)99 b(Atom)25 b(with)g(a)g(Single) g(Bound)h(State)257 1692 y Fv(F)o(or)d(the)g(atom)g(to)g(be)g (described)g(we)g(mak)o(e)g(the)g(simplifying)e(assumptions)g(that)h (it)h(consists)257 1813 y(of)30 b(a)f(single,)g(spinless)f(electron)h (bound)f(in)h(a)h(potential)e(well)g(which)h(admits)f(e)o(xactly)g(one) 257 1933 y(bound)36 b(state)h(of)g(ener)n(gy)g Fs(e)1234 1948 y Fl(0)1324 1933 y Fs(<)50 b Fp(0)p Fv(.)66 b(More)37 b(speci\002cally)-6 b(,)39 b(we)e(assume)f(the)h(Hamiltonian)257 2053 y(generating)25 b(the)f(dynamics)g(of)h(the)g(electron)g(to)f(be)h (gi)n(v)o(en)e(in)i(diagonal)f(form)g(as)1124 2306 y Fs(H)1205 2321 y Fr(el)1316 2306 y Fp(=)52 b Fo(\000)p Fp(\001)23 b Fo(\010)g Fs(e)1770 2321 y Fl(0)1862 2306 y Fp(=)1990 2166 y Fi(\022)2105 2245 y Fo(\000)p Fp(\001)102 b(0)2160 2365 y(0)138 b Fs(e)2392 2380 y Fl(0)2473 2166 y Fi(\023)2585 2306 y Fs(;)709 b Fv(\(I.3\))257 2555 y(acting)25 b(on)917 2753 y Fo(H)1001 2768 y Fr(el)1113 2753 y Fp(:=)52 b Fs(L)1334 2712 y Fl(2)1374 2753 y Fp(\()p Fh(R)1478 2712 y Fl(3)1523 2753 y Fp(\))22 b Fo(\010)h Fh(C)79 b Fp(=)52 b Fo(H)2020 2768 y Fr(ac)2092 2753 y Fp(\()p Fs(H)2211 2768 y Fr(el)2270 2753 y Fp(\))22 b Fo(\010)h(H)2514 2768 y Fr(d)2554 2753 y Fp(\()p Fs(H)2673 2768 y Fr(el)2732 2753 y Fp(\))f Fs(;)502 b Fv(\(I.4\))257 2951 y(and)25 b Fo(H)510 2966 y Fr(sc)578 2951 y Fp(\()p Fs(H)697 2966 y Fr(el)755 2951 y Fp(\))j(=)f Fo(f)p Fp(0)p Fo(g)p Fv(.)j(W)-8 b(e)26 b(further)f(denote)f(by)789 3207 y Fs(P)852 3222 y Fr(c)939 3207 y Fp(:=)1095 3066 y Fi(\022)1209 3146 y Fg(1)84 b Fp(0)1213 3266 y(0)j(0)1439 3066 y Fi(\023)1647 3207 y Fv(and)118 b Fs(P)1972 3222 y Fr(d)2065 3207 y Fp(:=)52 b Fs(P)2297 3166 y Ff(?)2283 3232 y Fr(c)2408 3207 y Fp(=)2537 3066 y Fi(\022)2652 3146 y Fp(0)82 b(0)2652 3266 y(0)g(1)2874 3066 y Fi(\023)3321 3207 y Fv(\(I.5\))257 3456 y(the)22 b(projections)f(onto)h(the)f (continuous)g(subspace)h(and)g(the)g(discrete)g(subspace)g(\(of)g (dimen-)257 3576 y(sion)28 b(one\),)i(respecti)n(v)o(ely)-6 b(.)40 b(The)28 b(Hamiltonian)f Fs(H)2013 3591 y Fr(el)2100 3576 y Fv(can)i(be)g(deri)n(v)o(ed)e(from)h(a)h(Schr)8 b(\250)-41 b(odinger)257 3697 y(operator)26 b Fo(\000)p Fp(\001)e Fo(\000)f Fs(V)e Fp(\()p Fs(x)p Fp(\))26 b Fv(with)f(a)h(short-range)f(potential)g Fs(V)c Fv(.)33 b(An)26 b(appropriate)f(choice)h(of)g Fs(V)257 3817 y Fv(guarantees)j(that)f Fo(\000)p Fp(\001)d Fo(\000)g Fs(V)d Fp(\()p Fs(x)p Fp(\))29 b Fv(has)f(a)h(single)e(bound)g(state)h (only)-6 b(,)28 b(and)h(then)f Fo(\000)p Fp(\001)d Fo(\000)g Fs(V)d Fp(\()p Fs(x)p Fp(\))257 3938 y Fv(and)27 b Fs(H)509 3953 y Fr(el)595 3938 y Fv(are)h(unitarily)e(equi)n(v)n(alent,)g(as)h (can)g(be)h(seen)f(by)g(conjugating)e Fo(\000)p Fp(\001)g Fo(\000)f Fs(V)e Fp(\()p Fs(x)p Fp(\))27 b Fv(with)257 4058 y(the)e(w)o(a)n(v)o(e)g(operators.)257 4321 y Fk(I.1.2)99 b(Photons)257 4508 y Fv(W)-8 b(e)40 b(couple)g(the)f(atom)g(described)g (abo)o(v)o(e)g(to)g(the)g(quantized)g(photon)g(\002eld)g(which,)k(for) 257 4629 y(notational)21 b(con)l(v)o(enience,)i(is)f(assumed)f(to)i(be) f(scalar)-5 b(.)30 b(F)o(or)22 b(our)h(study)e(of)i(the)f (photoelectric)257 4749 y(ef)n(fect,)37 b(the)c(dif)n(ference)i(is)e (not)g(rele)n(v)n(ant.)57 b(The)34 b(Hilbert)f(space)h Fo(F)10 b Fv(,)36 b(carrying)e(the)f(photon)257 4869 y(de)o(grees)38 b(of)g(freedom)g(is)f(the)g(bosonic)g(F)o(ock)h(space)g Fo(F)61 b Fp(=)52 b Fo(F)2528 4884 y Fr(b)2562 4869 y Fp([)p Fs(L)2655 4833 y Fl(2)2695 4869 y Fp(\()p Fh(R)2798 4833 y Fl(3)2844 4869 y Fp(\)])38 b Fv(o)o(v)o(er)f(the)g(one-)257 4990 y(photon)24 b(Hilbert)g(space)h Fs(L)1178 4954 y Fl(2)1218 4990 y Fp(\()p Fh(R)1322 4954 y Fl(3)1367 4990 y Fp(\))p Fv(,)g(i.e.,)1529 5264 y Fo(F)62 b Fp(=)1831 5139 y Ff(1)1791 5169 y Fi(M)1800 5378 y Fr(n)p Fl(=0)1975 5264 y Fo(F)2057 5222 y Fl(\()p Fr(n)p Fl(\))2180 5264 y Fs(;)1114 b Fv(\(I.6\))p eop %%Page: 5 6 5 5 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2409 b Fv(5)257 573 y(where)38 b Fo(F)620 537 y Fl(\()p Fr(n)p Fl(\))759 573 y Fv(is)f(the)g(state)g(space)g(of)h(all)f Fs(n)p Fv(-photon)f(states,)k(the)d Fs(n)p Fv(-photon)g(sector)-5 b(.)67 b(The)257 693 y(v)n(acuum)27 b(sector)l(,)i Fo(F)968 657 y Fl(\(0\))1062 693 y Fv(,)g(is)e(one-dimensional)f(and)i(spanned)g (by)f(the)h(normalized)f(v)n(acuum)257 814 y(v)o(ector)l(,)39 b Fp(\012)p Fv(,)h(i.e.,)g Fo(F)969 777 y Fl(\(0\))1112 814 y Fp(:=)49 b Fh(C)37 b Fp(\012)p Fv(.)73 b(F)o(or)36 b Fs(n)50 b Fo(\025)g Fp(1)p Fv(,)39 b(the)e Fs(n)p Fv(-photon)e (sector)i(is)f(the)h(subspace)257 934 y(of)c Fs(L)439 898 y Fl(2)478 934 y Fp([\()p Fh(R)609 898 y Fl(3)655 934 y Fp(\))693 898 y Fr(n)740 934 y Fp(])f Fv(containing)f(all)g (totally)g(symmetric)g(v)o(ectors.)52 b(Note)32 b(that)f(we)i(are)f (making)257 1054 y(the)i(simplifying)e(assumption)f(of)j Fq(scalar)f Fv(photons,)i(since)f(the)g(polarization)e(de)o(grees)i(of) 257 1175 y(freedom)j(play)g(no)g(role)g(in)f(our)h(analysis.)66 b(The)37 b(Hamiltonian)f(on)g Fo(F)47 b Fv(representing)36 b(the)257 1295 y(ener)n(gy)26 b(of)e(the)h(free)h(photon)d(\002eld)j (is)e(gi)n(v)o(en)f(by)1245 1533 y Fs(H)1326 1548 y Fr(f)1424 1533 y Fp(=)1552 1398 y Fi(Z)1668 1533 y Fs(d)1719 1492 y Fl(3)1759 1533 y Fs(k)c(!)t Fp(\()p Fs(k)s Fp(\))e Fs(a)2092 1492 y Ff(\003)2131 1533 y Fp(\()p Fs(k)s Fp(\))p Fs(a)p Fp(\()p Fs(k)s Fp(\))22 b Fs(;)830 b Fv(\(I.7\))257 1775 y(where)40 b Fs(!)t Fp(\()p Fs(k)s Fp(\))53 b(:=)i Fo(j)p Fs(k)s Fo(j)38 b Fv(is)h(the)g(photon)f(dispersion,)j(and)e Fs(a)2388 1739 y Ff(\003)2428 1775 y Fv(,)k Fs(a)c Fv(are)h(the)f (usual)g(standard)257 1895 y(creation-)22 b(and)f(annihilation)e (operators)i(on)f Fo(F)31 b Fv(representing)21 b(the)f(canonical)h (commutation)257 2016 y(relations,)j Fp([)p Fs(a)p Fp(\()p Fs(k)s Fp(\))p Fs(;)17 b(a)p Fp(\()p Fs(k)1045 1980 y Ff(0)1069 2016 y Fp(\)])28 b(=)g([)p Fs(a)1344 1980 y Ff(\003)1384 2016 y Fp(\()p Fs(k)s Fp(\))p Fs(;)17 b(a)1609 1980 y Ff(\003)1648 2016 y Fp(\()p Fs(k)1740 1980 y Ff(0)1763 2016 y Fp(\)])28 b(=)g(0)p Fv(,)d Fp([)p Fs(a)p Fp(\()p Fs(k)s Fp(\))p Fs(;)17 b(a)2362 1980 y Ff(\003)2401 2016 y Fp(\()p Fs(k)2493 1980 y Ff(0)2517 2016 y Fp(\)])28 b(=)g Fs(\016)t Fp(\()p Fs(k)d Fo(\000)d Fs(k)3028 1980 y Ff(0)3052 2016 y Fp(\))p Fv(,)j Fs(a)p Fp(\()p Fs(k)s Fp(\)\012)j(=)257 2136 y(0)p Fv(,)d(in)f(the)h(sense)g(of)g(operator)n (-v)n(alued)f(distrib)n(utions.)257 2398 y Fk(I.1.3)99 b(The)26 b(Atom-Photon)g(System)257 2586 y Fv(The)f(Hilbert)f(space)i (of)e(states)h(of)g(the)f(atom-photon)f(system)h(is)g(the)h(tensor)f (product)h(space)1524 2779 y Fo(H)53 b Fp(:=)g Fo(H)1901 2794 y Fr(el)1982 2779 y Fo(\012)23 b(F)31 b Fs(;)1109 b Fv(\(I.8\))257 2972 y(and)25 b(its)f(dynamics)g(is)g(generated)h(by)g (the)g(Hamiltonian)1445 3165 y Fs(H)1526 3180 y Fr(g)1618 3165 y Fp(:=)53 b Fs(H)1855 3180 y Fl(0)1944 3165 y Fp(+)d Fs(g)19 b(W)36 b(;)1030 b Fv(\(I.9\))257 3359 y(where)1262 3552 y Fs(H)1343 3567 y Fl(0)1434 3552 y Fp(:=)53 b Fs(H)1671 3567 y Fr(el)1752 3552 y Fo(\012)22 b Fg(1)1907 3567 y Fr(f)2003 3552 y Fp(+)49 b Fg(1)2184 3567 y Fr(el)2265 3552 y Fo(\012)23 b Fs(H)2446 3567 y Fr(f)3271 3552 y Fv(\(I.10\))257 3745 y(is)h(the)h(non-interacting)e(Hamiltonian,)g Fp(0)k Fs(<)h(g)i Fo(\034)e Fp(1)c Fv(is)g(a)h(small)e(coupling)h (parameter)l(,)g(and)886 3983 y Fs(W)66 b Fp(:=)1200 3848 y Fi(Z)1316 3983 y Fs(d)1367 3942 y Fl(3)1406 3983 y Fs(k)1460 3873 y Fi(n)1526 3983 y Fs(G)p Fp(\()p Fs(k)s Fp(\))22 b Fo(\012)h Fs(a)1906 3942 y Ff(\003)1946 3983 y Fp(\()p Fs(k)s Fp(\))49 b(+)h Fs(G)2328 3942 y Ff(\003)2368 3983 y Fp(\()p Fs(k)s Fp(\))22 b Fo(\012)g Fs(a)p Fp(\()p Fs(k)s Fp(\))2800 3873 y Fi(o)3271 3983 y Fv(\(I.11\))257 4229 y(is)j(the)h(interaction)f(operator)-5 b(.)32 b(Here,)27 b Fs(G)i Fo(2)g Fs(L)1845 4193 y Fl(2)1885 4229 y Fp([)p Fh(R)1978 4193 y Fl(3)2023 4229 y Fp(;)17 b Fo(B)s Fp(\()p Fo(H)2257 4244 y Fr(el)2316 4229 y Fp(\)])26 b Fv(is)f(a)h(square-inte) o(grable)f(func-)257 4349 y(tion)f(with)g(v)n(alues)g(in)h(the)f (bounded)g(operators)h(on)g Fo(H)2138 4364 y Fr(el)2196 4349 y Fv(,)g(gi)n(v)o(en)e(by)1253 4597 y Fs(G)p Fp(\()p Fs(k)s Fp(\))52 b(:=)1668 4457 y Fi(\022)1787 4536 y Fs(B)5 b Fp(\()p Fs(k)s Fp(\))88 b Fs(p)2133 4551 y Ff(")2172 4536 y Fp(\()p Fs(k)s Fp(\))1783 4657 y Fs(p)1832 4672 y Ff(#)1871 4657 y Fp(\()p Fs(k)s Fp(\))168 b(0)2344 4457 y Fi(\023)2456 4597 y Fs(;)788 b Fv(\(I.12\))257 4853 y(for)31 b Fs(k)41 b Fo(2)e Fh(R)667 4817 y Fl(3)712 4853 y Fv(,)32 b(a.e.)48 b(F)o(or)31 b(the)f(formulation)f(of)i (assumptions)d(about)i Fs(G)g Fv(it)g(is)g(con)l(v)o(enient)g(to)257 4973 y(introduce)382 5166 y Fs(J)436 5181 y Fr(K)q(;\015)560 5166 y Fp(\()p Fs(k)s Fp(\))53 b(:=)64 b(max)899 5233 y Ff(j)p Fr(\013)p Ff(j\024)p Fr(K)1119 5082 y Fi(\015)1119 5142 y(\015)1174 5166 y Fs(@)1230 5125 y Fr(\013)1225 5191 y(k)1281 5166 y Fs(G)p Fp(\()p Fs(k)s Fp(\))1488 5082 y Fi(\015)1488 5142 y(\015)1582 5166 y Fp(+)1696 5082 y Fi(\015)1696 5142 y(\015)1751 5166 y Fp(\()p Fo(j)p Fs(x)p Fo(j)1900 5125 y Fr(\015)1967 5166 y Fo(\010)22 b Fp(1\))17 b Fs(G)p Fp(\()p Fs(k)s Fp(\))2377 5082 y Fi(\015)2377 5142 y(\015)2471 5166 y Fp(+)2585 5082 y Fi(\015)2585 5142 y(\015)2641 5166 y Fs(G)p Fp(\()p Fs(k)s Fp(\))g(\()p Fo(j)p Fs(x)p Fo(j)3014 5125 y Fr(\015)3079 5166 y Fo(\010)23 b Fp(1\))3266 5082 y Fi(\015)3266 5142 y(\015)3343 5166 y Fs(:)3271 5352 y Fv(\(I.13\))p eop %%Page: 6 7 6 6 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2409 b Fv(6)257 573 y(Note)21 b(that,)f(for)h Fs(J)858 588 y Fr(K)q(;\015)982 573 y Fp(\()p Fs(k)s Fp(\))28 b Fs(<)f Fo(1)p Fv(,)21 b(we)g(necessarily)f(assume)g(a)g(decay)h(of)g(the)f(coupling)f(matrix) 257 693 y Fs(G)p Fp(\()p Fs(k)s Fp(\))33 b Fv(as)g(least)g(as)g Fo(j)p Fs(x)p Fo(j)1056 657 y Ff(\000)p Fr(\015)1155 693 y Fv(,)i(as)e Fs(x)43 b Fo(!)f Fs(inf)11 b(ty)t Fv(.)54 b(This)33 b(assumption)d(is)j(not)f(satis\002ed,)j(e.g.,)f(if)257 814 y(we)k(consider)e(electrons)h(minimally)e(coupled)h(to)h(the)g (quantized)f(radiation)h(\002eld.)67 b(The)257 934 y(e)o(xtension)22 b(of)h(our)g(results)f(to)g(this)g(case)i(of)f(main)f(physical)g (interest)g(will)g(be)h(the)g(subject)f(of)257 1054 y(a)k(forthcoming)d (paper)-5 b(.)404 1175 y(Further)25 b(note)f(that)274 1339 y Fs(J)328 1354 y Fr(K)q(;\015)452 1339 y Fp(\()p Fs(k)s Fp(\))52 b Fo(\024)h Fs(J)827 1298 y Ff(0)818 1363 y Fr(K)q(;\015)943 1339 y Fp(\()p Fs(k)s Fp(\))f(:=)64 b(max)1281 1405 y Ff(j)p Fr(\013)p Ff(j\024)p Fr(K)1501 1254 y Fi(\015)1501 1314 y(\015)1556 1339 y Fs(@)1612 1298 y Fr(\013)1607 1363 y(k)1663 1339 y Fs(B)5 b Fp(\()p Fs(k)s Fp(\))1872 1254 y Fi(\015)1872 1314 y(\015)1966 1339 y Fp(+)2080 1254 y Fi(\015)2080 1314 y(\015)2158 1339 y Fo(j)p Fs(x)p Fo(j)2269 1298 y Fr(\015)2330 1339 y Fs(B)g Fp(\()p Fs(k)s Fp(\))2539 1254 y Fi(\015)2539 1314 y(\015)2633 1339 y Fp(+)2747 1254 y Fi(\015)2747 1314 y(\015)2803 1339 y Fs(B)g Fp(\()p Fs(k)s Fp(\))17 b Fo(j)p Fs(x)p Fo(j)3140 1298 y Fr(\015)3206 1254 y Fi(\015)3206 1314 y(\015)3271 1339 y Fv(\(I.14\))456 1550 y Fp(+)44 b(max)565 1616 y Ff(j)p Fr(\013)p Ff(j\024)p Fr(K)785 1465 y Fi(\015)785 1525 y(\015)841 1550 y Fs(@)897 1508 y Fr(\013)892 1574 y(k)947 1550 y Fs(p)996 1565 y Ff(")1036 1550 y Fp(\()p Fs(k)s Fp(\))1166 1465 y Fi(\015)1166 1525 y(\015)1260 1550 y Fp(+)49 b(max)1374 1616 y Ff(j)p Fr(\013)p Ff(j\024)p Fr(K)1594 1465 y Fi(\015)1594 1525 y(\015)1650 1550 y Fs(@)1706 1508 y Fr(\013)1701 1574 y(k)1756 1550 y Fs(p)1805 1565 y Ff(#)1845 1550 y Fp(\()p Fs(k)s Fp(\))1975 1465 y Fi(\015)1975 1525 y(\015)2069 1550 y Fp(+)2183 1465 y Fi(\015)2183 1525 y(\015)2261 1550 y Fo(j)p Fs(x)p Fo(j)2372 1508 y Fr(\015)2433 1550 y Fs(p)2482 1565 y Ff(")2521 1550 y Fp(\()p Fs(k)s Fp(\))2651 1465 y Fi(\015)2651 1525 y(\015)2745 1550 y Fp(+)2860 1465 y Fi(\015)2860 1525 y(\015)2915 1550 y Fs(p)2964 1565 y Ff(#)3003 1550 y Fp(\()p Fs(k)s Fp(\))17 b Fo(j)p Fs(x)p Fo(j)3261 1508 y Fr(\015)3327 1465 y Fi(\015)3327 1525 y(\015)3405 1550 y Fs(;)257 1769 y Fv(and,)36 b(con)l(v)o(ersely) -6 b(,)33 b Fs(J)1004 1733 y Ff(0)995 1794 y Fr(K)q(;\015)1120 1769 y Fp(\()p Fs(k)s Fp(\))43 b Fo(\024)h Fp(3)p Fs(J)1517 1784 y Fr(K)q(;\015)1641 1769 y Fp(\()p Fs(k)s Fp(\))p Fv(.)55 b(W)-8 b(e)34 b(shall)f(mak)o(e)g(use)g(of)g(the)g(follo)n (wing)e(Hy-)257 1889 y(pothesis)24 b(throughout)f(the)i(paper)-5 b(.)257 2045 y Fk(Hypothesis)25 b(1.)49 b Fq(Ther)l(e)30 b(is)e(an)h(inte)l(g)o(er)f Fs(K)42 b(>)34 b Fp(1)29 b Fq(and)f(a)h(r)l(eal)f(number)h Fs(\015)39 b(>)c Fp(3)p Fs(=)p Fp(2)28 b Fq(suc)o(h)g(that)257 2165 y Fs(G)g Fo(2)g Fs(C)533 2129 y Fr(K)601 2165 y Fp([)p Fh(R)694 2129 y Fl(3)740 2165 y Fp(;)17 b Fo(B)s Fp(\()p Fo(H)974 2180 y Fr(el)1033 2165 y Fp(\)])j Fq(is)f Fs(K)27 b Fq(times)20 b(dif)n(fer)l(entiable)e(and)i(satis\002es)e(the)i(following)e (estimate)o(,)1099 2242 y Fi(Z)1216 2267 y(\020)1275 2378 y Fp(1)k(+)g Fs(!)t Fp(\()p Fs(k)s Fp(\))1639 2336 y Ff(\000)p Fl(1)1733 2267 y Fi(\021)1809 2378 y Fo(j)p Fs(J)1891 2393 y Fr(K)q(;\015)2015 2378 y Fp(\()p Fs(k)s Fp(\))p Fo(j)2173 2336 y Fl(2)2228 2378 y Fs(d)2279 2336 y Fl(3)2318 2378 y Fs(k)56 b Fo(\024)d Fp(1)22 b Fs(:)618 b Fv(\(I.15\))404 2583 y(W)-8 b(e)26 b(remark)g(that,)f(although)g(the) g(requirement)h(\(I.15\))g(with)f(fractional)g(deri)n(v)n(ati)n(v)o(es) e(of)257 2703 y Fs(G)p Fv(,)30 b(for)e(real)h Fs(K)42 b(>)34 b Fp(1)p Fv(,)c(is)e(presumably)f(suf)n(\002cient,)i(we)g(do)f (not)g(try)g(to)g(optimize)f(our)i(result)257 2824 y(in)c(this)f (respect)h(and)f(w)o(ork)h(with)f(classical)h(deri)n(v)n(ati)n(v)o(es)d (in)i(this)g(paper)-5 b(.)404 2944 y(Since,)25 b(for)g(a.e.)31 b Fs(k)g Fo(2)d Fh(R)1227 2908 y Fl(3)1272 2944 y Fv(,)777 3108 y Fs(p)826 3123 y Ff(")866 3108 y Fp(\()p Fs(k)s Fp(\))f(:)h Fh(C)81 b Fo(!)55 b Fs(L)1426 3067 y Fl(2)1466 3108 y Fp(\()p Fh(R)1570 3067 y Fl(3)1615 3108 y Fp(\))118 b Fv(and)g Fs(p)2082 3123 y Ff(#)2122 3108 y Fp(\()p Fs(k)s Fp(\))27 b(:)h Fs(L)2400 3067 y Fl(2)2440 3108 y Fp(\()p Fh(R)2544 3067 y Fl(3)2589 3108 y Fp(\))56 b Fo(!)f Fh(C)48 b Fs(;)312 b Fv(\(I.16\))257 3272 y(there)25 b(e)o(xist)f Fs(\032)p Fp(\()17 b Fo(\001)g Fs(;)g(k)s Fp(\))p Fs(;)g(\021)t Fp(\()g Fo(\001)g Fs(k)s Fp(\))25 b Fo(2)j Fs(L)1452 3236 y Fl(2)1492 3272 y Fp(\()p Fh(R)1596 3236 y Fl(3)1641 3272 y Fp(\))p Fv(,)d(such)f(that)273 3404 y Fi(\002)315 3484 y Fs(p)364 3499 y Ff(")403 3484 y Fp(\()p Fs(k)s Fp(\))p Fs(z)582 3404 y Fi(\003)624 3484 y Fp(\()p Fs(x)p Fp(\))51 b(=)e Fs(z)22 b(\021)t Fp(\()p Fs(x;)17 b(k)s Fp(\))70 b Fv(and)h Fs(p)1613 3499 y Ff(#)1652 3484 y Fp(\()p Fs(k)s Fp(\))p Fs( )54 b Fp(=)c Fo(h)p Fs(\032)p Fp(\()17 b Fo(\001)g Fs(;)g(k)s Fp(\))p Fo(j)p Fs( )t Fo(i)48 b Fp(=)2657 3349 y Fi(Z)p 2774 3398 280 4 v 2774 3484 a Fs(\032)p Fp(\()p Fs(x;)17 b(k)s Fp(\))p Fs( )t Fp(\()p Fs(x)p Fp(\))g Fs(d)3319 3443 y Fl(3)3358 3484 y Fs(x)23 b(;)3271 3663 y Fv(\(I.17\))257 3830 y(for)44 b(all)f Fs(z)67 b Fo(2)c Fh(C)20 b Fv(,)54 b Fs( )66 b Fo(2)c Fs(L)1269 3794 y Fl(2)1309 3830 y Fp(\()p Fh(R)1413 3794 y Fl(3)1458 3830 y Fp(\))p Fv(,)48 b(and)c Fp(\()p Fs(x;)17 b(k)s Fp(\))62 b Fo(2)g Fh(R)2242 3794 y Fl(3)2324 3830 y Fo(\002)36 b Fh(R)2503 3794 y Fl(3)2549 3830 y Fv(,)48 b(a.e.)87 b(Furthermore,)48 b(in)257 3950 y(man)o(y)37 b(applications)e(\(e.g.,)41 b(the)c(Dipole)g(approximation\))e Fs(B)5 b Fp(\()p Fs(k)s Fp(\))51 b Fo(2)g(B)s Fp([)p Fs(L)2933 3914 y Fl(2)2974 3950 y Fp(\()p Fh(R)3078 3914 y Fl(3)3123 3950 y Fp(\)])37 b Fv(acts)h(in)257 4070 y(the)f(Schr)8 b(\250)-41 b(odinger)38 b(representation)e(as)h(a)h(multiplication)c(operator)-5 b(.)67 b(That)37 b(is,)j(there)d(is)g(a)257 4191 y(function)24 b Fs(M)39 b Fo(2)28 b Fs(L)907 4155 y Fl(2)947 4191 y Fp(\()p Fh(R)1051 4155 y Fl(3)1118 4191 y Fo(\002)23 b Fh(R)1284 4155 y Fl(3)1329 4191 y Fp(\))i Fv(such)g(that)1233 4274 y Fi(\002)1275 4355 y Fs(B)5 b Fp(\()p Fs(k)s Fp(\))p Fs( )1551 4274 y Fi(\003)1592 4355 y Fp(\()p Fs(x)p Fp(\))53 b(=)g Fs(M)10 b Fp(\()p Fs(x;)17 b(k)s Fp(\))g Fs( )t Fp(\()p Fs(x)p Fp(\))23 b Fs(;)768 b Fv(\(I.18\))257 4519 y(for)22 b Fp(\()p Fs(x;)17 b(k)s Fp(\))28 b Fo(2)g Fh(R)812 4483 y Fl(3)869 4519 y Fo(\002)11 b Fh(R)1023 4483 y Fl(3)1069 4519 y Fv(,)22 b(a.e.)30 b(In)22 b(this)f(case,)i Fs(M)10 b Fv(,)23 b Fs(p)1975 4534 y Ff(")2015 4519 y Fv(,)f(and)g Fs(p)2277 4534 y Ff(#)2338 4519 y Fv(determine)g Fs(G)p Fv(,)g(and)g(we)g(observ)o(e)257 4639 y(that)403 4803 y Fs(J)466 4762 y Ff(0)457 4828 y Fr(K)q(;\015)582 4803 y Fp(\()p Fs(k)s Fp(\))52 b(=)183 b(sup)893 4890 y Fr(x)p Ff(2)p Fe(R)1028 4871 y Fd(3)1062 4890 y Fr(;)15 b Ff(j)p Fr(\013)p Ff(j\024)p Fr(K)1317 4718 y Fi(\014)1317 4778 y(\014)1351 4803 y Fs(@)1407 4762 y Fr(\013)1402 4828 y(k)1457 4803 y Fs(M)10 b Fp(\()p Fs(x;)17 b(k)s Fp(\))1790 4718 y Fi(\014)1790 4778 y(\014)1863 4803 y Fp(+)38 b(2)28 b(sup)2043 4890 y Fr(x)p Ff(2)p Fe(R)2178 4871 y Fd(3)2229 4718 y Fi(\014)2229 4778 y(\014)2284 4803 y Fo(j)p Fs(x)p Fo(j)2395 4762 y Fr(\015)2456 4803 y Fs(M)10 b Fp(\()p Fs(x;)17 b(k)s Fp(\))2789 4718 y Fi(\014)2789 4778 y(\014)586 5088 y Fp(+)44 b(max)695 5154 y Ff(j)p Fr(\013)p Ff(j\024)p Fr(K)915 4977 y Fi(n)998 4952 y(Z)1115 5003 y(\014)1115 5063 y(\014)1148 5088 y Fs(@)1204 5047 y Fr(\013)1199 5112 y(k)1254 5088 y Fs(\032)p Fp(\()p Fs(x;)17 b(k)s Fp(\))1533 5003 y Fi(\014)1533 5063 y(\014)1567 5029 y Fl(2)1623 5088 y Fs(d)1674 5047 y Fl(3)1713 5088 y Fs(x)1768 4977 y Fi(o)1835 5000 y Fl(1)p Fr(=)p Fl(2)1984 5088 y Fp(+)49 b(max)2098 5154 y Ff(j)p Fr(\013)p Ff(j\024)p Fr(K)2318 4977 y Fi(n)2401 4952 y(Z)2518 5003 y(\014)2518 5063 y(\014)2551 5088 y Fs(@)2607 5047 y Fr(\013)2602 5112 y(k)2657 5088 y Fs(\021)t Fp(\()p Fs(x;)17 b(k)s Fp(\))2938 5003 y Fi(\014)2938 5063 y(\014)2971 5029 y Fl(2)3027 5088 y Fs(d)3078 5047 y Fl(3)3117 5088 y Fs(x)3172 4977 y Fi(o)3239 5000 y Fl(1)p Fr(=)p Fl(2)586 5352 y Fp(+)679 5242 y Fi(n)761 5217 y(Z)878 5352 y Fo(j)p Fs(x)p Fo(j)989 5311 y Fl(2)p Fr(\015)1090 5352 y Fo(j)p Fs(\032)p Fp(\()p Fs(x;)g(k)s Fp(\))p Fo(j)1425 5311 y Fl(2)1481 5352 y Fs(d)1532 5311 y Fl(3)1571 5352 y Fs(x)1626 5242 y Fi(o)1693 5264 y Fl(1)p Fr(=)p Fl(2)1842 5352 y Fp(+)1956 5242 y Fi(n)2040 5217 y(Z)2156 5352 y Fo(j)p Fs(x)p Fo(j)2267 5311 y Fl(2)p Fr(\015)2368 5352 y Fo(j)p Fs(\021)t Fp(\()p Fs(x;)g(k)s Fp(\))p Fo(j)2705 5311 y Fl(2)2761 5352 y Fs(d)2812 5311 y Fl(3)2851 5352 y Fs(x)2906 5242 y Fi(o)2973 5264 y Fl(1)p Fr(=)p Fl(2)3271 5352 y Fv(\(I.19\))p eop %%Page: 7 8 7 7 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2409 b Fv(7)257 573 y(W)-8 b(e)25 b(remark)g(that,)f(for)h(a)f(gi)n(v)o(en)f(function)h Fs(M)10 b Fv(,)25 b(a)g(physically)e(natural)h(choice)h(for)f Fs(\021)k Fv(and)d Fs(\032)g Fv(is)1053 792 y Fs(\021)t Fp(\()p Fs(x;)17 b(k)s Fp(\))52 b(:=)h Fs(\032)p Fp(\()p Fs(x;)17 b(k)s Fp(\))53 b(:=)f Fs(M)10 b Fp(\()p Fs(x;)17 b(k)s Fp(\))g Fs(')2443 807 y Fr(el)2503 792 y Fp(\()p Fs(x)p Fp(\))22 b Fs(;)588 b Fv(\(I.20\))257 1010 y(where)32 b Fs(')596 1025 y Fr(el)693 1010 y Fo(2)39 b Fs(L)864 974 y Fl(2)904 1010 y Fp(\()p Fh(R)1008 974 y Fl(3)1053 1010 y Fp(\))p Fv(,)32 b Fo(k)p Fs(')1262 1025 y Fr(el)1321 1010 y Fo(k)38 b Fp(=)g(1)p Fv(,)32 b(is)f(the)f(normalized)g(w)o(a)n (v)o(e)h(function)f(of)g(the)h(atomic)257 1131 y(bound)19 b(state.)29 b(In)20 b(this)f(case,)i(it)e(is)h(easy)g(to)f(see)h(that)g (\(I.19\))g(can)g(be)g(reduced)g(to)g(the)f(follo)n(wing)257 1251 y(simpli\002ed)24 b(condition,)350 1470 y Fs(J)413 1428 y Ff(0)404 1494 y Fr(K)q(;\015)528 1470 y Fp(\()p Fs(k)s Fp(\))53 b Fo(\024)g Fs(J)904 1428 y Ff(00)895 1494 y Fr(K)q(;\015)1019 1470 y Fp(\()p Fs(k)s Fp(\))g(:=)f(3)147 b(sup)1423 1557 y Fr(x)p Ff(2)p Fe(R)1558 1538 y Fd(3)1592 1557 y Fr(;)15 b Ff(j)p Fr(\013)p Ff(j\024)p Fr(K)1848 1385 y Fi(\014)1848 1445 y(\014)1881 1470 y Fs(@)1937 1428 y Fr(\013)1932 1494 y(k)1987 1470 y Fs(M)10 b Fp(\()p Fs(x;)17 b(k)s Fp(\))2320 1385 y Fi(\014)2320 1445 y(\014)2393 1470 y Fp(+)39 b(4)27 b(sup)2574 1557 y Fr(x)p Ff(2)p Fe(R)2709 1538 y Fd(3)2759 1385 y Fi(\014)2759 1445 y(\014)2814 1470 y Fo(j)p Fs(x)p Fo(j)2925 1428 y Fr(\015)2986 1470 y Fs(M)10 b Fp(\()p Fs(x;)17 b(k)s Fp(\))3319 1385 y Fi(\014)3319 1445 y(\014)3375 1470 y Fs(:)3271 1676 y Fv(\(I.21\))257 1893 y(Note)25 b(that)f(the)h(special)g(form)f (\(I.12\))h(of)g Fs(G)p Fp(\()p Fs(k)s Fp(\))g Fv(implies)e(that)1291 2112 y Fp(\()p Fs(P)1392 2127 y Fr(d)1454 2112 y Fo(\012)g Fg(1)p Fp(\))28 b Fs(W)41 b Fp(\()p Fs(P)1910 2127 y Fr(d)1972 2112 y Fo(\012)23 b Fg(1)p Fp(\))53 b(=)f(0)22 b Fs(;)826 b Fv(\(I.22\))257 2330 y(since)25 b(the)g(lo)n(wer)f(right)g (matrix)g(element)g(of)h Fs(G)p Fp(\()p Fs(k)s Fp(\))g Fv(v)n(anishes.)404 2451 y(W)-8 b(e)30 b(remark)h(that)f(if)g(the)h (electron)f(Hamiltonian)f(is)h(a)g(Schr)8 b(\250)-41 b(odinger)31 b(operator)l(,)h(as)e(de-)257 2571 y(scribed)e(belo)n(w)f (Eq.)g(\(I.5\),)i(a)f Fq(single)f Fv(diagonal)g(matrix)g(element)g(can) h(al)o(w)o(ays)g(be)g(\223gauged)257 2692 y(a)o(w)o(ay\224)22 b(by)f(means)g(of)h(a)f(Bogoliubo)o(v)f(transformation.)28 b(The)22 b(construction)e(of)h(this)g(Bogoli-)257 2812 y(ubo)o(v)k(transformation)g(is)g(gi)n(v)o(en)f(in)i(Appendix)e(A.)34 b(That)26 b(is,)f(for)h(our)g(model)f(with)g(a)h Fq(single)257 2932 y Fv(atomic)g(bound)g(state,)g(we)h(do)g(not)f(loose)f(an)o(y)h (generality)h(by)f(assuming)f(\(I.22\).)36 b(As)26 b(\(I.22\))257 3053 y(is)33 b(an)h(important)e(assumption)g(for)h(the)h(present)f (paper)l(,)j(we)e(point)e(out)h(that,)i(in)e(case)h(the)257 3173 y(atom)26 b(has)h(more)g(than)f(one)h(bound)f(state,)g Fs(P)1817 3188 y Fr(d)1884 3173 y Fv(w)o(ould)g(be)h(the)g(orthogonal)e (projection)h(onto)257 3293 y(these,)h(and)f(\(I.22\))g(w)o(ould)f (mean)h(that)g(we)g(assume)f(these)h(bound)g(states)f(not)h(to)f (interact)h(at)257 3414 y(all.)257 3680 y Fk(I.1.4)99 b(Selfadjointness,)26 b(Semiboundedness,)h(and)f(Binding)257 3867 y Fv(Ne)o(xt,)20 b(we)f(discuss)f(self)o(adjointness,)g (semiboundedness)f(and)i(e)o(xistence)f(of)h(a)g(ground)g(state)257 3988 y(\(binding\))24 b(of)g(the)h(Hamiltonian)d Fs(H)1510 4003 y Fr(g)1575 3988 y Fv(of)i(the)h(system.)k(W)-8 b(e)25 b(basically)f(in)l(v)n(ok)o(e)g(the)g(theorems)257 4108 y(and)h(methods)f(for)h(their)f(proof)h(established)f(in)g([2,)h (Sects.)g(II)g(and)g(III].)404 4229 y(Assuming)h(Hypothesis)h(1,)i (self)o(adjointness)e(and)h(semiboundedness,)f(for)i(an)o(y)f(v)n(alue) 257 4349 y(of)39 b(the)f(coupling)f(parameter)i Fs(g)56 b(>)c Fp(0)p Fv(,)42 b(are)d(a)g(simple)e(consequence)h(of)h(standard)f (Kato)257 4469 y(perturbation)33 b(theory;)38 b(we)c(appeal)g(to)g([2,) i(Cor)-5 b(.)34 b(I.7)g(and)g(Lemma)f(I.5])h(which)g(yield)f(that)257 4590 y Fs(H)338 4605 y Fr(g)402 4590 y Fv(is)24 b(self)o(adjoint)f(on)g (its)h(natural)f(domain,)g Fp(dom[)p Fs(H)2120 4605 y Fr(g)2160 4590 y Fp(])k(=)h(dom)o([)p Fs(H)2610 4605 y Fl(0)2649 4590 y Fp(])p Fv(,)d(and)f(that)f(the)h(ground)257 4710 y(state)h(ener)n(gy)g(is)f(gi)n(v)o(en)g(by)1190 4929 y Fs(E)1262 4944 y Fl(0)1354 4929 y Fp(:=)53 b(inf)22 b Fs(\033)t Fp([)p Fs(H)1811 4944 y Fr(g)1851 4929 y Fp(])53 b Fo(\025)g Fs(e)2106 4944 y Fl(0)2168 4929 y Fp(+)22 b Fo(O)s Fp(\()p Fs(g)2437 4888 y Fl(2)2476 4929 y Fp(\))g Fs(:)708 b Fv(\(I.23\))257 5147 y(In)32 b(the)g(present)f (paper)h(we)g(additionally)e(assume)h(that)g Fs(E)2332 5162 y Fl(0)2403 5147 y Fv(is)g(an)h(eigen)l(v)n(alue)f(of)h Fs(H)3280 5162 y Fr(g)3351 5147 y Fv(and)257 5268 y(that)25 b(the)g(corresponding)g(eigen)l(v)o(ectors)f(ha)n(v)o(e)h(a)h(lar)n(ge) g(component)e(in)h(the)g(v)n(acuum)f(sector)257 5388 y Fo(F)339 5352 y Fl(\(0\))433 5388 y Fv(.)31 b(More)25 b(precisely)-6 b(,)24 b(we)h(require)p eop %%Page: 8 9 8 8 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2409 b Fv(8)257 573 y Fk(Hypothesis)25 b(2.)49 b Fq(The)20 b(Hamiltonian)c Fs(H)1649 588 y Fr(g)1708 573 y Fq(possesses)i(a)h(\(normalizable\))e (gr)l(ound)g(state)h Fp(\010)3368 588 y Fl(gs)3464 573 y Fo(2)257 693 y(H)q Fq(,)25 b Fo(k)p Fp(\010)512 708 y Fl(gs)580 693 y Fo(k)i Fp(=)h(1)p Fq(,)c(i.e)o(.,)g Fs(E)1101 708 y Fl(0)1166 693 y Fq(is)g(an)h(eig)o(en)l(value)f(with)h (corr)l(esponding)e(eig)o(en)l(vector)i Fp(\010)3187 708 y Fl(gs)3255 693 y Fq(,)1491 886 y Fs(H)1572 901 y Fr(g)1628 886 y Fp(\010)1698 901 y Fl(gs)1818 886 y Fp(=)53 b Fs(E)2019 901 y Fl(0)2075 886 y Fp(\010)2145 901 y Fl(gs)2235 886 y Fs(:)1009 b Fv(\(I.24\))257 1080 y Fq(Mor)l(eo)o(ver)-11 b(,)25 b(denoting)f Fs(P)1121 1095 y Fl(\012)1204 1080 y Fp(:=)j Fo(j)p Fp(\012)p Fo(ih)p Fp(\012)p Fo(j)h(\021)g Fg(1)22 b Fo(\012)h(j)p Fp(\012)p Fo(ih)p Fp(\012)p Fo(j)p Fq(,)h(the)h(gr)l(ound)e(state)h Fp(\010)2980 1095 y Fl(gs)3073 1080 y Fq(obe)m(ys)1474 1188 y Fi(\015)1474 1248 y(\015)1530 1273 y Fs(P)1607 1232 y Ff(?)1593 1298 y Fl(\012)1693 1273 y Fp(\010)1763 1288 y Fl(gs)1831 1188 y Fi(\015)1831 1248 y(\015)1938 1273 y Fo(\024)54 b Fs(C)23 b(g)j(;)1009 b Fv(\(I.25\))257 1466 y Fq(for)25 b(some)f(constant)g Fs(C)34 b(<)28 b Fo(1)p Fq(.)404 1647 y Fv(The)39 b(e)o(xistence)g(\(I.24\))g(of)h(a)f (ground)g(state)g Fp(\010)2090 1662 y Fl(gs)2197 1647 y Fv(and)g(the)h(o)o(v)o(erlap)e(bound)g(\(I.25\))i(is)257 1767 y(pro)o(v)o(ed)26 b(in)g([2,)h(Thm.)e(I.1])i(under)g(the)f (assumption)e(of)j(a)g(some)n(what)e(stronger)h(bound)g(than)257 1887 y(Eq.)f(\(I.15\))g(in)f(Hypothesis)f(1,)i(namely)-6 b(,)1112 1990 y Fi(Z)1228 2015 y(\020)1288 2126 y Fp(1)22 b(+)g Fs(!)t Fp(\()p Fs(k)s Fp(\))1652 2085 y Ff(\000)p Fl(2)1745 2015 y Fi(\021)1821 2126 y Fo(j)p Fs(J)1903 2141 y Fl(0)p Fr(;\015)2002 2126 y Fp(\()p Fs(k)s Fp(\))p Fo(j)2160 2085 y Fl(2)2216 2126 y Fs(d)2267 2085 y Fl(3)2306 2126 y Fs(k)56 b Fo(\024)d Fp(1)22 b Fs(:)630 b Fv(\(I.26\))257 2361 y(In)37 b(this)f(paper)l(,)k(we)d(do)g(not)f(use)h(this)f (estimate)g(b)n(ut)g(only)g(its)g(consequence)h(in)g(form)f(of)257 2481 y(Hypothesis)24 b(2.)404 2601 y(W)-8 b(e)25 b(remark)g(that)g(the) g(noninteracting)e(Hamiltonian)h Fs(H)2393 2616 y Fl(0)2457 2601 y Fv(has)h(the)g(unique)f(\(non-dege-)257 2722 y(nerate\))d (ground)e(state)h Fp(0)t Fo(\010)t Fp(\012)27 b Fo(2)h(H)21 b Fv(corresponding)e(to)h(its)f(ground)g(state)h(ener)n(gy)g Fs(e)3130 2737 y Fl(0)3169 2722 y Fv(,)h(and)f(we)257 2842 y(note)j(in)f(parentheses)h(that,)f(by)h(the)f(results)g(of)h ([2],)h(Eq.)e(\(I.25\))h(holds)f(for)h Fq(any)g Fv(ground)f(state)257 2963 y(of)30 b Fs(H)451 2978 y Fr(g)491 2963 y Fv(.)45 b(Consequently)-6 b(,)29 b(the)g(interacting)g(ground)g(state)g Fp(\010)2359 2978 y Fl(gs)2456 2963 y Fv(of)h Fs(H)2650 2978 y Fr(g)2720 2963 y Fv(is)f(unique,)h(pro)o(vided)257 3083 y Fs(g)h(>)d Fp(0)d Fv(is)f(suf)n(\002ciently)g(small.)404 3203 y(In)33 b(Lemma)g(IV)-13 b(.2)33 b(we)h(strengthen)e(\(I.25\))h (and)h(sho)n(w)e(that,)j(for)e(an)o(y)g Fs(\013)44 b Fo(\025)g Fp(1)33 b Fv(and)g(an)o(y)257 3324 y Fp(\003)28 b Fs(<)f Fo(1)p Fv(,)566 3432 y Fi(\015)566 3492 y(\015)621 3517 y Fp(\()p Fs(H)748 3466 y Fl(\(\003\))740 3545 y Fr(f)878 3517 y Fp(+)22 b Fg(1)p Fp(\))1070 3476 y Fr(\013)1136 3517 y Fs(P)1199 3532 y Fr(c)1250 3517 y Fp(\010)1320 3532 y Fl(gs)1388 3432 y Fi(\015)1388 3492 y(\015)1487 3517 y Fp(+)1607 3432 y Fi(\015)1607 3492 y(\015)1663 3517 y Fp(\()p Fs(H)1790 3466 y Fl(\(\003\))1782 3545 y Fr(f)1919 3517 y Fp(+)g Fg(1)p Fp(\))2111 3476 y Fr(\013)2177 3517 y Fs(P)2254 3476 y Ff(?)2240 3542 y Fl(\012)2330 3517 y Fp(\010)2400 3532 y Fl(gs)2467 3432 y Fi(\015)2467 3492 y(\015)2575 3517 y Fo(\024)53 b(O)s Fp(\()p Fs(g)t Fp(\))22 b Fs(:)308 b Fv(\(I.27\))257 3740 y(Here,)31 b Fs(H)595 3689 y Fl(\(\003\))587 3767 y Fr(f)731 3740 y Fv(denotes)e(a)g(free)h(photon)e(Hamiltonian)f(acting)h(only)g(on)h (photon)f(states)g(with)257 3860 y(ener)n(gy)e(less)e(than)g Fp(\003)k Fs(<)f Fo(1)p Fv(,)401 4098 y Fs(H)490 4047 y Fl(\(\003\))482 4126 y Fr(f)650 4098 y Fp(=)778 3963 y Fi(Z)894 4098 y Fs(d)945 4057 y Fl(3)985 4098 y Fs(k)19 b(!)1116 4113 y Fl(\003)1169 4098 y Fp(\()p Fs(k)s Fp(\))e Fs(a)1367 4057 y Ff(\003)1406 4098 y Fp(\()p Fs(k)s Fp(\))p Fs(a)p Fp(\()p Fs(k)s Fp(\))53 b(=)1898 3963 y Fi(Z)1954 4188 y Ff(f)p Fr(!)r Fl(\()p Fr(k)r Fl(\))p Ff(\024)p Fl(\003)p Ff(g)2289 4098 y Fs(d)2340 4057 y Fl(3)2379 4098 y Fs(k)19 b(!)t Fp(\()p Fs(k)s Fp(\))e Fs(a)2712 4057 y Ff(\003)2751 4098 y Fp(\()p Fs(k)s Fp(\))p Fs(a)p Fp(\()p Fs(k)s Fp(\))22 b Fs(;)160 b Fv(\(I.28\))257 4361 y(where)26 b Fs(!)587 4376 y Fl(\003)639 4361 y Fp(\()p Fs(k)s Fp(\))i(:=)g Fs(!)t Fp(\()p Fs(k)s Fp(\))17 b Fg(1)1196 4376 y Ff(f)p Fr(!)r Fl(\()p Fr(k)r Fl(\))p Fr(<)p Fl(\003)p Ff(g)1513 4361 y Fv(.)257 4652 y Fj(I.2)119 b(Main)30 b(Results)g(and)h(Discussion)257 4839 y Fk(I.2.1)99 b(Char)o(ge)26 b(transported)g(to)f(In\002nity)257 5027 y Fv(Ha)n(ving)f(introduced)f(the)g(quantum)g(mechanical)g(frame)n(w)o (ork)h(in)f(terms)g(of)h(Hilbert)f(spaces)257 5147 y(and)k (Hamiltonians)d(and)i(established)g(some)f(basic)i(f)o(acts)f(such)g (as)h(self)o(adjointness,)e(semi-)257 5268 y(boundedness,)d(and)g (binding,)f(we)h(proceed)h(to)f(de\002ning)g(the)g Fq(c)o(har)l(g)o(e)f (tr)o(ansported)f(to)i(in\002n-)257 5388 y(ity)k Fv(or)g(simply)f(the)h Fq(tr)o(ansported)d(c)o(har)l(g)o(e)p Fv(.)34 b(T)-8 b(o)26 b(this)f(end,)i(we)f(introduce)f(the)h(projection)f Fs(F)3437 5403 y Fr(R)p eop %%Page: 9 10 9 9 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2409 b Fv(9)257 573 y(onto)24 b(the)h(functions)f(with)g(support)g(outside)f(the)i (ball)f(of)h(radius)g Fs(R)j(>)g Fp(0)p Fv(.)i(More)25 b(precisely)-6 b(,)1211 848 y Fs(F)1274 863 y Fr(R)1384 848 y Fp(:=)1540 707 y Fi(\022)1655 787 y Fg(1)1711 802 y Ff(fj)p Fr(x)p Ff(j\025)p Fr(R)p Ff(g)2056 787 y Fp(0)1789 907 y(0)218 b(0)2146 707 y Fi(\023)2258 848 y Fo(\012)39 b Fg(1)2430 863 y Fr(f)2498 848 y Fs(;)746 b Fv(\(I.29\))257 1126 y(where)31 b Fg(1)587 1141 y Ff(fj)p Fr(x)p Ff(j\025)p Fr(R)p Ff(g)887 1126 y Fp(:=)38 b Fg(1)1084 1142 y Fe(R)1132 1124 y Fd(3)1167 1142 y Ff(n)p Fr(B)s Fl(\(0)p Fr(;R)p Fl(\))1426 1126 y Fp([)p Fs(x)p Fp(])p Fv(.)49 b(Similarly)-6 b(,)30 b(we)h(introduce)e(the)i(projection)e Fs(T)3227 1141 y Ff(T)3318 1126 y Fv(onto)257 1246 y(the)24 b(particle)g(states)g (with)f(momentum)f(in)i(a)h(measurable)e(set)h Fo(T)54 b(\032)28 b Fh(R)2713 1210 y Fl(3)2758 1246 y Fv(,)c(i.e.,)g(the)g (functions)257 1367 y(whose)h(F)o(ourier)g(transform)f(is)g(supported)g (in)g Fo(T)i Fv(.)31 b(That)24 b(is,)1245 1641 y Fs(T)1302 1656 y Ff(T)1415 1641 y Fp(:=)1571 1501 y Fi(\022)1686 1580 y Fg(1)1742 1596 y Ff(f)p Fr(p)p Ff(2T)17 b(g)2038 1580 y Fp(0)1796 1701 y(0)193 b(0)2128 1501 y Fi(\023)2241 1641 y Fo(\012)39 b Fg(1)2413 1656 y Fr(f)2480 1641 y Fs(:)764 b Fv(\(I.30\))257 1919 y(where)32 b Fg(1)588 1935 y Ff(f)p Fr(p)p Ff(2T)17 b(g)832 1919 y Fv(acts)30 b(as)h(a)g(F)o(ourier)g(multiplier)e(with)g Fg(1)2216 1934 y Ff(T)2277 1919 y Fp([)p Fs(p)p Fp(])p Fv(.)48 b(Gi)n(v)o(en)30 b(a)h(state)f Fp(\011)38 b Fo(2)h(H)q Fv(,)32 b(the)257 2040 y(corresponding)e(\(least)h(and)f(most\))g (transported)g(char)n(ges)h(with)f(momentum)f(in)h Fo(T)64 b(\032)39 b Fh(R)3450 2004 y Fl(3)257 2160 y Fv(are)26 b(de\002ned)f(to)g(be)838 2380 y Fs(Q)915 2339 y Fl(inf)915 2405 y Ff(T)1005 2380 y Fp(\(\011\))83 b(:=)g(lim)17 b(inf)1464 2442 y Fr(R)p Ff(!1)1735 2380 y Fp(lim)g(inf)1787 2440 y Fr(t)p Ff(!1)2022 2296 y 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Fp(\(\011\))f Fv(and)g Fs(Q)847 4145 y Fl(sup)847 4219 y Ff(T)957 4192 y Fp(\(\011\))g Fv(to)g(leading)f(order)h(in)f Fs(g)t Fv(.)404 4313 y(Finally)-6 b(,)31 b(we)g(remark)g(that)g(it)f (is)h(important)e(to)i(observ)o(e)f(the)h(order)g(of)g(the)g(tw)o(o)f (limits)257 4433 y Fs(R)f Fo(!)e(1)d Fv(and)g Fs(t)k Fo(!)g(1)p Fv(,)23 b(as)i(interchanging)e(these)h(limits)e(w)o(ould,)h (indeed,)h(yield)g(the)g(tri)n(vial)257 4553 y(result)h Fp(lim)639 4568 y Fr(t)p Ff(!1)827 4553 y Fp(lim)962 4568 y Fr(R)p Ff(!1)1177 4553 y Fo(k)p Fs(F)1290 4568 y Fr(R)1365 4553 y Fs(e)1410 4517 y Ff(\000)p Fr(itH)1572 4525 y Fc(g)1629 4553 y Fp(\011)p Fo(k)1755 4517 y Fl(2)1822 4553 y Fp(=)j(0)p Fv(,)c(for)h(all)g Fp(\011)i Fo(2)h(H)q Fv(.)j(Lik)o(e)n(wise,)24 b(we)h(ha)n(v)o(e)1088 4773 y Fp(lim)1059 4835 y Fr(R)p Ff(!1)1270 4773 y Fp(sup)1285 4853 y Fr(t>)p Fl(0)1433 4689 y Fi(\015)1433 4749 y(\015)1489 4773 y Fs(F)1552 4788 y Fr(R)1626 4773 y Fg(1)1682 4788 y Fr(pp)1757 4773 y Fp(\()p Fs(H)1876 4788 y Fr(g)1916 4773 y Fp(\))17 b Fs(e)2016 4732 y Ff(\000)p Fr(itH)2178 4740 y Fc(g)2235 4773 y Fp(\011)2311 4689 y Fi(\015)2311 4749 y(\015)2450 4773 y Fp(=)82 b(0)22 b Fs(;)565 b Fv(\(I.33\))1059 4986 y Fp(lim)1030 5048 y Fr(R)p Ff(!1)1241 4986 y Fp(sup)1256 5066 y Fr(t>)p Fl(0)1404 4902 y Fi(\015)1404 4962 y(\015)1460 4986 y Fs(F)1523 5001 y Fr(R)1597 4986 y Fg(1)1653 5014 y Fe(R)1701 4984 y Fb(\000)1701 5035 y Fd(0)1757 4986 y Fp(\()p Fs(H)1876 5001 y Fr(g)1916 4986 y Fp(\))17 b Fs(e)2016 4945 y Ff(\000)p Fr(itH)2178 4953 y Fc(g)2235 4986 y Fp(\011)2311 4902 y Fi(\015)2311 4962 y(\015)2450 4986 y Fp(=)82 b(0)22 b Fs(;)565 b Fv(\(I.34\))257 5256 y(for)25 b(all)g Fp(\011)j Fo(2)g(H)q Fv(.)p eop %%Page: 10 11 10 10 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(10)257 573 y Fk(I.2.2)99 b(Gr)n(ound)26 b(State)g(and)f(Photon)h (Cloud)257 760 y Fv(Our)f(choice)g(for)g(the)g(initial)e(state)i Fp(\011)g Fv(is)f(of)h(the)f(form)1474 980 y Fp(\011)53 b(:=)f Fs(A)p Fp(\()p Fs(\034)6 b(;)17 b(f)p 1961 1016 59 4 v 11 w Fp(\))g(\010)2145 995 y Fl(gs)2234 980 y Fs(;)1010 b Fv(\(I.35\))257 1200 y(where)765 1421 y Fs(A)p Fp(\(0)p Fs(;)17 b(f)p 969 1456 V 10 w Fp(\))52 b Fo(\021)h Fs(A)p Fp(\()p Fs(f)p 1358 1456 V 11 w Fp(\))g(:=)f Fg(1)1719 1436 y Fr(el)1817 1421 y Fo(\012)39 b Fs(a)1984 1379 y Ff(\003)2024 1421 y Fp(\()p Fs(f)2110 1436 y Fl(1)2149 1421 y Fp(\))17 b Fs(a)2255 1379 y Ff(\003)2294 1421 y Fp(\()p Fs(f)2380 1436 y Fl(2)2420 1421 y Fp(\))33 b Fo(\001)17 b(\001)g(\001)31 b Fs(a)2691 1379 y Ff(\003)2731 1421 y Fp(\()p Fs(f)2817 1436 y Fr(N)2884 1421 y Fp(\))22 b Fs(;)300 b Fv(\(I.36\))257 1641 y(and)994 1861 y Fs(A)p Fp(\()p Fs(\034)6 b(;)17 b(f)p 1197 1896 V 11 w Fp(\))52 b(:=)h Fs(e)1547 1819 y Ff(\000)p Fr(i\034)8 b(H)1723 1827 y Fc(g)1780 1861 y Fs(e)1825 1819 y Fr(i\034)g(H)1946 1828 y Fd(0)2002 1861 y Fs(A)p Fp(\()p Fs(f)p 2113 1896 V 10 w Fp(\))17 b Fs(e)2271 1819 y Ff(\000)p Fr(i\034)8 b(H)2447 1828 y Fd(0)2502 1861 y Fs(e)2547 1819 y Fr(i\034)g(H)2668 1827 y 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y(S.)f(De)f(Bi)6 b(\036)-39 b(evre,)26 b(M.)e(Griesemer)l(,)i(H.)e(Spohn)h(and)g(especially)g(S.)g(T)-7 b(eufel)25 b(for)h(clarifying)e(this)257 2682 y(point)31 b(to)g(us.\))50 b(From)31 b(Eq.)g(\(II.8\))h(belo)n(w)e(we)i(sho)n(w)e (that,)i(in)f(particular)l(,)i Fs(A)p Fp(\()p Fo(1)p Fs(;)17 b(f)p 3119 2718 V 10 w Fp(\)\010)3285 2697 y Fl(gs)3392 2682 y Fp(:=)257 2803 y(lim)393 2818 y Fr(\034)8 b Ff(!1)594 2803 y Fs(A)p Fp(\()p Fs(\034)e(;)17 b(f)p 797 2838 V 10 w Fp(\)\010)963 2818 y Fl(gs)1058 2803 y Fv(e)o(xists,)25 b(thus)h(representing)g(an)g(incoming)f(photon)h (scattering)g(state.)257 2923 y(In)i(contrast,)g(choosing)e Fp(\011)i Fv(according)g(to)f(\(I.35\))h(and)f(with)g Fs(\034)44 b Fp(=)33 b(0)28 b Fv(amounts)e(to)h(adding)g(at)257 3044 y(time)d Fs(t)k Fp(=)g(0)d Fv(the)f(photon)g(cloud)g Fs(A)p Fp(\()p Fs(f)p 1505 3079 V 11 w Fp(\))h Fv(to)f(the)h (\(interacting\))f(atom)g(at)h(rest.)404 3164 y(W)-8 b(e)42 b(henceforth)g(often)g(lea)n(v)o(e)f(out)h(tri)n(vial)e(tensor)h 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y Fl(3)2937 4485 y Fs(k)25 b(;)231 b Fv(\(I.38\))731 4750 y Fo(h)p Fs(L)836 4703 y Fr(p)836 4774 y Fl(0)892 4750 y Fs(;)33 b(f)11 b Fo(i)83 b Fp(:=)1319 4614 y Fi(Z)1482 4683 y Fo(\000)p Fs(i)26 b Fp(^)-58 b Fs(\032)q Fp(\()p Fs(p;)17 b(k)s Fp(\))g Fs(f)11 b Fp(\()p Fs(k)s Fp(\))17 b Fs(d)2157 4646 y Fl(3)2194 4683 y Fs(k)p 1445 4727 841 4 v 1445 4818 a(p)1494 4790 y Fl(2)1556 4818 y Fo(\000)22 b Fs(E)1727 4833 y Fl(0)1789 4818 y Fo(\000)h Fs(!)t Fp(\()p Fs(k)s Fp(\))e(+)h Fs(i)p Fp(0)3271 4750 y Fv(\(I.39\))1146 5012 y Fp(=)97 b Fo(\000)p Fs(\031)1472 4877 y Fi(Z)1588 5012 y Fs(\016)1635 4931 y Fi(\000)1681 5012 y Fs(p)1730 4971 y Fl(2)1791 5012 y Fo(\000)23 b Fs(E)1963 5027 y Fl(0)2025 5012 y Fo(\000)f Fs(!)t Fp(\()p Fs(k)s Fp(\))2319 4931 y Fi(\001)2390 5012 y Fp(^)-58 b Fs(\032)p Fp(\()p Fs(p;)17 b(k)s Fp(\))g Fs(f)11 b Fp(\()p Fs(k)s Fp(\))17 b Fs(d)2928 4971 y Fl(3)2966 5012 y Fs(k)254 b Fv(\(I.40\))1319 5277 y Fo(\000)17 b Fs(i)g Fh(P)1539 5142 y Fi(Z)1675 5210 y Fp(^)-57 b Fs(\032)p Fp(\()p Fs(p;)17 b(k)s Fp(\))g Fs(f)11 b Fp(\()p Fs(k)s Fp(\))17 b Fs(d)2214 5174 y Fl(3)2251 5210 y Fs(k)p 1667 5255 640 4 v 1668 5346 a(p)1717 5317 y Fl(2)1778 5346 y Fo(\000)22 b Fs(E)1949 5361 y Fl(0)2011 5346 y Fo(\000)h Fs(!)t Fp(\()p Fs(k)s Fp(\))2338 5277 y Fs(;)p eop %%Page: 11 12 11 11 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(11)257 573 y Fh(P)333 493 y Fi(R)430 573 y Fv(denoting)28 b(the)h(Cauchy)f(principal)h(v)n(alue)f(and)38 b Fp(^)-58 b Fs(\032)p Fp(\()p Fs(p;)17 b(k)s Fp(\))29 b Fv(denoting)e(the)i (partial)f(F)o(ourier)257 693 y(transform)g(of)h Fs(\032)p Fp(\()p Fs(x;)17 b(k)s Fp(\))29 b Fv(\(as)g(a)g(square-inte)o(grable)e (function\))h(with)g(respect)h(to)f(the)g(particle)257 814 y(position)23 b(v)n(ariable)i Fs(x)p Fv(.)404 934 y(No)n(w)f(we)h(are)g(in)g(position)e(to)h(formulate)g(our)h(\002rst)g (main)f(result.)257 1137 y Fk(Theor)n(em)j(I.1.)49 b Fq(Assume)40 b(Hypotheses)g(1\2262,)k(\002x)c Fs(N)5 b(;)17 b(m)57 b Fo(2)g Fh(N)f Fq(with)40 b 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Fl(0)2960 3738 y Fs(;)33 b(')3084 3753 y Fr(j)3120 3738 y Fo(ij)3187 3697 y Fl(2)3226 3738 y Fs(d)3277 3697 y Fl(3)3316 3738 y Fs(p)22 b(:)257 4101 y Fq(Her)l(e)o(,)43 b Fs(L)585 4065 y Fr(p)585 4126 y Ff(1)698 4101 y Fq(and)38 b Fs(L)952 4054 y Fr(p)952 4126 y Fl(0)1031 4101 y Fq(ar)l(e)g(the)g(distrib)n (utions)e(de\002ned)i(in)g(Eqs.)g(\(I.38\)\226\(I.40\),)j(and)d Fs(\026)53 b Fo(\021)257 4222 y Fs(\026)p Fp(\()p Fs(K)7 b Fp(\))28 b(:=)g(1)22 b Fo(\000)h Fs(K)902 4186 y Ff(\000)p Fl(1)1024 4222 y Fs(>)28 b Fp(0)p Fq(,)c(wher)l(e)i Fs(K)32 b Fq(is)25 b(the)g(de)l(gr)l(ee)g(of)g(dif)n(fer)l(entiability)d(in)j (Hypothesis)e(1)257 4342 y(and)i(2.)404 4545 y Fv(Note)k(that)g(the)h (coef)n(\002cient)f Fs(Q)1482 4509 y Fr(\034)1482 4571 y Ff(T)1543 4545 y Fp(\()p Fs(')p 1581 4581 V Fp(\))g Fv(of)h(the)f(leading)g(order)h(contrib)n(ution)e(to)h(the)h(pho-)257 4666 y(toelectric)38 b(ef)n(fect)g(is)g Fq(additive)e Fv(\(counting)h(multiplicities\))e(in)j(the)f(single)g(photon)g(states) 257 4786 y Fs(')321 4801 y Fl(1)361 4786 y Fs(;)17 b(:)g(:)g(:)32 b(;)17 b(')660 4801 y Fr(m)726 4786 y Fv(.)50 b(This)30 b(implies)g(that)h(the)g(contrib)n(ution)e(of)i(each)h Fs(')2576 4801 y Fr(j)2643 4786 y Fv(is)f(independent)f(of)h(the)257 4907 y(other)26 b(single)g(photon)f(states)h Fs(')1368 4922 y Fr(i)1396 4907 y Fv(,)g Fs(i)31 b Fo(6)p Fp(=)f Fs(j)6 b Fv(,)27 b(in)f(the)g(photon)f(cloud,)h(in)g(contrast)g(to)g (the)g(classi-)257 5027 y(cal)k(model)f(for)h(the)g(photoelectric)f(ef) n(fect)h(which)g(w)o(ould)f(ha)n(v)o(e)g(predicted)h(a)g(dependence)257 5147 y(of)25 b Fs(Q)442 5111 y Fr(\034)442 5173 y Ff(T)503 5147 y Fp(\()p Fs(')p 541 5183 V Fp(\))g Fv(only)f(on)g(the)h Fq(total)e(ener)l(gy)j Fv(of)f(the)f(photon)g(cloud.)404 5268 y(While,)j(for)g(lar)n(ge)g(v)n(alues)g(of)g Fs(\034)11 b Fv(,)28 b(i.e.,)f(the)g(case)h(\(i\))f(in)f(Theorem)h(I.1)g(abo)o(v)o (e,)g(the)f(trans-)257 5388 y(ported)37 b(char)n(ge)g(directly)g (yields)f(Einstein')-5 b(s)35 b(predictions)g(\(I.1\))i(and)g(\(I.2\),) j(there)d(is)g(still)p eop %%Page: 12 13 12 12 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(12)257 573 y(an)35 b(inconsistenc)o(y)d(between)i(\(I.43\))g(and)h (Einstein')-5 b(s)32 b(predictions)h(\(I.1\))i(and)f(\(I.2\),)i(since) 257 693 y Fo(h)p Fs(L)362 646 y Fr(p)362 718 y Fl(0)419 693 y Fs(;)d(')543 708 y Fr(j)579 693 y Fo(i)23 b Fv(may)g(be)g(non-v)n (anishing,)d(e)n(v)o(en)i(if)h Fs(')1923 708 y Fr(j)1982 693 y Fv(is)g(supported)f(in)g(a)h(re)o(gion)f(where)h Fs(!)t Fp(\()p Fs(k)s Fp(\))k Fs(<)257 814 y(p)306 777 y Fl(2)362 814 y Fo(\000)16 b Fs(E)527 829 y Fl(0)567 814 y Fv(.)31 b(It)23 b(turns)f(out)h(that)g(\(I.43\))g(is)g(in)g (accordance)h(with)e(\(I.1\))i(and)f(\(I.2\))h(for)f Fq(monoc)o(hr)l(o-)257 934 y(matic)k(light)p Fv(,)g(i.e.,)h(for)g(a)g (photon)e(cloud)h(with)g(sharply)g(localized)g(momentum)e(on)j(\(one)f (or)257 1054 y(more\))e(photon)f(ener)n(gy)h(shell\(s\).)404 1175 y(The)19 b(photon)g(cloud)g(represented)h(by)g Fs(A)p Fp(\()p Fo(1)p Fs(;)d(')p 1970 1210 64 4 v -1 w Fp(\))j Fv(may)f(actually)g(be)h(physically)e(closer)i(to)257 1295 y(the)j(e)o(xperimental)f(situation)f(described)h(by)h(Einstein') -5 b(s)21 b(theory)h(than)h(the)f(one)h(represented)257 1416 y(by)29 b Fs(A)p Fp(\(0)p Fs(;)17 b(')p 590 1451 V -1 w Fp(\))p Fv(.)43 b(Indeed,)30 b(the)f(photons)e(in)i Fs(A)p Fp(\()p Fo(1)p Fs(;)17 b(')p 1942 1451 V -1 w Fp(\))29 b Fv(are)h(localized)e(at)h(spatial)f(in\002nity)g(\(and)257 1536 y(temporal)35 b(ne)o(gati)n(v)o(e)d(in\002nity\),)37 b(when)e(added)g(to)g(the)g(system.)60 b(Since)36 b(the)f(interaction)f (is)257 1656 y(switched)25 b(of)n(f)f(at)h(lar)n(ge)g(distances)g(from) f(the)h(origin,)f(the)h(sum)f(of)h(the)g(free)g(ener)n(gies)h(of)e(the) 257 1777 y(photon)d(cloud)h(and)g(the)g(atom)g(coincides)g(with)f(the)h (total)f(ener)n(gy)i(of)f(the)g(interacting)g(atom-)257 1897 y(photon)j(system.)34 b(In)27 b(contrast,)f(the)g(photons)f(in)h Fs(A)p Fp(\(0)p Fs(;)17 b(')p 2172 1932 V -1 w Fp(\))26 b Fv(can)h(be)f(vie)n(wed)g(as)g(added)h(to)e(the)257 2017 y(system)f(at)h Fs(t)j Fp(=)f(0)p Fv(,)e(hence)g(discontinuously)d (changing)i(the)h(total)f(ener)n(gy)h(of)g(the)g(system.)257 2283 y Fk(I.2.4)99 b(Monochr)n(omatic)26 b(Light)257 2470 y Fv(In)i(this)f(subsection,)h(we)g(further)g(analyze)g(the)g (leading)g(order)g(contrib)n(ution)e(to)i(the)f(trans-)257 2591 y(ported)e(char)n(ge)g(gi)n(v)o(en)f(by)g Fs(g)1243 2555 y Fl(2)1307 2591 y Fv(times)484 2879 y Fs(Q)561 2838 y Fl(0)561 2904 y Ff(T)622 2798 y Fi(\000)667 2879 y Fs(A)p Fp(\(0)p Fs(;)17 b(')p 871 2914 V Fp(\)\010)1043 2894 y Fl(gs)1110 2798 y Fi(\001)1209 2879 y Fp(=)1337 2798 y Fi(\000)1383 2879 y Fs(n)1441 2894 y Fl(1)1480 2879 y Fp(!)g Fs(n)1582 2894 y Fl(2)1622 2879 y Fp(!)g Fo(\001)g(\001)g(\001)d Fs(n)1856 2894 y Fr(m)1923 2879 y Fp(!)1950 2798 y Fi(\001)2070 2754 y Fr(m)2029 2784 y Fi(X)2040 2994 y Fr(j)t Fl(=1)2190 2879 y Fs(n)2248 2894 y Fr(j)2318 2743 y Fi(Z)2373 2969 y Ff(T)2450 2879 y Fo(jh)p Fs(L)2583 2832 y Fr(p)2583 2903 y Fl(0)2639 2879 y Fs(;)33 b(')2763 2894 y Fr(j)2800 2879 y Fo(ij)2867 2838 y Fl(2)2905 2879 y Fs(d)2956 2838 y Fl(3)2996 2879 y Fs(p)226 b Fv(\(I.44\))257 3188 y(for)34 b(the)f(special)h(case,)i (in)d(which)g(each)h(single)f(photon)f(w)o(a)n(v)o(e)h(function)g Fs(')2960 3203 y Fr(j)3030 3188 y Fv(has)g(support)257 3309 y(in)27 b(a)h(small)e(neighborhood)g(of)i(an)f(ener)n(gy)h(shell)f (in)g(momentum)e(space.)39 b(More)27 b(precisely)-6 b(,)257 3429 y(for)39 b Fs(N)5 b(;)17 b(m)54 b Fo(2)f Fh(N)9 b Fv(,)48 b(with)38 b Fs(m)54 b Fo(\024)f Fs(N)10 b Fv(,)43 b(we)c(\002x)g(photon)e(ener)n(gies)i Fs(!)2621 3444 y Fl(1)2660 3429 y Fs(;)17 b(!)2765 3444 y Fl(2)2804 3429 y Fs(;)g(:)g(:)g(:)32 b(;)17 b(!)3100 3444 y Fr(m)3220 3429 y Fo(2)53 b Fh(R)3405 3393 y Fl(+)3470 3429 y Fv(,)257 3550 y Fs(L)323 3513 y Fl(2)363 3550 y Fp(\()p Fh(R)5 b Fp(\))p Fv(-normalized)30 b(functions)23 b Fs(\037)1473 3565 y Fl(1)1513 3550 y Fs(;)17 b(\037)1618 3565 y Fl(2)1657 3550 y Fs(;)g(:)g(:)g(:)f(\037)1893 3565 y Fr(m)1987 3550 y Fo(2)28 b Fs(C)2158 3513 y Ff(1)2151 3574 y Fl(0)2233 3550 y Fp([\()p Fo(\000)2385 3510 y Fl(1)p 2385 3527 36 4 v 2385 3584 a(2)2431 3550 y Fs(;)2484 3510 y Fl(1)p 2484 3527 V 2484 3584 a(2)2530 3550 y Fp(\)])p Fv(,)c(and)g(normalized) g(func-)257 3670 y(tions)42 b(on)g(the)g(tw)o(o-sphere,)47 b Fs(\024)1370 3685 y Fl(1)1410 3670 y Fs(;)17 b(\024)1510 3685 y Fl(2)1549 3670 y Fs(;)g(:)g(:)g(:)f(\024)1780 3685 y Fr(m)1907 3670 y Fo(2)61 b Fs(C)2111 3634 y Ff(1)2185 3670 y Fp([)p Fh(S)2274 3634 y Fl(2)2307 3670 y Fp(])p Fv(,)47 b(for)c(which)f(we)h(assume)f(that)257 3790 y Fo(h)p Fs(\024)352 3805 y Fr(i)380 3790 y Fo(j)p Fs(\024)464 3805 y Fr(j)501 3790 y Fo(i)61 b Fp(=)h(0)p Fv(,)48 b(whene)n(v)o(er)43 b Fs(!)1349 3805 y Fr(i)1438 3790 y Fp(=)62 b Fs(!)1637 3805 y Fr(j)1673 3790 y Fv(.)87 b(Then,)47 b(for)d Fp(0)62 b Fs(<)f(\016)66 b(<)c Fp(1)p Fv(,)48 b(we)43 b(de\002ne)h(func-)257 3911 y(tions)37 b Fs(')553 3926 y Fl(1)p Fr(;\016)646 3911 y Fs(;)17 b(')754 3926 y Fl(2)p Fr(;\016)846 3911 y Fs(;)g(:)g(:)g(:)33 b(;)17 b(')1146 3926 y Fr(m;\016)1318 3911 y Fo(2)52 b Fs(C)1513 3875 y Ff(1)1506 3935 y Fl(0)1588 3911 y Fp([)p Fh(R)1681 3875 y Fl(3)1758 3911 y Fo(n)32 b(f)p Fp(0)p Fo(g)p Fp(])p Fv(,)41 b(in)d(spherical)g(coordinates)f Fp(\()p Fs(r)m(;)17 b Fp(\012)3286 3926 y Fr(k)3329 3911 y Fp(\))52 b(=)257 4031 y(\()p Fo(j)p Fs(k)s Fo(j)p Fs(;)17 b(k)s(=)p Fo(j)p Fs(k)s Fo(j)p Fp(\))24 b Fv(by)934 4285 y Fs(')998 4300 y Fr(j;\016)1083 4285 y Fp(\()p Fs(r)m(;)17 b Fp(\012)1276 4300 y Fr(k)1319 4285 y Fp(\))53 b(:=)g Fs(!)1631 4244 y Ff(\000)p Fl(1)1627 4311 y Fr(j)1741 4285 y Fs(\016)1788 4244 y Ff(\000)p Fl(1)p Fr(=)p Fl(2)1969 4285 y Fs(\037)2030 4300 y Fr(j)2067 4174 y Fi(\020)2136 4218 y Fs(r)25 b Fo(\000)e Fs(!)2366 4233 y Fr(j)p 2136 4262 266 4 v 2246 4353 a Fs(\016)2412 4174 y Fi(\021)2488 4285 y Fs(\024)2544 4300 y Fr(j)2581 4285 y Fp(\(\012)2689 4300 y Fr(k)2732 4285 y Fp(\))f Fs(:)452 b Fv(\(I.45\))257 4526 y(In)30 b(order)f(to)g(ensure)g(the)h(disjointness)c(of)k Fp(supp)q Fs(')2034 4541 y Fr(i;\016)2144 4526 y Fv(and)g Fp(supp)q Fs(')2583 4541 y Fr(j;\016)2669 4526 y Fv(,)g(whene)n(v)o(er) f Fs(!)3198 4541 y Fr(i)3261 4526 y Fo(6)p Fp(=)36 b Fs(!)3434 4541 y Fr(j)3470 4526 y Fv(,)257 4646 y(we)25 b(additionally)e(require)i(that)1208 4897 y Fs(\016)56 b(<)d Fp(min)1615 4786 y Fi(n)1691 4829 y Fs(!)1752 4844 y Fr(i)1802 4829 y Fo(\000)23 b Fs(!)1963 4844 y Fr(j)p 1691 4874 308 4 v 1821 4965 a Fp(2)2037 4782 y Fi(\014)2037 4842 y(\014)2037 4902 y(\014)2095 4897 y Fs(!)2156 4912 y Fr(i)2212 4897 y Fo(6)p Fp(=)k Fs(!)2376 4912 y Fr(j)2412 4786 y Fi(o)2501 4897 y Fs(;)743 b Fv(\(I.46\))257 5137 y(and,)25 b(furthermore,)1295 5352 y Fs(\016)57 b(<)52 b Fp(min)o Fo(f)p Fs(!)1797 5367 y Fl(1)1836 5352 y Fs(;)17 b(!)1941 5367 y Fl(2)1979 5352 y Fs(;)g(:)g(:)g(:)33 b(;)17 b(!)2276 5367 y Fr(m)2342 5352 y Fo(g)22 b Fs(;)830 b Fv(\(I.47\))p eop %%Page: 13 14 13 13 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(13)257 573 y(to)25 b(ensure)h(that)f Fs(')885 588 y Fr(j;\016)996 573 y Fv(is)g(supported)g(a)o(w)o(ay)g(from)g(zero)h (momentum.)31 b(Note)25 b(that)g(the)g(v)o(ectors)257 693 y Fs(')321 708 y Fl(1)p Fr(;\016)414 693 y Fs(;)17 b(')522 708 y Fl(2)p Fr(;\016)614 693 y Fs(;)g(:)g(:)g(:)33 b(;)17 b(')914 708 y Fr(m;\016)1058 693 y Fv(are)26 b(orthogonal)d(and) i(asymptotically)e(orthonormal,)1432 913 y Fp(lim)1430 976 y Fr(\016)r Ff(!)p Fl(0)1569 913 y Fo(h)p Fs(')1672 928 y Fr(i;\016)1753 913 y Fo(j)p Fs(')1845 928 y Fr(j;\016)1931 913 y Fo(i)52 b Fp(=)g Fs(\016)2193 928 y Fr(i;j)2296 913 y Fs(:)948 b Fv(\(I.48\))257 1162 y(Under)22 b(these)g(assumptions) e(and)i(choosing)f Fs(n)1876 1177 y Fl(1)1915 1162 y Fs(;)c(n)2017 1177 y Fl(2)2057 1162 y Fs(;)g(:)g(:)g(:)32 b(;)17 b(n)2350 1177 y Fr(m)2444 1162 y Fo(2)28 b Fh(N)9 b Fv(,)29 b(with)21 b Fs(n)2915 1177 y Fl(1)2966 1162 y Fp(+)12 b Fs(:)17 b(:)g(:)10 b Fp(+)i Fs(n)3325 1177 y Fr(m)3419 1162 y Fp(=)257 1283 y 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y Fr(j)1966 2112 y Fp(^)-58 b Fs(\032)p Fp(\()p Fs(p;)17 b(r)s(\033)t Fp(\))g Fs(\024)2355 2127 y Fr(j)2391 2112 y Fp(\()p Fs(\033)t Fp(\))g Fs(d)2594 2070 y Fl(2)2632 2112 y Fs(\033)26 b(:)531 b Fv(\(I.50\))257 2385 y(W)-8 b(e)37 b(require)g(an)g(additional)e(condition)h(on)45 b Fp(^)-58 b Fs(\032)p Fv(,)40 b(the)c(F)o(ourier)h(transform)f(with)g (respect)h(to)257 2505 y(the)32 b(electron)f(v)n(ariable)h Fs(x)g Fv(of)f Fs(\032)p Fv(,)j(de\002ned)e(in)f(\(I.17\).)52 b(More)31 b(speci\002cally)-6 b(,)33 b(we)f(require)f(the)257 2626 y(e)o(xistence)24 b(of)498 2873 y Fs(J)561 2832 y Ff(000)552 2897 y Fl(2)p Fr(;\015)651 2873 y Fp(\()p Fs(k)s Fp(\))53 b(:=)f(max)993 2939 y Ff(j)p Fr(\013)p Ff(j\024)p Fl(2)1187 2762 y Fi(n)1270 2737 y(Z)1387 2792 y(\000)1432 2873 y Fo(j)p Fs(x)p Fo(j)1543 2832 y Fl(2)p Fr(\015)1645 2873 y Fp(+)22 b(1)1792 2792 y Fi(\001)1854 2788 y(\014)1854 2848 y(\014)1887 2873 y Fs(@)1943 2832 y Fr(\013)1938 2897 y(k)1994 2873 y Fs(\032)p Fp(\()p Fs(x;)17 b(k)s 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Fs(\032)p Fp(\()p Fs(p;)17 b(k)s Fp(\))p Fo(j)2170 3533 y Fi(\011)2280 3613 y Fo(\024)53 b Fs(J)2473 3572 y Ff(000)2464 3638 y Fl(2)p Fr(;\015)2563 3613 y Fp(\()p Fs(k)s Fp(\))22 b Fs(;)529 b Fv(\(I.52\))257 3910 y(for)25 b(all)g Fs(k)31 b Fo(2)d Fh(R)765 3874 y Fl(3)810 3910 y Fv(.)j(W)-8 b(e)25 b(assume)f Fs(J)1402 3874 y Ff(0)q(00)1393 3935 y Fl(2)p Fr(;\015)1518 3910 y Fv(to)g(obe)o(y)g(the)h(follo)n (wing)e(bound.)257 4114 y Fk(Hypothesis)i(3.)49 b Fq(F)-10 b(or)24 b(some)h Fs(\015)33 b(>)27 b Fp(3)p Fs(=)p Fp(2)d Fq(and)h(all)f Fp(\003)j Fs(<)h Fo(1)p Fq(,)1233 4334 y Fp(sup)1397 4253 y Fi(\010)1455 4334 y Fs(J)1518 4293 y Ff(000)1509 4358 y Fl(2)p Fr(;\015)1608 4334 y Fp(\()p Fs(k)s Fp(\))1763 4249 y Fi(\014)1763 4309 y(\014)1821 4334 y Fo(j)p Fs(k)s Fo(j)f(\024)h Fp(\003)2131 4253 y Fi(\011)2242 4334 y Fs(<)52 b Fo(1)22 b Fs(:)752 b Fv(\(I.53\))404 4554 y(This)24 b(hypothesis)e(implies)i(the)g(follo)n (wing)f(bounds)h(on)g Fp(\002)2444 4569 y Fr(j)2481 4554 y Fv(,)h(for)g(all)f Fs(j)34 b Fp(=)27 b(1)p Fs(;)17 b(:)g(:)g(:)33 b(;)17 b(m)p Fv(,)412 4804 y Fp(sup)575 4693 y Fi(n)642 4804 y Fo(j)p Fp(\002)746 4819 y Fr(j)782 4804 y Fp(\()p Fs(p;)g(r)s Fp(\))p Fo(j)k Fp(+)h Fo(j)p Fs(@)1224 4819 y Fr(r)1262 4804 y Fp(\002)1338 4819 y Fr(j)1374 4804 y Fp(\()p Fs(p;)17 b(r)s Fp(\))p Fo(j)1642 4689 y Fi(\014)1642 4749 y(\014)1642 4809 y(\014)1700 4804 y Fs(p)28 b Fo(2)g Fh(R)1937 4763 y Fl(3)2010 4804 y Fs(;)41 b(r)31 b Fo(2)d Fp([0)17 b Fs(;)44 b(!)2472 4819 y Fr(j)2530 4804 y Fp(+)22 b(1])2704 4693 y Fi(o)2823 4804 y Fs(<)52 b Fo(1)22 b Fs(;)171 b Fv(\(I.54\))257 5045 y(which)25 b(we)g(require)g(for)g(the)g(proof)f(of)h(the)g(follo)n (wing)e(theorem.)p eop %%Page: 14 15 14 14 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(14)257 573 y Fk(Theor)n(em)27 b(I.2.)49 b Fq(Assume)30 b(Hypothesis)g(1\2263)g(and)g(Eqs.)g(\(I.45\)\226\(I.49\),)i(and)e (\002x)h(a)g(measur)n(-)257 693 y(able)23 b(set)g Fo(T)53 b(\022)29 b Fh(R)864 657 y Fl(3)910 693 y Fq(.)h(Then)24 b(in)e(the)h(limit)f(of)h(monoc)o(hr)l(omatic)e(light,)h(the)h(contrib) n(ution)e(of)h(the)257 814 y(photon)i Fs(a)610 777 y Ff(\003)650 814 y Fp(\()p Fs(')752 829 y Fr(j;\016)837 814 y Fp(\))h Fq(for)f(the)h(\002r)o(st)f(or)l(der)h(term)f(of)h(the)f (tr)o(ansported)f(c)o(har)l(g)o(e)h(is)g(given)h(by:)549 1155 y Fp(lim)547 1218 y Fr(\016)r Ff(!)p Fl(0)703 1019 y Fi(Z)702 1303 y Ff(T)819 1155 y Fo(jh)p Fs(L)952 1108 y Fr(p)952 1179 y Fl(0)1008 1155 y Fs(;)33 b(')1132 1170 y Fr(j;\016)1218 1155 y Fo(ij)1285 1114 y Fl(2)1341 1155 y Fs(d)1392 1114 y Fl(3)1431 1155 y Fs(p)52 b Fp(=)1661 951 y Fi(8)1661 1040 y(<)1661 1220 y(:)1938 1034 y Fp(0)230 b Fq(if)83 b Fs(E)2428 1049 y Fl(0)2489 1034 y Fp(+)22 b Fs(!)2648 1049 y Fr(j)2712 1034 y Fs(<)28 b Fp(0)22 b Fs(;)1801 1154 y Fp(\002)1877 1169 y Fl(0)p Fr(;j)1990 1154 y Fs(I)2033 1169 y Fl(0)p Fr(;j)2217 1154 y Fq(if)83 b Fs(E)2428 1169 y Fl(0)2489 1154 y Fp(+)22 b Fs(!)2648 1169 y Fr(j)2712 1154 y Fp(=)28 b(0)22 b Fs(;)1791 1274 y Fp(\002)1867 1289 y Fl(0)p Fr(;j)1980 1274 y Fs(I)2023 1289 y Fl(+)p Fr(;j)2217 1274 y Fq(if)83 b Fs(E)2428 1289 y Fl(0)2489 1274 y Fp(+)22 b Fs(!)2648 1289 y Fr(j)2712 1274 y Fs(>)28 b Fp(0)22 b Fs(:)3271 1155 y Fv(\(I.55\))257 1486 y Fq(Her)l(e)o(,)316 1742 y Fp(\002)392 1757 y Fl(0)p Fr(;j)536 1742 y Fp(:=)692 1607 y Fi(Z)747 1832 y Fe(S)790 1813 y Fd(2)839 1742 y Fg(1)895 1757 y Ff(T)956 1662 y Fi(\000)1002 1657 y(p)p 1101 1657 330 4 v 1101 1742 a Fs(E)1173 1757 y Fl(0)1235 1742 y Fp(+)g Fs(!)1394 1757 y Fr(j)1430 1742 y Fs(\033)t Fp(\))1555 1657 y Fi(p)p 1654 1657 352 4 v 1654 1742 a Fs(E)1726 1757 y Fl(0)1788 1742 y Fp(+)g Fs(!)1947 1757 y Fr(j)2033 1658 y Fi(\014)2033 1717 y(\014)2067 1742 y Fp(\002\()2181 1657 y Fi(p)p 2280 1657 330 4 v 2280 1742 a Fs(E)2352 1757 y Fl(0)2414 1742 y Fp(+)g Fs(!)2573 1757 y Fr(j)2626 1742 y Fs(\033)k(;)39 b(!)2834 1757 y Fr(j)2870 1742 y Fp(\))2908 1658 y Fi(\014)2908 1717 y(\014)2941 1684 y Fl(2)2997 1742 y Fs(d)3048 1701 y Fl(2)3087 1742 y Fs(\033)26 b(;)76 b Fv(\(I.56\))257 2008 y Fq(and)1110 2279 y Fs(I)1153 2294 y Fl(0)p Fr(;j)1327 2279 y Fp(:=)1513 2143 y Fi(Z)1613 2169 y Ff(1)1568 2369 y Fl(0)1704 2134 y Fi(\014)1704 2194 y(\014)1704 2254 y(\014)1704 2314 y(\014)1754 2143 y(Z)1853 2169 y Ff(1)1809 2369 y(\0001)2013 2211 y Fs(\037)2074 2226 y Fr(j)2111 2211 y Fp(\()p Fs(x)p Fp(\))p Fs(dx)p 1965 2256 431 4 v 1965 2347 a(y)26 b Fo(\000)c Fs(x)h Fp(+)f Fs(i)p Fp(0)2406 2134 y Fi(\014)2406 2194 y(\014)2406 2254 y(\014)2406 2314 y(\014)2439 2161 y Fl(2)2495 2279 y Fs(dy)j(;)625 b Fv(\(I.57\))1090 2570 y Fs(I)1133 2585 y Fl(+)p Fr(;j)1327 2570 y Fp(:=)1513 2434 y Fi(Z)1613 2460 y Ff(1)1568 2660 y(\0001)1715 2425 y Fi(\014)1715 2485 y(\014)1715 2545 y(\014)1715 2605 y(\014)1764 2434 y(Z)1864 2460 y Ff(1)1820 2660 y(\0001)2024 2502 y Fs(\037)2085 2517 y Fr(j)2121 2502 y Fp(\()p Fs(x)p Fp(\))p Fs(dx)p 1976 2547 V 1976 2638 a(y)25 b Fo(\000)e Fs(x)f Fp(+)g Fs(i)p Fp(0)2416 2425 y Fi(\014)2416 2485 y(\014)2416 2545 y(\014)2416 2605 y(\014)2450 2452 y Fl(2)2506 2570 y Fs(dy)j(;)614 b Fv(\(I.58\))404 2843 y(Einstein')-5 b(s)21 b(predictions)h(\(I.1\))i (and)f(\(I.2\))h(are)g(no)n(w)f(manifest)f(in)h(the)g(right)g(hand)g (side)g(of)257 2963 y(\(I.55\).)37 b(The)27 b(coef)n(\002cient)g Fp(lim)1306 2978 y Fr(\016)r Ff(!)p Fl(0)1466 2963 y Fs(Q)1543 2927 y Fl(0)1543 2988 y Ff(T)1604 2963 y Fp(\()p Fs(A)1715 2978 y Fr(\016)1752 2963 y Fp(\010)1822 2978 y Fl(gs)1890 2963 y Fp(\))g Fv(may)g(still)e(v)n(anish,as)h(the)g(f)o (actor)h Fp(\002)3205 2978 y Fl(0)p Fr(;j)3324 2963 y Fv(may)257 3083 y(turn)32 b(out)g(to)g(be)g(zero,)j(for)d(all)g Fs(j)48 b Fp(=)41 b(1)p Fs(;)17 b(:)g(:)g(:)32 b(;)17 b(m)p Fv(.)53 b(This)32 b(case)g(is)g(in)g(accordance)i(with)d(\(I.1\)) 257 3204 y(and)d(\(I.2\))f(and)h(merely)f(re\003ects)h(the)f(physical)f (f)o(act)i(that)e(some)h(transitions)f(\(here:)36 b(from)27 b(a)257 3324 y(bound)d(into)g(a)h(scattering)g(state\))f(are)i (forbidden)e(in)h(lo)n(west)e(order)i(perturbation)f(theory)-6 b(.)p eop %%Page: 15 16 15 15 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(15)257 573 y Fu(II)143 b(Dyson)35 b(Series)f(and)h(Photoelectric)e (Effect)257 796 y Fv(In)27 b(this)f(section)h(we)g(study)f(the)g (asymptotics,)g(as)h Fs(t)32 b Fo(!)f(1)p Fv(,)c(for)g Fs(e)2559 760 y Ff(\000)p Fr(itH)2721 768 y Fc(g)2762 796 y Fs(A)p Fp(\()p Fs(\034)6 b(;)17 b(f)p 2965 832 59 4 v 11 w Fp(\)\010)3132 811 y Fl(gs)3199 796 y Fv(,)28 b(where)257 917 y Fs(H)338 932 y Fr(g)422 917 y Fp(=)44 b Fs(H)623 932 y Fl(0)691 917 y Fp(+)28 b Fs(g)t(W)47 b Fv(is)33 b(the)g(Hamiltonian)f(generating)h(the)h(dynamics)e(of)i (the)f(interacting)257 1037 y(system,)i(which)e(is)g(applied)f(to)h(an) h(initial)e(state)h Fs(A)p Fp(\()p Fs(\034)6 b(;)17 b(f)p 2222 1072 V 11 w Fp(\)\010)2389 1052 y Fl(gs)2490 1037 y Fv(for)34 b Fp(0)43 b Fo(\024)h Fs(\034)55 b Fo(\024)44 b Fs(g)3122 1001 y Fr(\026)3202 1037 y Fv(and)33 b(for)257 1181 y Fs(\034)40 b Fo(\025)28 b Fs(g)495 1144 y Ff(\000)571 1117 y Fd(1)p 559 1129 55 3 v 559 1170 a Fc(K)627 1181 y Fv(.)j(The)25 b(initial)f(state)g(is)g(composed)g(of)h(a)g(photon)f (cloud)257 1372 y Fs(A)k Fo(\021)g Fs(A)p Fp(\()p Fs(f)p 574 1407 59 4 v 11 w Fp(\))50 b(=)g Fs(a)898 1331 y Ff(\003)937 1372 y Fp(\()p Fs(f)1023 1387 y Fl(1)1063 1372 y Fp(\))p Fs(a)1152 1331 y Ff(\003)1191 1372 y Fp(\()p Fs(f)1277 1387 y Fl(2)1317 1372 y Fp(\))17 b Fo(\001)g(\001)g(\001)d Fs(a)1555 1331 y Ff(\003)1595 1372 y Fp(\()p Fs(f)1681 1387 y Fr(N)1748 1372 y Fp(\))22 b Fs(;)64 b Fv(with)46 b Fs(f)2171 1387 y Fl(1)2211 1372 y Fs(;)17 b(f)2303 1387 y Fl(2)2342 1372 y Fs(;)g(:)g(:)g(:)33 b(;)17 b(f)2626 1387 y Fr(N)2737 1372 y Fo(2)44 b Fs(C)2924 1331 y Ff(1)2917 1397 y Fl(0)2999 1372 y Fp(\()p Fs(B)3111 1387 y Fl(\003)3186 1372 y Fo(n)22 b(f)p Fp(0)p Fo(g)p Fp(\))g Fs(;)3303 1504 y Fv(\(II.1\))257 1687 y(of)j Fs(N)36 b Fv(photons)23 b(and)i(its)f(time)g(e)n(v)n(olution)494 1878 y Fs(A)p Fp(\()p Fs(\034)11 b Fp(\))17 b(\010)783 1893 y Fl(gs)903 1878 y Fo(\021)53 b Fs(A)p Fp(\()p Fs(\034)6 b(;)17 b(f)p 1236 1913 V 11 w Fp(\)\010)1403 1893 y Fl(gs)1523 1878 y Fp(:=)53 b Fs(e)1724 1837 y Ff(\000)p Fr(i\034)8 b(H)1900 1845 y Fc(g)1957 1878 y Fs(e)2002 1837 y Fr(i\034)g(H)2123 1846 y Fd(0)2184 1878 y Fs(A)p Fp(\()p Fs(f)p 2295 1913 V 11 w Fp(\))22 b Fs(e)2459 1837 y Ff(\000)p Fr(i\034)8 b(H)2635 1846 y Fd(0)2690 1878 y Fs(e)2735 1837 y Fr(i\034)g(H)2856 1845 y Fc(g)2913 1878 y Fp(\010)2983 1893 y Fl(gs)3288 1878 y Fv(\(II.2\))257 2069 y(applied)25 b(to)f(a)h(ground)f(state)h(v) o(ector)f Fp(\010)1604 2084 y Fl(gs)1672 2069 y Fv(,)1482 2260 y Fs(H)1563 2275 y Fr(g)1620 2260 y Fp(\010)1690 2275 y Fl(gs)1810 2260 y Fp(=)53 b Fs(E)2011 2275 y Fl(0)2067 2260 y Fp(\010)2137 2275 y Fl(gs)2227 2260 y Fs(;)1034 b Fv(\(II.3\))257 2452 y(representing)31 b(the)g(interacting)g(atom)g (at)g(rest.)50 b(Note)31 b(that)g(each)h(photon)f(orbital)f Fs(f)3213 2467 y Fr(j)3281 2452 y Fv(is)h(as-)257 2572 y(sumed)26 b(to)f(be)h(smooth)f(and)h(compactly)f(supported)g(a)o(w)o (ay)h(from)f(zero)i(momentum)d(and)i(in)257 2692 y(the)f(ball)f(of)h (radius)g Fp(\003)i Fs(<)h Fo(1)c Fv(about)g(the)h(origin,)788 2884 y Fp(supp)17 b Fs(f)1053 2899 y Fr(j)1143 2884 y Fo(\022)53 b Fs(B)1347 2899 y Fl(\003)1422 2884 y Fo(n)22 b(f)p Fp(0)p Fo(g)52 b Fp(:=)1850 2803 y Fi(\010)1909 2884 y Fs(k)30 b Fo(2)e Fh(R)2150 2842 y Fl(3)2223 2799 y Fi(\014)2223 2859 y(\014)2284 2884 y Fp(0)g Fs(<)f(!)t Fp(\()p Fs(k)s Fp(\))g Fs(<)h Fp(\003)2858 2803 y Fi(\011)2938 2884 y Fs(:)323 b Fv(\(II.4\))257 3075 y(W)-8 b(e)29 b(no)n(w)e(gi)n(v)o(e)f(a)j(list)e(of)h(assumptions)d(used)j(to)f(deri) n(v)o(e)g(Theorem)h(II.1,)h(belo)n(w)-6 b(.)39 b(Gi)n(v)o(en)26 b(an)257 3195 y(operator)f Fs(X)33 b Fv(on)24 b Fo(H)i Fv(and)f Fs(t)j Fo(2)g Fh(R)36 b Fv(we)25 b(denote)g(its)f Fq(fr)l(ee)h Fv(Heisenber)n(g)g(time)f(e)n(v)n(olution)f(by)1412 3386 y Fs(X)1493 3401 y Fr(t)1576 3386 y Fp(:=)52 b Fs(e)1776 3345 y Ff(\000)p Fr(itH)1938 3354 y Fd(0)1994 3386 y Fs(X)24 b(e)2144 3345 y Fr(itH)2251 3354 y Fd(0)2313 3386 y Fs(:)948 b Fv(\(II.5\))257 3577 y(Ne)o(xt,)24 b(we)h(recall)h(from)e(\(I.27\))h(the)g(estimate)566 3684 y Fi(\015)566 3744 y(\015)621 3769 y Fp(\()p Fs(H)748 3718 y Fl(\(\003\))740 3796 y Fr(f)878 3769 y Fp(+)d Fg(1)p Fp(\))1070 3727 y Fr(\013)1136 3769 y Fs(P)1199 3784 y Fr(c)1250 3769 y Fp(\010)1320 3784 y Fl(gs)1388 3684 y Fi(\015)1388 3744 y(\015)1487 3769 y Fp(+)1608 3684 y Fi(\015)1608 3744 y(\015)1663 3769 y Fp(\()p Fs(H)1790 3718 y Fl(\(\003\))1782 3796 y Fr(f)1919 3769 y Fp(+)g Fg(1)p Fp(\))2111 3727 y Fr(\013)2177 3769 y Fs(P)2254 3727 y Ff(?)2240 3793 y Fl(\012)2330 3769 y Fp(\010)2400 3784 y Fl(gs)2467 3684 y Fi(\015)2467 3744 y(\015)2575 3769 y Fo(\024)53 b(O)s Fp(\()p Fs(g)t Fp(\))22 b Fs(;)325 b Fv(\(II.6\))257 3989 y(for)35 b(an)o(y)g Fs(\013)46 b Fo(\025)h Fp(1)35 b Fv(and)f(an)o(y)g Fp(\003)46 b Fs(<)g Fo(1)p Fv(,)37 b(where)e Fs(H)2022 3938 y Fl(\(\003\))2014 4017 y Fr(f)2165 3989 y Fv(is)f(de\002ned)h(in)f(\(I.28\).)61 b(Eq.)35 b(\(II.6\))g(is)257 4109 y(pro)o(v)o(ed)24 b(in)g(Lemma)h(IV) -13 b(.2,)24 b(belo)n(w)-6 b(.)30 b(Furthermore,)25 b(we)g(recall)g (from)f(\(I.22\))h(that)1035 4301 y Fs(P)1098 4316 y Fr(d)1155 4301 y Fs(W)30 b(P)1340 4316 y Fr(d)1433 4301 y Fo(\021)53 b Fp(\()p Fs(P)1664 4316 y Fr(d)1727 4301 y Fo(\012)22 b Fg(1)p Fp(\))28 b Fs(W)41 b Fp(\()p Fs(P)2182 4316 y Fr(d)2245 4301 y Fo(\012)22 b Fg(1)p Fp(\))53 b(=)f(0)22 b Fs(:)571 b Fv(\(II.7\))257 4492 y(In)25 b(Lemma)g(IV)-13 b(.3,)24 b(we)h(demonstrate)f(that,)h(for)g(an)o(y)f Fs(s)j Fo(2)i Fh(R)5 b Fv(,)795 4608 y Fi(\015)795 4668 y(\015)867 4693 y Fp([)p Fs(W)30 b(;)k(A)1150 4708 y Fr(s)1187 4693 y Fp(])17 b(\()p Fs(H)1358 4642 y Fl(\(\003\))1350 4721 y Fr(f)1487 4693 y Fp(+)22 b Fg(1)p Fp(\))1679 4652 y Ff(\000)p Fl(1)p Ff(\000)p Fl(\()p Fr(N)q(=)p Fl(2\))2011 4608 y Fi(\015)2011 4668 y(\015)2119 4693 y Fo(\024)53 b(O)2331 4612 y Fi(\000)2377 4693 y Fp(\(1)21 b(+)h Fo(j)p Fs(s)p Fo(j)p Fp(\))2723 4652 y Ff(\000)p Fr(K)2846 4612 y Fi(\001)2914 4693 y Fs(;)347 b Fv(\(II.8\))257 4884 y(where)38 b Fs(K)58 b(>)51 b Fp(1)37 b Fv(is)g(the)g(de)o(gree)h(of)f (dif)n(ferentiability)e(in)i(Hypothesis)f(1.)68 b(Since)38 b Fp(\010)3310 4899 y Fl(gs)3429 4884 y Fo(2)257 5005 y Fp(dom[\()p Fs(H)588 5020 y Fr(f)659 5005 y Fp(+)26 b Fg(1)p Fp(\))855 4968 y Fl(1+\()p Fr(N)q(=)p Fl(2\))1132 5005 y Fp(])p Fv(,)32 b(for)e(an)o(y)g Fs(N)48 b Fo(2)38 b Fh(N)1831 5020 y Fl(0)1876 5005 y Fv(,)32 b(the)e(estimate)f (\(II.8\))i(and)f(the)g(f)o(act)g(that)g Fs(H)3450 5020 y Fr(f)257 5125 y Fv(and)25 b Fs(P)489 5140 y Fr(d)554 5125 y Fv(commute)f(imply)g(the)g(norm-con)l(v)o(er)n(gence)h(of)1112 5316 y Fp(\010)1182 5331 y Ff(1)1257 5235 y Fi(\000)1303 5316 y Fs(A)p Fp(\()p Fs(f)p 1414 5351 V 10 w Fp(\))p Fs(;)17 b(\034)1607 5235 y Fi(\001)1697 5316 y Fp(:=)73 b(lim)1845 5378 y Fr(T)10 b Ff(!1)2053 5316 y Fp(\010)2123 5331 y Fr(T)2179 5235 y Fi(\000)2224 5316 y Fs(A)p Fp(\()p Fs(f)p 2335 5351 V 11 w Fp(\))p Fs(;)17 b(\034)2529 5235 y Fi(\001)2597 5316 y Fs(;)664 b Fv(\(II.9\))p eop %%Page: 16 17 16 16 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(16)257 573 y(where)354 1011 y Fp(\010)424 1026 y Fr(T)479 1011 y Fp(\()p Fs(X)r(;)17 b(\034)11 b Fp(\))45 b(:=)927 658 y Fi(8)927 747 y(>)927 777 y(>)927 807 y(>)927 837 y(>)927 867 y(>)927 897 y(<)927 1076 y(>)927 1106 y(>)927 1136 y(>)927 1166 y(>)927 1196 y(>)927 1226 y(:)1057 675 y(Z)1157 701 y Fr(T)1113 901 y Fl(0)1229 811 y Fs(ds)17 b(e)1388 770 y Fr(is)p Fl(\()p Fr(H)1530 779 y Fd(0)1564 770 y Ff(\000)p Fr(E)1671 779 y Fd(0)1705 770 y Fl(\))1753 811 y Fp([)p Fs(W)30 b(;)k(X)2044 826 y Fr(s)2080 811 y Fp(])17 b Fs(P)2187 826 y Fr(d)2244 811 y Fp(\010)2314 826 y Fl(gs)2465 811 y Fv(if)83 b Fp(0)27 b Fo(\024)h Fs(\034)39 b Fo(\024)29 b Fs(g)3028 775 y Fr(\026)1107 997 y(T)1060 1017 y Fi(Z)1035 1300 y Ff(\000)p Fr(T)1180 1152 y Fs(ds)17 b(e)1339 1111 y Fr(is)p Fl(\()p Fr(H)1481 1120 y Fd(0)1515 1111 y Ff(\000)p Fr(E)1622 1120 y Fd(0)1656 1111 y Fl(\))1704 1152 y Fp([)p Fs(W)30 b(;)k(X)1995 1167 y Fr(s)2032 1152 y Fp(])17 b Fs(P)2139 1167 y Fr(d)2195 1152 y Fp(\010)2265 1167 y Fl(gs)2465 1152 y Fv(if)83 b Fs(\034)39 b Fo(\025)28 b Fs(g)2846 1115 y Ff(\000)2922 1088 y Fd(1)p 2910 1100 55 3 v 2910 1141 a Fc(K)3238 1011 y Fv(\(II.10\))257 1467 y(pro)o(vided)f Fs(K)41 b(>)34 b Fp(1)p Fv(.)40 b(Furthermore,)29 b(we)g(sho)n(w)e(in)g(Lemma)h (IV)-13 b(.4)28 b(that,)h(for)f(an)o(y)g Fs(r)36 b Fo(2)e Fh(R)39 b Fv(and)257 1588 y Fs(s)28 b Fo(2)g Fh(R)491 1546 y Fl(+)491 1612 y(0)556 1588 y Fv(,)435 1697 y Fi(\015)435 1756 y(\015)507 1781 y Fs(W)i(e)674 1740 y Ff(\000)p Fr(ir)r(H)845 1749 y Fd(0)900 1781 y Fs(P)963 1796 y Fr(c)1015 1781 y Fp([)p Fs(W)g(;)j(A)1297 1796 y Fr(s)1334 1781 y Fp(])17 b(\()p Fs(H)1497 1796 y Fr(f)1564 1781 y Fp(+)22 b Fg(1)p Fp(\))1756 1740 y Ff(\000)p Fl(2)p Ff(\000)p Fl(\()p Fr(N)q(=)p Fl(2\))2088 1697 y Fi(\015)2088 1756 y(\015)2196 1781 y Fo(\024)53 b(O)2408 1701 y Fi(\000)2454 1781 y Fp(\(1)22 b(+)g Fo(j)p Fs(r)s Fo(j)p Fp(\))2802 1740 y Ff(\000)p Fl(3)p Fr(=)p Fl(2)2965 1701 y Fi(\001)3033 1781 y Fs(:)178 b Fv(\(II.11\))257 1975 y(Under)25 b(these)g (assumptions,)d(we)j(pro)o(v)o(e)f(the)h(follo)n(wing)e(Theorem.)257 2156 y Fk(Theor)n(em)k(II.1.)49 b Fq(Assume)36 b(Hypotheses)h(1\2262)g (and)f(\(II.1\)\226\(II.11\).)69 b(Then)37 b(ther)l(e)h(e)n(xists)e(a) 257 2277 y(constant)24 b Fs(C)35 b(<)27 b Fo(1)e Fq(suc)o(h)f(that,)f (for)i(all)f Fs(t)k Fo(\025)g Fs(g)1854 2240 y Ff(\000)p Fl(1)p Fr(=K)2047 2277 y Fq(,)316 2386 y Fi(\015)316 2445 y(\015)372 2470 y Fs(e)417 2429 y Ff(\000)p Fr(it)p Fl(\()p Fr(H)606 2437 y Fc(g)643 2429 y Ff(\000)p Fr(E)750 2438 y Fd(0)784 2429 y Fl(\))832 2470 y Fs(A)p Fp(\()p Fs(\034)11 b Fp(\))17 b(\010)1121 2485 y Fl(gs)1239 2470 y Fo(\000)50 b Fs(A)1439 2485 y Fr(t)1486 2470 y Fp(\010)1556 2485 y Fl(gs)1673 2470 y Fp(+)g Fs(ig)t(e)1928 2429 y Ff(\000)p Fr(it)p Fl(\()p Fr(H)2117 2438 y Fd(0)2152 2429 y Ff(\000)p Fr(E)2259 2438 y Fd(0)2293 2429 y Fl(\))2341 2470 y Fp(\010)2411 2485 y Ff(1)2487 2470 y Fp(\()p Fs(A;)17 b(\034)11 b Fp(\))2733 2386 y Fi(\015)2733 2445 y(\015)2841 2470 y Fo(\024)53 b Fs(C)23 b(g)3115 2429 y Fl(1+)p Fr(\026)p Fl(\()p Fr(K)5 b Fl(\))3393 2470 y Fs(;)3238 2591 y Fv(\(II.12\))257 2805 y Fq(pr)l(o)o(vided)26 b(that)f(either)h Fp(0)31 b Fo(\024)g Fs(\034)42 b Fo(\024)31 b Fs(g)1504 2768 y Fr(\026)1577 2805 y Fq(or)26 b Fs(\034)42 b Fo(\025)31 b Fs(g)1935 2768 y Ff(\000)2012 2741 y Fd(1)p 2000 2753 V 2000 2794 a Fc(K)2068 2805 y Fq(,)c(and)f(wher)l(e)h Fs(\026)p Fp(\()p Fs(K)7 b Fp(\))31 b(:=)g(1)23 b Fo(\000)h Fs(K)3215 2768 y Ff(\000)p Fl(1)3340 2805 y Fs(>)30 b Fp(0)257 2925 y Fq(and)25 b Fs(K)35 b(>)27 b Fp(1)e Fq(is)f(the)h(de)l (gr)l(ee)g(of)f(dif)n(fer)l(entiability)f(in)h(Hypothesis)g(1.)257 3106 y(Pr)l(oof)o(.)47 b Fv(First)25 b(we)g(compute)f(the)g(Heisenber)n (g)h(time)f(e)n(v)n(olution)f(of)h(the)h(photon)f(cloud)g Fs(A)p Fp(\()p Fs(\034)11 b Fp(\))257 3226 y Fv(on)25 b Fp(Ran[\()p Fs(H)703 3241 y Fr(g)765 3226 y Fo(\000)d Fs(i)p 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Fp(=)g Fs(A)p Fp(\(0\))f(=)h Fs(A)p Fp(\(0)p Fs(;)17 b(f)p 1314 4422 59 4 v 10 w Fp(\))35 b Fv(and)f(a)g(change)h(of)f(inte)o(gration)e(v)n(ariables,)k Fs(s)45 b Fo(7!)g Fs(t)29 b Fo(\000)g Fs(s)p Fv(.)257 4507 y(Applying)24 b(\(II.13\))h(to)f(the)h(ground)f(state)h Fp(\010)1775 4522 y Fl(gs)1842 4507 y Fv(,)g(we)g(obtain)704 4700 y Fo(\000)p Fs(ig)20 b Fp(\011\()p Fs(t;)d(\034)11 b Fp(\))84 b(:=)f Fs(e)1480 4659 y Ff(\000)p Fr(it)p Fl(\()p Fr(H)1669 4667 y Fc(g)1706 4659 y Ff(\000)p Fr(E)1813 4668 y Fd(0)1847 4659 y Fl(\))1895 4700 y Fs(A)p Fp(\()p Fs(\034)11 b Fp(\))17 b(\010)2184 4715 y Fl(gs)2296 4700 y Fo(\000)45 b Fs(A)2491 4715 y Fr(t)2538 4700 y Fp(\010)2608 4715 y Fl(gs)3238 4700 y Fv(\(II.14\))1262 4912 y Fp(=)97 b Fo(\000)p Fs(ig)1612 4776 y Fi(Z)1712 4802 y Fr(t)1668 5002 y Ff(\000)p Fr(\034)1782 4912 y Fs(ds)17 b(e)1941 4871 y Ff(\000)p Fr(i)p Fl(\()p Fr(t)p Ff(\000)p Fr(s)p Fl(\)\()p Fr(H)2272 4879 y Fc(g)2309 4871 y Ff(\000)p Fr(E)2416 4880 y Fd(0)2450 4871 y Fl(\))2499 4912 y Fp([)p Fs(W)30 b(;)j(A)2781 4927 y Fr(s)2818 4912 y Fp(])17 b(\010)2932 4927 y Fl(gs)3022 4912 y Fs(;)257 5158 y Fv(and)25 b(our)g(goal)f(is)h(to)f(sho)n(w)g(that)870 5267 y Fi(\015)870 5327 y(\015)926 5352 y Fp(\011\()p Fs(t;)17 b(\034)11 b Fp(\))50 b Fo(\000)g Fs(e)1432 5311 y Ff(\000)p Fr(it)p Fl(\()p Fr(H)1621 5320 y Fd(0)1657 5311 y Ff(\000)p Fr(E)1764 5320 y Fd(0)1798 5311 y Fl(\))1846 5352 y Fp(\010)1916 5367 y Ff(1)1991 5352 y Fp(\()p Fs(A)p Fp(\()p Fs(\034)11 b Fp(\))p Fs(;)17 b(\034)11 b Fp(\)\))2404 5267 y Fi(\015)2404 5327 y(\015)2513 5352 y Fo(\024)53 b Fs(C)23 b(g)2787 5311 y Fr(\026)2855 5352 y Fs(:)356 b Fv(\(II.15\))p eop %%Page: 17 18 17 17 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(17)404 573 y(T)-8 b(o)23 b(this)h(end,)g(we)g(\002rst)g(pro)o(v)o(e) f(\(II.15\))h(for)g(the)g(special)g(case)g Fs(\034)40 b Fp(=)27 b(0)p Fv(.)j(Gi)n(v)o(en)23 b Fs(T)41 b Fo(\025)29 b Fp(1)23 b Fv(and)257 693 y Fs(K)35 b(>)28 b Fp(1)p Fv(,)c(and)h(assuming)e(that)i Fs(t)j Fo(\025)g Fs(T)14 b Fv(,)24 b(we)h(observ)o(e)g(that)1038 773 y Fi(\015)1038 833 y(\015)1094 858 y Fp(\010)1164 873 y Ff(1)1239 858 y Fp(\()p Fs(A;)17 b Fp(0\))49 b Fo(\000)h Fp(\010)1727 873 y Fr(T)1783 858 y Fp(\()p Fs(A;)17 b Fp(0\))2025 773 y Fi(\015)2025 833 y(\015)2132 858 y Fo(\024)53 b(O)2344 777 y Fi(\000)2390 858 y Fs(T)2461 822 y Fl(1)p Ff(\000)p Fr(K)2619 777 y Fi(\001)2687 858 y Fs(;)524 b Fv(\(II.16\))625 950 y Fi(\015)625 1010 y(\015)625 1069 y(\015)625 1129 y(\015)681 1094 y Fp(\011\()p Fs(t;)17 b Fp(0\))49 b Fo(\000)1137 1014 y Fi(R)1204 1040 y Fr(T)1184 1129 y Fl(0)1275 1094 y Fs(ds)17 b(e)1434 1058 y Fr(i)p Fl(\()p Fr(t)p Ff(\000)p Fr(s)p Fl(\)\()p Fr(H)1710 1066 y Fc(g)1747 1058 y Ff(\000)p Fr(E)1854 1067 y Fd(0)1888 1058 y Fl(\))1937 1094 y Fp([)p Fs(W)30 b(;)j(A)2219 1109 y Fr(s)2256 1094 y Fp(])17 b(\010)2370 1109 y Fl(gs)2438 950 y Fi(\015)2438 1010 y(\015)2438 1069 y(\015)2438 1129 y(\015)2545 1094 y Fo(\024)54 b(O)2758 1014 y Fi(\000)2803 1094 y Fs(T)2874 1058 y Fl(1)p Ff(\000)p Fr(K)3033 1014 y Fi(\001)3100 1094 y Fs(:)111 b Fv(\(II.17\))257 1310 y(Ne)o(xt,)24 b(we)g(use)g(that)g Fs(H)1051 1325 y Fl(0)1114 1310 y Fv(and)g Fs(A)h Fv(commute)e(with)g Fs(P)2039 1325 y Fr(d)2104 1310 y Fv(and)h(that)f Fs(P)2508 1325 y Fr(d)2565 1310 y Fs(W)30 b(P)2750 1325 y Fr(d)2818 1310 y Fp(=)e(0)c Fv(which)g(imply)1237 1475 y Fp([)p Fs(W)31 b(;)i(A)1520 1490 y Fr(s)1557 1475 y Fp(])17 b Fs(P)1664 1490 y Fr(d)1757 1475 y Fp(=)52 b Fs(P)1948 1490 y Fr(c)1999 1475 y Fp([)p Fs(W)31 b(;)i(A)2282 1490 y Fr(s)2319 1475 y Fp(])17 b Fs(P)2426 1490 y Fr(d)2488 1475 y Fs(:)723 b Fv(\(II.18\))257 1640 y(Hence)26 b(we)f(ha)n(v)o(e)f(that)450 1720 y Fi(\015)450 1779 y(\015)505 1804 y Fs(e)550 1763 y Ff(\000)p Fr(itH)712 1771 y Fc(g)770 1804 y Fp(\010)840 1819 y Fr(T)895 1804 y Fp(\()p Fs(A;)17 b Fp(0\))49 b Fo(\000)i Fs(e)1359 1763 y Ff(\000)p Fr(itH)1521 1772 y Fd(0)1577 1804 y Fp(\010)1647 1819 y Fr(T)1702 1804 y Fp(\()p Fs(A;)17 b Fp(0\))1944 1720 y Fi(\015)1944 1779 y(\015)3238 1804 y Fv(\(II.19\))534 2023 y Fp(=)83 b Fs(g)744 1878 y Fi(\015)744 1938 y(\015)744 1998 y(\015)744 2058 y(\015)815 1887 y(Z)915 1913 y Fr(t)871 2113 y Fl(0)961 2023 y Fs(dr)19 b(e)1120 1982 y Ff(\000)p Fr(ir)r(H)1291 1990 y Fc(g)1348 2023 y Fs(W)30 b(e)1515 1982 y Ff(\000)p Fr(i)p Fl(\()p Fr(t)p Ff(\000)p Fr(r)r Fl(\))p Fr(H)1820 1991 y Fd(0)1877 2023 y Fp(\010)1947 2038 y Fr(T)2002 2023 y Fp(\()p Fs(A;)17 b Fp(0\))2244 1878 y Fi(\015)2244 1938 y(\015)2244 1998 y(\015)2244 2058 y(\015)534 2305 y Fp(=)83 b Fs(g)744 2161 y Fi(\015)744 2221 y(\015)744 2280 y(\015)744 2340 y(\015)815 2170 y(Z)915 2196 y Fr(t)871 2395 y Fl(0)961 2305 y Fs(dr)1075 2170 y Fi(Z)1175 2196 y Fr(T)1131 2395 y Fl(0)1247 2305 y Fs(ds)17 b(e)1406 2264 y Ff(\000)p Fr(isE)1570 2273 y Fd(0)1624 2305 y Fs(e)1669 2264 y Ff(\000)p Fr(ir)r(H)1840 2272 y Fc(g)1897 2305 y Fs(W)30 b(e)2064 2264 y Ff(\000)p Fr(i)p Fl(\()p Fr(t)p Ff(\000)p Fr(r)r Ff(\000)p Fr(s)p Fl(\))p Fr(H)2457 2273 y Fd(0)2513 2305 y Fs(P)2576 2320 y Fr(c)2610 2305 y Fp([)p Fs(W)h(;)i(A)2893 2320 y Fr(s)2930 2305 y Fp(])17 b Fs(P)3037 2320 y Fr(d)3094 2305 y Fp(\010)3164 2320 y Fl(gs)3231 2161 y Fi(\015)3231 2221 y(\015)3231 2280 y(\015)3231 2340 y(\015)533 2524 y Fo(\024)83 b Fs(g)744 2439 y Fi(\015)744 2499 y(\015)799 2524 y Fp(\()p Fs(H)918 2539 y Fr(f)985 2524 y Fp(+)22 b Fg(1)p Fp(\))1177 2483 y Fl(2+\()p Fr(N)q(=)p Fl(2\))1454 2524 y Fp(\010)1524 2539 y Fl(gs)1592 2439 y Fi(\015)1592 2499 y(\015)811 2611 y(Z)911 2637 y Fr(t)867 2837 y Fl(0)957 2747 y Fs(dr)1071 2611 y Fi(Z)1171 2637 y Fr(T)1127 2837 y Fl(0)1242 2747 y Fs(ds)1356 2662 y Fi(\015)1356 2722 y(\015)1411 2747 y Fs(W)30 b(e)1578 2706 y Ff(\000)p Fr(i)p Fl(\()p Fr(t)p Ff(\000)p Fr(r)r Ff(\000)p Fr(s)p Fl(\))p Fr(H)1971 2715 y Fd(0)2027 2747 y Fs(P)2090 2762 y Fr(c)2125 2747 y Fp([)p Fs(W)g(;)k(A)2408 2762 y Fr(s)2444 2747 y Fp(])17 b(\()p Fs(H)2607 2762 y Fr(f)2674 2747 y Fp(+)22 b Fg(1)p Fp(\))2866 2706 y Ff(\000)p Fl(2)p Ff(\000)p Fl(\()p Fr(N)q(=)p Fl(2\))3198 2662 y Fi(\015)3198 2722 y(\015)3276 2747 y Fs(;)257 2957 y Fv(and)j(\(II.11\))g(and)g(a)g (change)g(of)g(v)n(ariable)f Fs(r)31 b Fo(7!)c Fs(t)22 b Fo(\000)h Fs(r)28 b Fv(no)n(w)c(imply)f(that)338 3102 y Fi(\015)338 3162 y(\015)393 3187 y Fs(e)438 3145 y Ff(\000)p Fr(itH)600 3153 y Fc(g)658 3187 y Fp(\010)728 3202 y Fr(T)783 3187 y Fp(\()p Fs(A;)17 b Fp(0\))50 b Fo(\000)g Fs(e)1247 3145 y Ff(\000)p Fr(itH)1409 3154 y Fd(0)1465 3187 y Fp(\010)1535 3202 y Fr(T)1590 3187 y Fp(\()p Fs(A;)17 b Fp(0\))1832 3102 y Fi(\015)1832 3162 y(\015)1970 3187 y Fo(\024)84 b Fs(C)2201 3202 y Fl(1)2256 3187 y Fs(g)2340 3051 y Fi(Z)2440 3077 y Fr(T)2395 3276 y Fl(0)2511 3187 y Fs(ds)2625 3051 y Fi(Z)2724 3077 y Fr(t)2680 3276 y Fl(0)3044 3119 y Fs(dr)p 2780 3164 625 4 v 2780 3255 a Fp(\(1)22 b(+)g Fo(j)p Fs(r)j Fo(\000)d Fs(s)p Fo(j)p Fp(\))3295 3226 y Fl(3)p Fr(=)p Fl(2)1970 3405 y Fo(\024)84 b Fs(C)2201 3420 y Fl(1)2256 3405 y Fs(C)2326 3420 y Fl(2)2382 3405 y Fs(g)20 b(T)36 b(;)669 b Fv(\(II.20\))257 3570 y(where)34 b Fs(C)604 3585 y Fl(1)687 3570 y Fs(<)44 b Fo(1)33 b Fv(and)g Fs(C)1187 3585 y Fl(2)1270 3570 y Fp(:=)1416 3489 y Fi(R)1483 3516 y Ff(1)1464 3604 y(\0001)1593 3570 y Fp(\(1)28 b(+)h Fo(j)p Fs(r)s Fo(j)p Fp(\))1954 3533 y Ff(\000)p Fl(3)p Fr(=)p Fl(2)2118 3570 y Fs(dr)46 b(<)d Fo(1)p Fv(.)56 b(Applying)32 b(the)h(estimates)257 3690 y(\(II.16\)\226\(II.20\))26 b(to)e(the)h(left)f(side)h(of)g(\(II.15\),)g(we)g(hence)g(obtain)f (that)297 3770 y Fi(\015)297 3830 y(\015)353 3855 y Fp(\011\()p Fs(t;)17 b Fp(0\))49 b Fo(\000)h Fs(e)854 3814 y Ff(\000)p Fr(it)p Fl(\()p Fr(H)1043 3823 y Fd(0)1079 3814 y Ff(\000)p Fr(E)1186 3823 y Fd(0)1220 3814 y Fl(\))1268 3855 y Fp(\010)1338 3870 y Ff(1)1413 3855 y Fp(\()p Fs(A;)17 b Fp(0\))1655 3770 y Fi(\015)1655 3830 y(\015)3238 3855 y Fv(\(II.21\))380 4078 y Fo(\024)541 3963 y Fi(\015)541 4023 y(\015)541 4083 y(\015)596 4078 y Fp(\011\()p Fs(t;)g Fp(0\))38 b Fo(\000)1031 3942 y Fi(Z)1130 3968 y Fr(T)1086 4168 y Fl(0)1202 4078 y Fs(ds)17 b(e)1361 4036 y Ff(\000)p Fr(i)p Fl(\()p Fr(t)p Ff(\000)p Fr(s)p Fl(\)\()p Fr(H)1692 4044 y Fc(g)1729 4036 y Ff(\000)p Fr(E)1836 4045 y Fd(0)1870 4036 y Fl(\))1918 4078 y Fp([)p Fs(W)31 b(;)i(A)2201 4093 y Fr(s)2238 4078 y Fp(])17 b(\010)2352 4093 y Fl(gs)2419 3963 y Fi(\015)2419 4023 y(\015)2419 4083 y(\015)569 4355 y Fp(+)672 4241 y Fi(\015)672 4301 y(\015)672 4360 y(\015)744 4220 y(Z)844 4246 y Fr(T)799 4445 y Fl(0)915 4355 y Fs(ds)g(e)1074 4314 y Ff(\000)p Fr(i)p Fl(\()p Fr(t)p Ff(\000)p Fr(s)p Fl(\)\()p Fr(H)1405 4322 y Fc(g)1442 4314 y Ff(\000)p Fr(E)1549 4323 y Fd(0)1583 4314 y Fl(\))1632 4355 y Fp([)p Fs(W)30 b(;)j(A)1914 4370 y Fr(s)1951 4355 y Fp(])17 b(\010)2065 4370 y Fl(gs)2171 4355 y Fo(\000)40 b Fs(e)2333 4314 y Ff(\000)p Fr(it)p Fl(\()p Fr(H)2522 4322 y Fc(g)2559 4314 y Ff(\000)p Fr(E)2666 4323 y Fd(0)2700 4314 y Fl(\))2732 4355 y Fp(\010)2802 4370 y Fr(T)2857 4355 y Fp(\()p Fs(A;)17 b Fp(0\))3099 4271 y Fi(\015)3099 4331 y(\015)563 4570 y Fp(+)661 4485 y Fi(\015)661 4545 y(\015)716 4489 y(\000)762 4570 y Fs(e)807 4528 y Ff(\000)p Fr(it)p Fl(\()p Fr(H)996 4536 y Fc(g)1033 4528 y Ff(\000)p Fr(E)1140 4537 y Fd(0)1175 4528 y Fl(\))1245 4570 y Fo(\000)39 b Fs(e)1406 4528 y Ff(\000)p Fr(it)p Fl(\()p Fr(H)1595 4537 y Fd(0)1631 4528 y Ff(\000)p Fr(E)1738 4537 y Fd(0)1772 4528 y Fl(\))1804 4489 y Fi(\001)1849 4570 y Fp(\010)1919 4585 y Fr(T)1975 4570 y Fp(\()p Fs(A;)17 b Fp(0\))2217 4485 y Fi(\015)2217 4545 y(\015)2316 4570 y Fp(+)2436 4485 y Fi(\015)2436 4545 y(\015)2491 4570 y Fp(\010)2561 4585 y Fr(T)2617 4570 y Fp(\()p Fs(A;)g Fp(0\))38 b Fo(\000)h Fp(\010)3083 4585 y Ff(1)3158 4570 y Fp(\()p Fs(A;)17 b Fp(0\))3400 4485 y Fi(\015)3400 4545 y(\015)380 4792 y Fo(\024)541 4678 y Fi(\015)541 4738 y(\015)541 4797 y(\015)613 4657 y(Z)713 4683 y Fr(T)668 4882 y Fl(0)784 4792 y Fs(ds)g(e)943 4751 y Fr(is)p Fl(\()p Fr(H)1085 4759 y Fc(g)1121 4751 y Ff(\000)p Fr(E)1228 4760 y Fd(0)1262 4751 y Fl(\))1310 4792 y Fp([)p Fs(W)31 b(;)i(A)1593 4807 y Fr(s)1630 4792 y Fp(])17 b(\010)1744 4807 y Fl(gs)1861 4792 y Fo(\000)50 b Fp(\010)2058 4807 y Fr(T)2114 4792 y Fp(\()p Fs(A;)17 b Fp(0\))2356 4678 y Fi(\015)2356 4738 y(\015)2356 4797 y(\015)2461 4792 y Fp(+)49 b Fo(O)2668 4712 y Fi(\000)2714 4792 y Fs(T)2785 4751 y Fl(1)p Ff(\000)p Fr(K)2965 4792 y Fp(+)22 b Fs(g)t(T)3185 4712 y Fi(\001)380 5070 y Fo(\024)541 4956 y Fi(\015)541 5015 y(\015)541 5075 y(\015)613 4935 y(Z)713 4961 y Fr(T)668 5160 y Fl(0)784 5070 y Fs(ds)898 4989 y Fi(\000)943 5070 y Fs(e)988 5029 y Fr(is)p Fl(\()p Fr(H)1130 5037 y Fc(g)1167 5029 y Ff(\000)p Fr(E)1274 5038 y Fd(0)1308 5029 y Fl(\))1361 5070 y Fo(\000)h Fs(e)1506 5029 y Fr(is)p Fl(\()p Fr(H)1648 5038 y Fd(0)1683 5029 y Ff(\000)p Fr(E)1790 5038 y Fd(0)1824 5029 y Fl(\))1856 4989 y Fi(\001)1918 5070 y Fp([)p Fs(W)30 b(;)k(A)2201 5085 y Fr(s)2238 5070 y Fp(])17 b Fs(P)2345 5085 y Fr(d)2401 5070 y Fp(\010)2471 5085 y Fl(gs)2539 4956 y Fi(\015)2539 5015 y(\015)2539 5075 y(\015)569 5348 y Fp(+)672 5233 y Fi(\015)672 5293 y(\015)672 5353 y(\015)744 5212 y(Z)844 5239 y Fr(T)799 5438 y Fl(0)915 5348 y Fs(ds)g(e)1074 5307 y Fr(is)p Fl(\()p Fr(H)1216 5315 y Fc(g)1252 5307 y Ff(\000)p Fr(E)1359 5316 y Fd(0)1393 5307 y Fl(\))1442 5348 y Fp([)p Fs(W)30 b(;)j(A)1724 5363 y Fr(s)1761 5348 y Fp(])17 b Fs(P)1868 5363 y Fr(c)1919 5348 y Fp(\010)1989 5363 y Fl(gs)2057 5233 y Fi(\015)2057 5293 y(\015)2057 5353 y(\015)2162 5348 y Fp(+)50 b Fo(O)2370 5267 y Fi(\000)2415 5348 y Fs(T)2486 5307 y Fl(1)p Ff(\000)p Fr(K)2667 5348 y Fp(+)22 b Fs(g)t(T)2887 5267 y Fi(\001)2953 5348 y Fs(;)p eop %%Page: 18 19 18 18 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(18)257 573 y(using)29 b(that)g Fs(P)745 588 y Fr(d)811 573 y Fp(+)c Fs(P)975 588 y Fr(c)1046 573 y Fp(=)37 b Fg(1)p Fv(.)45 b(It)29 b(remains)g(to)g(bound)g(the)g(tw)o(o)g(inte)o (grals)g(in)g(the)g(last)g(line)g(of)257 693 y(\(II.21\).)i(The)25 b(second)g(of)g(these)f(is)h(bounded)f(by)411 863 y Fi(\015)411 923 y(\015)411 982 y(\015)483 842 y(Z)582 868 y Fr(T)538 1067 y Fl(0)654 977 y Fs(ds)17 b(e)813 936 y Fr(is)p Fl(\()p Fr(H)955 944 y Fc(g)991 936 y Ff(\000)p Fr(E)1098 945 y Fd(0)1132 936 y Fl(\))1180 977 y Fp([)p Fs(W)30 b(;)k(A)1463 992 y Fr(s)1499 977 y Fp(])17 b Fs(P)1606 992 y Fr(c)1657 977 y Fp(\010)1727 992 y Fl(gs)1795 863 y Fi(\015)1795 923 y(\015)1795 982 y(\015)3238 977 y Fv(\(II.22\))494 1255 y Fo(\024)654 1171 y Fi(\015)654 1230 y(\015)709 1255 y Fp(\()p Fs(H)836 1204 y Fl(\(\003\))828 1283 y Fr(f)966 1255 y Fp(+)22 b Fg(1)p Fp(\))1158 1214 y Fl(1+\()p Fr(N)q(=)p Fl(2\))1452 1255 y Fs(P)1515 1270 y Fr(c)1566 1255 y Fp(\010)1636 1270 y Fl(gs)1704 1171 y Fi(\015)1704 1230 y(\015)1803 1120 y(Z)1903 1146 y Fr(T)1858 1345 y Fl(0)1974 1255 y Fs(ds)2071 1171 y Fi(\015)2071 1230 y(\015)2143 1255 y Fp([)p Fs(W)30 b(;)k(A)2426 1270 y Fr(s)2463 1255 y Fp(])17 b(\()p Fs(H)2634 1204 y Fl(\(\003\))2626 1283 y Fr(f)2763 1255 y Fp(+)22 b Fg(1)p Fp(\))2955 1214 y Ff(\000)p Fl(1)p Ff(\000)p Fl(\()p Fr(N)q(=)p Fl(2\))3287 1171 y Fi(\015)3287 1230 y(\015)494 1453 y Fo(\024)83 b(O)s Fp(\()p Fs(g)t Fp(\))52 b Fo(\024)h(O)s Fp(\()p Fs(g)t(T)14 b Fp(\))21 b Fs(;)257 1673 y Fv(due)26 b(to)g(\(II.6\))g (and)g(\(II.8\),)h(since)f Fs(K)36 b(>)30 b Fp(1)c Fv(and)g Fs(T)43 b Fo(\025)30 b Fp(1)p Fv(.)k(T)-8 b(o)26 b(estimate)f(the)h (\002rst)g(inte)o(gral,)f(we)257 1794 y(use)g(again)f(the)h(DuHamel)f (formula)g(as)h(in)g(\(II.19\).)31 b(This)24 b(yields)549 1963 y Fi(\015)549 2023 y(\015)549 2083 y(\015)621 1942 y(Z)721 1968 y Fr(T)677 2168 y Fl(0)793 2078 y Fs(ds)907 1997 y Fi(\000)951 2078 y Fs(e)996 2037 y Fr(is)p Fl(\()p Fr(H)1138 2045 y Fc(g)1175 2037 y Ff(\000)p Fr(E)1282 2046 y Fd(0)1316 2037 y Fl(\))1370 2078 y Fo(\000)f Fs(e)1515 2037 y Fr(is)p Fl(\()p Fr(H)1657 2046 y Fd(0)1691 2037 y Ff(\000)p Fr(E)1798 2046 y Fd(0)1833 2037 y Fl(\))1864 1997 y Fi(\001)1926 2078 y Fp([)p Fs(W)31 b(;)i(A)2209 2093 y Fr(s)2246 2078 y Fp(])17 b Fs(P)2353 2093 y Fr(d)2410 2078 y Fp(\010)2480 2093 y Fl(gs)2547 1963 y Fi(\015)2547 2023 y(\015)2547 2083 y(\015)3238 2078 y Fv(\(II.23\))632 2356 y Fo(\024)84 b Fs(g)844 2211 y Fi(\015)844 2271 y(\015)844 2331 y(\015)844 2390 y(\015)915 2220 y(Z)1015 2246 y Fr(T)970 2446 y Fl(0)1086 2356 y Fs(ds)17 b(e)1245 2314 y Ff(\000)p Fr(isE)1409 2323 y Fd(0)1463 2220 y Fi(Z)1563 2246 y Fr(s)1519 2446 y Fl(0)1616 2356 y Fs(dr)i(e)1775 2314 y Fr(ir)r(H)1891 2322 y Fc(g)1948 2356 y Fs(W)31 b(e)2116 2314 y Fr(i)p Fl(\()p Fr(s)p Ff(\000)p Fr(r)r Fl(\))p Fr(H)2374 2323 y Fd(0)2429 2356 y Fs(P)2492 2371 y Fr(c)2527 2356 y Fp([)p Fs(W)f(;)k(A)2810 2371 y Fr(s)2847 2356 y Fp(])17 b Fs(P)2954 2371 y Fr(d)3010 2356 y Fp(\010)3080 2371 y Fl(gs)3148 2211 y Fi(\015)3148 2271 y(\015)3148 2331 y(\015)3148 2390 y(\015)632 2574 y Fo(\024)84 b Fs(g)844 2490 y Fi(\015)844 2549 y(\015)898 2574 y Fp(\()p Fs(H)1017 2589 y Fr(f)1085 2574 y Fp(+)22 b Fg(1)p Fp(\))1277 2533 y Fl(2+\()p Fr(N)q(=)p Fl(2\))1554 2574 y Fp(\010)1624 2589 y Fl(gs)1691 2490 y Fi(\015)1691 2549 y(\015)911 2662 y(Z)1011 2688 y 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4718 y Fl(gs)2573 4703 y Fo(\000)32 b Fs(A)p Fp(\010)2825 4718 y Fl(gs)2893 4703 y Fo(k)52 b(\024)h(O)s Fp(\()p Fs(g)3296 4667 y Fl(1+)p Fr(\026)3432 4703 y Fp(\))p Fv(,)257 4823 y(pro)o(ving)24 b(the)g(asymptotic)g(e)o(xpansion)f (\(II.12\))i(in)f(the)h(case)g Fs(\034)40 b Fo(2)28 b Fp([0)p Fs(;)17 b(g)2701 4787 y Fr(\026)2746 4823 y Fp(])p Fv(.)404 4944 y(In)26 b(order)h(to)e(deal)i(with)e(the)h(case)h Fs(\034)42 b Fo(2)30 b Fp([)p Fs(g)1853 4907 y Ff(\000)p Fl(1)p Fr(=K)2046 4944 y Fs(;)17 b Fo(1)p Fp(\))p Fv(,)26 b(we)h(observ)o(e)e(that)h(an)g(application)257 5073 y(of)f(\(II.25\),)g(for)g Fs(t)j Fo(\025)g Fs(g)1032 5036 y Ff(\000)1109 5009 y Fd(1)p 1097 5021 55 3 v 1097 5062 a Fc(K)1165 5073 y Fv(,)d(in)f(combination)f(with)i(the)f(decay)h (property)g(\(II.8\))g(establishes)873 5293 y Fo(k)p Fs(e)968 5252 y Ff(\000)p Fr(itH)1130 5260 y Fc(g)1171 5293 y Fs(A)p Fp(\()p Fs(t)p Fp(\)\010)1425 5308 y Fl(gs)1532 5293 y Fo(\000)39 b Fs(e)1693 5252 y Ff(\000)p Fr(itH)1855 5260 y Fc(g)1896 5293 y Fs(A)p Fp(\()p Fs(\034)11 b Fp(\)\010)2168 5308 y Fl(gs)2236 5293 y Fo(k)52 b(\024)i(O)s Fp(\()p Fs(g)2640 5252 y Fl(1+)p Fr(\026)2776 5293 y Fp(\))22 b Fs(;)375 b Fv(\(II.26\))p eop %%Page: 19 20 19 19 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(19)257 573 y(and)25 b(we)g(may)f(replace)h(the)f(time-e)n(v)n(olv)o (ed)e(state)j Fs(e)2018 537 y Fr(itH)2125 545 y Fc(g)2166 573 y Fs(A)p Fp(\()p Fs(\034)11 b Fp(\)\010)2438 588 y Fl(gs)2531 573 y Fv(by)24 b Fs(e)2700 537 y Ff(\000)p Fr(itH)2862 545 y Fc(g)2903 573 y Fs(A)p Fp(\()p Fs(t)p Fp(\)\010)3157 588 y Fl(gs)3249 573 y Fv(in)g(this)257 693 y(case.)32 b(Using)24 b Fs(\034)39 b Fp(=)27 b Fs(t)e Fv(in)g(Eq.)f(\(II.13\),)h(we)g(furthermore)g(obtain)387 973 y Fs(e)432 932 y Ff(\000)p Fr(it)p Fl(\()p Fr(H)621 940 y Fc(g)659 932 y Ff(\000)p Fr(E)766 941 y Fd(0)800 932 y Fl(\))831 973 y Fs(A)p Fp(\()p Fs(t)p Fp(\)\010)1085 988 y Fl(gs)1206 973 y Fp(=)52 b Fs(A)1407 988 y Fr(t)1437 973 y Fp(\010)1507 988 y Fl(gs)1625 973 y Fo(\000)e Fs(ig)1852 837 y Fi(Z)1952 864 y Fr(t)1908 1063 y Ff(\000)p Fr(t)2009 973 y Fs(ds)17 b(e)2168 932 y Ff(\000)p Fr(i)p Fl(\()p Fr(t)p Ff(\000)p Fr(s)p Fl(\))p Fr(H)2472 940 y Fc(g)2529 973 y Fp([)p Fs(W)m(;)g(A)2762 988 y Fr(s)2799 973 y Fp(])g Fs(e)2888 932 y Fr(i)p Fl(\()p Fr(t)p Ff(\000)p Fr(s)p Fl(\))p Fr(H)3137 940 y Fc(g)3178 973 y Fp(\010)3248 988 y Fl(gs)3338 973 y Fs(:)3238 1162 y Fv(\(II.27\))257 1374 y(Using)958 1654 y Fp(\011\()p Fs(t;)g(t)p Fp(\))52 b(=)1405 1518 y Fi(Z)1504 1545 y Fr(t)1460 1744 y Ff(\000)p Fr(t)1561 1654 y Fs(ds)17 b(e)1720 1613 y Ff(\000)p Fr(i)p Fl(\()p Fr(t)p Ff(\000)p Fr(s)p Fl(\)\()p Fr(H)2051 1621 y Fc(g)2088 1613 y Ff(\000)p Fr(E)2195 1622 y Fd(0)2229 1613 y Fl(\))2277 1654 y Fp([)p Fs(W)m(;)g(A)2510 1669 y Fr(s)2548 1654 y Fp(])g(\010)2662 1669 y Fl(gs)2751 1654 y Fs(;)460 b Fv(\(II.28\))257 1933 y(computations)34 b(running)g(along)h(the)h(lines)f(of)g(\(II.16\)\226\(II.23\))h (substituting)d Fp(\010)3103 1948 y Ff(1)3178 1933 y Fp(\()p Fs(A;)17 b Fo(1)p Fp(\))p Fv(,)257 2054 y Fp(\010)327 2069 y Fr(T)383 2054 y Fp(\()p Fs(A;)g Fo(1)p Fp(\))p Fv(,)37 b Fp(\011\()p Fs(t;)17 b Fo(1)p Fp(\))35 b Fv(for)g Fp(\010)1325 2069 y Ff(1)1401 2054 y Fp(\()p Fs(A;)17 b Fp(0\))p Fv(,)37 b Fp(\010)1775 2069 y Fr(T)1831 2054 y Fp(\()p Fs(A;)17 b Fp(0\))p Fv(,)37 b Fp(\011\()p Fs(t;)17 b Fp(0\))35 b Fv(and)g(replacing)g(inte)o(grals)f(of)257 2243 y(the)25 b(form)640 2142 y Fr(T)623 2162 y Fi(R)628 2333 y Fl(0)706 2243 y Fs(ds)f Fv(by)h(those)f(of)h(the)f(form)1697 2142 y Fr(T)1680 2162 y Fi(R)1650 2335 y Ff(\000)p Fr(T)1782 2243 y Fs(ds)h Fv(pro)o(v)o(es)e(\(II.12\))j(for)f Fs(\034)39 b Fo(\025)28 b Fs(g)2852 2206 y Ff(\000)2928 2179 y Fd(1)p 2916 2191 55 3 v 2916 2232 a Fc(K)2985 2243 y Fv(,)d(as)g(well.)78 b Fa(\003)404 2428 y Fv(Theorem)34 b(II.1)i(is)e(the)h(main)f(tool)h (for)g(the)g(analysis)f(of)h(the)g(char)n(ges)g(transported)g(to)257 2549 y(in\002nity)-6 b(,)23 b Fs(Q)665 2512 y Fl(inf)665 2574 y Ff(T)755 2549 y Fp(\()p Fs(A)p Fp(\()p Fs(\034)11 b Fp(\)\010)1065 2564 y Fl(gs)1133 2549 y Fp(\))25 b Fv(and)f Fs(Q)1441 2501 y Fl(sup)1441 2576 y Ff(T)1551 2549 y Fp(\()p Fs(A)p Fp(\()p Fs(\034)11 b Fp(\)\010)1861 2564 y Fl(gs)1930 2549 y Fp(\))p Fv(,)24 b(because)g(it)g(allo)n(ws)f (us)h(to)g(tak)o(e)g(the)g(limits)257 2669 y Fs(t)k Fo(!)g(1)c Fv(and)h Fs(R)k Fo(!)e(1)p Fv(,)d(as)h(sho)n(wn)f(in)g(the)h(follo)n (wing)e(lemma.)257 2839 y Fk(Lemma)j(II.2.)49 b Fq(Assume)22 b(Hypotheses)h(1\2262,)g(and)g(let)g Fs(A)28 b Fp(=)f Fs(a)2395 2803 y Ff(\003)2435 2839 y Fp(\()p Fs(f)2521 2854 y Fl(1)2560 2839 y Fp(\))17 b Fs(a)2666 2803 y Ff(\003)2706 2839 y Fp(\()p Fs(f)2792 2854 y Fl(2)2831 2839 y Fp(\))g Fo(\001)g(\001)g(\001)d Fs(a)3069 2803 y Ff(\003)3109 2839 y Fp(\()p Fs(f)3195 2854 y Fr(N)3262 2839 y Fp(\))24 b Fq(with)257 2960 y Fs(f)305 2975 y Fl(1)345 2960 y Fs(;)17 b(:::;)g(f)562 2975 y Fr(N)667 2960 y Fo(2)38 b Fs(C)848 2923 y Ff(1)841 2984 y Fl(0)923 2960 y Fp(\()p Fh(R)1027 2923 y Fl(3)1098 2960 y Fo(n)26 b(f)p Fp(0)p Fo(g)p Fp(\))p Fq(.)47 b(Then)30 b(ther)l(e)h(e)n(xists)f(a)g(constant) f Fs(C)45 b(<)37 b Fo(1)p Fq(,)32 b(suc)o(h)d(that)g(the)257 3080 y(c)o(har)l(g)o(es)c(tr)o(ansported)d(to)i(in\002nity)g(obe)m(y) 266 3310 y Fs(Q)343 3262 y Fl(sup)343 3337 y Ff(T)454 3310 y Fp(\()p Fs(A)p Fp(\()p Fs(\034)11 b Fp(\)\010)764 3325 y Fl(gs)832 3310 y Fp(\))39 b Fo(\000)g Fs(C)7 b(g)1153 3268 y Fl(2+)p Fr(\026)1342 3310 y Fo(\024)53 b Fs(g)1523 3268 y Fl(2)1562 3225 y Fi(\015)1562 3285 y(\015)1617 3310 y Fs(T)1674 3325 y Ff(T)1751 3310 y Fp(\010)1821 3325 y Ff(1)1896 3310 y Fp(\()p Fs(A;)17 b(\034)11 b Fp(\))2142 3225 y Fi(\015)2142 3285 y(\015)2198 3251 y Fl(2)2290 3310 y Fo(\024)53 b Fs(Q)2497 3268 y Fl(inf)2497 3334 y Ff(T)2587 3310 y Fp(\()p Fs(A)p Fp(\()p Fs(\034)11 b Fp(\)\010)2897 3325 y Fl(gs)2965 3310 y Fp(\))39 b(+)f Fs(C)7 b(g)3284 3268 y Fl(2+)p Fr(\026)3442 3310 y Fs(;)3238 3430 y Fv(\(II.29\))257 3670 y Fq(for)28 b Fp(0)35 b Fo(\024)g Fs(\034)46 b Fo(\024)35 b Fs(g)849 3634 y Fr(\026)923 3670 y Fq(and)28 b Fs(\034)46 b Fo(\025)35 b Fs(g)1352 3633 y Ff(\000)1429 3606 y Fd(1)p 1417 3618 V 1417 3659 a Fc(K)1485 3670 y Fq(,)30 b(wher)l(e)f Fs(\026)p Fp(\()p Fs(K)7 b Fp(\))34 b(:=)h(1)25 b Fo(\000)g Fs(K)2471 3634 y Ff(\000)p Fl(1)2600 3670 y Fs(>)34 b Fp(0)29 b Fq(and)f Fs(K)42 b(>)34 b Fp(1)28 b Fq(is)g(the)257 3790 y(de)l(gr)l(ee)e(of)e (dif)n(fer)l(entiability)e(in)j(Hypothesis)f(1.)257 3961 y(Pr)l(oof:)31 b Fv(First)24 b(we)i(observ)o(e)e(that,)h(since)g Fs(F)1720 3976 y Fr(R)1806 3961 y Fo(\021)k Fs(F)1975 3976 y Fr(R)2055 3961 y Fo(\012)23 b Fg(1)2211 3976 y Fr(f)2282 3961 y Fv(commutes)g(with)i Fs(A)2992 3976 y Fr(t)3050 3961 y Fo(\021)j Fg(1)3211 3976 y Fr(el)3293 3961 y Fo(\012)22 b Fs(A)3465 3976 y Fr(t)257 4081 y Fv(and)j Fs(H)507 4096 y Fr(f)580 4081 y Fp(=)j Fg(1)740 4096 y Fr(el)820 4081 y Fo(\012)23 b Fs(H)1001 4096 y Fr(f)1046 4081 y Fv(,)i(we)g(ha)n(v)o(e)528 4216 y Fi(\015)528 4276 y(\015)583 4301 y Fs(T)640 4316 y Ff(T)717 4301 y Fs(F)780 4316 y Fr(R)855 4301 y Fs(A)928 4316 y Fr(t)974 4301 y Fp(\010)1044 4316 y Fl(gs)1112 4216 y Fi(\015)1112 4276 y(\015)1250 4301 y Fo(\024)1411 4216 y Fi(\015)1411 4276 y(\015)1466 4301 y Fs(F)1529 4316 y Fr(R)1603 4301 y Fs(A)1676 4316 y Fr(t)1723 4301 y Fp(\010)1793 4316 y Fl(gs)1860 4216 y Fi(\015)1860 4276 y(\015)1996 4301 y Fp(=)2152 4216 y Fi(\015)2152 4276 y(\015)2207 4301 y Fs(A)2280 4316 y Fr(t)2327 4301 y Fs(F)2390 4316 y Fr(R)2464 4301 y Fp(\010)2534 4316 y Fl(gs)2602 4216 y Fi(\015)2602 4276 y(\015)3238 4301 y Fv(\(II.30\))1250 4460 y Fo(\024)1411 4375 y Fi(\015)1411 4435 y(\015)1466 4460 y Fs(A)1539 4475 y Fr(t)1585 4460 y Fp(\()p Fs(H)1704 4475 y Fr(f)1771 4460 y Fp(+)e(1\))1957 4419 y Ff(\000)p Fr(N)q(=)p Fl(2)2143 4375 y Fi(\015)2143 4435 y(\015)2221 4460 y Fo(\001)2270 4375 y Fi(\015)2270 4435 y(\015)2326 4460 y Fs(F)2389 4475 y Fr(R)2463 4460 y Fp(\()p Fs(H)2582 4475 y Fr(f)2649 4460 y Fp(+)f(1\))2834 4419 y Fr(N)q(=)p Fl(2)2983 4460 y Fp(\010)3053 4475 y Fl(gs)3120 4375 y Fi(\015)3120 4435 y(\015)3198 4460 y Fs(:)257 4689 y Fv(Applying)27 b(Lemma)g(IV)-13 b(.1)28 b Fs(N)10 b Fv(-times,)28 b(we)g(observ)o(e)f(that)2238 4604 y Fi(\015)2238 4664 y(\015)2293 4689 y Fs(A)2366 4704 y Fr(t)2412 4689 y Fp(\()p Fs(H)2531 4704 y Fr(f)2601 4689 y Fp(+)d(1\))2788 4653 y Ff(\000)p Fr(N)q(=)p Fl(2)2975 4604 y Fi(\015)2975 4664 y(\015)3058 4689 y Fv(is)j(bounded)257 4809 y(by)e Fs(C)452 4824 y Fl(2)519 4809 y Fs(<)i Fo(1)p Fv(,)e(uniformly)e(in)i Fs(t)p Fv(,)273 5034 y Fi(\015)273 5094 y(\015)328 5119 y Fs(A)401 5134 y Fr(t)448 5119 y Fp(\()p Fs(H)567 5134 y Fr(f)634 5119 y Fp(+)d(1\))819 5077 y Ff(\000)p Fr(N)q(=)p Fl(2)1005 5034 y Fi(\015)1005 5094 y(\015)1113 5119 y Fo(\024)1276 4994 y Fr(N)1243 5024 y Fi(Y)1246 5234 y Fr(j)t Fl(=1)1387 5004 y Fi(\015)1387 5064 y(\015)1387 5124 y(\015)1443 5119 y Fp(\()p Fs(H)1562 5134 y Fr(f)1629 5119 y Fp(+)g(1\))1814 5077 y Fl(\()p Fr(j)t Ff(\000)p Fl(1\))p Fr(=)p Fl(2)2082 5119 y Fs(a)2133 5077 y Ff(\003)2173 5119 y Fp(\()p Fs(e)2256 5077 y Ff(\000)p Fr(it!)2411 5119 y Fs(f)2459 5134 y Fr(j)2496 5119 y Fp(\))17 b(\()p Fs(H)2670 5134 y Fr(f)2736 5119 y Fp(+)22 b(1\))2921 5077 y Ff(\000)p Fr(j)t(=)p Fl(2)3083 5004 y Fi(\015)3083 5064 y(\015)3083 5124 y(\015)3191 5119 y Fo(\024)53 b Fs(C)3391 5134 y Fl(2)3453 5119 y Fs(:)3238 5349 y Fv(\(II.31\))p eop %%Page: 20 21 20 20 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(20)257 573 y(Moreo)o(v)o(er)l(,)24 b Fp(\()p Fs(H)812 588 y Fr(f)879 573 y Fp(+)e(1\))1064 537 y Fr(N)q(=)p Fl(2)1196 573 y Fp(\010)1266 588 y Fl(gs)1361 573 y Fo(2)28 b(H)e Fv(and)f Fs(F)1797 588 y Fr(R)1883 573 y Fo(!)i Fp(0)e Fv(strongly)-6 b(,)23 b(as)i Fs(R)j Fo(!)g(1)p Fv(,)c(hence)1249 765 y Fp(lim)1220 827 y Fr(R)p Ff(!1)1447 765 y Fp(lim)1431 825 y Fr(t)p Ff(!1)1614 680 y Fi(\015)1614 740 y(\015)1670 765 y Fs(T)1727 780 y Ff(T)1804 765 y Fs(F)1867 780 y Fr(R)1941 765 y Fs(A)2014 780 y Fr(t)2061 765 y Fp(\010)2131 780 y Fl(gs)2198 680 y Fi(\015)2198 740 y(\015)2306 765 y Fp(=)53 b(0)22 b Fs(:)705 b Fv(\(II.32\))257 985 y(According)25 b(to)f(Theorem)h(II.1)g(we)g(ha)n(v)o(e)421 1176 y Fs(e)466 1135 y Ff(\000)p Fr(it)p Fl(\()p Fr(H)655 1143 y Fc(g)693 1135 y Ff(\000)p Fr(E)800 1144 y Fd(0)834 1135 y Fl(\))882 1176 y Fs(A)p Fp(\()p Fs(\034)11 b Fp(\))17 b(\010)1171 1191 y Fl(gs)1291 1176 y Fp(=)53 b Fs(A)1493 1191 y Fr(t)1539 1176 y Fp(\010)1609 1191 y Fl(gs)1721 1176 y Fo(\000)45 b Fs(ig)20 b(e)1988 1135 y Ff(\000)p Fr(it)p Fl(\()p Fr(H)2177 1144 y Fd(0)2213 1135 y Ff(\000)p Fr(E)2320 1144 y Fd(0)2354 1135 y Fl(\))2402 1176 y Fp(\010)2472 1191 y Ff(1)2547 1176 y Fp(\()p Fs(A;)d(\034)11 b Fp(\))45 b(+)f(Rem)o(\()p Fs(t)p Fp(\))23 b Fs(;)3238 1297 y Fv(\(II.33\))257 1492 y(where)30 b Fo(k)p Fp(Rem)o(\()p Fs(t)p Fp(\))p Fo(k)35 b(\024)g Fs(C)1154 1507 y Fl(1)1194 1492 y Fs(g)1245 1456 y Fl(1+)p Fr(\026)1381 1492 y Fv(,)30 b(for)f(some)f(constant)g Fs(C)2251 1507 y Fl(1)2325 1492 y Fs(<)35 b Fo(1)29 b Fv(independent)f(of)g Fs(t)36 b Fo(2)f Fh(R)3430 1456 y Fl(+)257 1612 y Fv(and)c Fs(\026)39 b Fp(=)h(1)26 b Fo(\000)h Fp(1)p Fs(=K)7 b Fv(.)50 b(Inserting)30 b(Eqs.)h(\(II.33\))g (and)g(\(II.32\))g(into)g(the)f(de\002nitions)g(\(I.31\)\226)257 1733 y(\(I.32\))25 b(of)g(the)g(transported)f(char)n(ges,)h(we)g (\002nd)g(that)295 1925 y Fs(Q)372 1877 y Fl(sup)372 1952 y Ff(T)483 1925 y Fp(\()p Fs(A)p Fp(\()p Fs(\034)11 b Fp(\)\010)793 1940 y Fl(gs)861 1925 y Fp(\))39 b Fo(\000)g Fs(C)1124 1940 y Fl(1)1163 1925 y Fs(g)1214 1883 y Fl(2+)p Fr(\026)1403 1925 y Fo(\024)53 b Fs(g)1584 1883 y Fl(2)1642 1899 y Fi(e)1623 1925 y Fs(Q)1700 1940 y Ff(T)1760 1925 y Fp(\()p Fs(A)p Fp(\()p Fs(\034)11 b Fp(\)\010)2070 1940 y Fl(gs)2138 1925 y Fp(\))53 b Fo(\024)g Fs(Q)2436 1883 y Fl(inf)2436 1949 y Ff(T)2526 1925 y Fp(\()p Fs(A)p Fp(\()p Fs(\034)11 b Fp(\)\010)2836 1940 y Fl(gs)2904 1925 y Fp(\))39 b(+)f Fs(C)3165 1940 y Fl(1)3205 1925 y Fs(g)3256 1883 y Fl(2+)p Fr(\026)3414 1925 y Fs(;)3238 2045 y Fv(\(II.34\))257 2229 y(where)522 2396 y Fi(e)503 2421 y Fs(Q)580 2436 y Ff(T)640 2421 y Fp(\()p Fs(A)p Fp(\()p Fs(\034)11 b Fp(\)\010)950 2436 y Fl(gs)1018 2421 y Fp(\))53 b(:=)82 b(lim)1264 2483 y Fr(R)p Ff(!1)1491 2421 y Fp(lim)1476 2481 y Fr(t)p Ff(!1)1659 2336 y Fi(\015)1659 2396 y(\015)1714 2421 y Fs(T)1771 2436 y Ff(T)1832 2421 y Fs(F)1895 2436 y Fr(R)1953 2421 y Fs(e)1998 2380 y Ff(\000)p Fr(it)p Fl(\()p Fr(H)2187 2389 y Fd(0)2222 2380 y Ff(\000)p Fr(E)2329 2389 y Fd(0)2363 2380 y Fl(\))2395 2421 y Fp(\010)2465 2436 y Ff(1)2540 2421 y Fp(\()p Fs(A;)17 b(\034)11 b Fp(\))2786 2340 y Fi(\001)2832 2336 y(\015)2832 2396 y(\015)2887 2362 y Fl(2)2949 2421 y Fs(;)262 b Fv(\(II.35\))257 2674 y(and)25 b(estimate)f(\(II.29\))h(w)o(ould)f(follo)n(w)g(from)1858 2649 y Fi(e)1839 2674 y Fs(Q)1916 2689 y Ff(T)1977 2674 y Fp(\()p Fs(A)p Fp(\()p Fs(\034)11 b Fp(\)\010)2287 2689 y Fl(gs)2355 2674 y Fp(\))28 b(=)2524 2589 y Fi(\015)2524 2649 y(\015)2580 2674 y Fs(T)2637 2689 y Ff(T)2697 2674 y Fp(\010)2767 2689 y Ff(1)2842 2674 y Fp(\()p Fs(A;)17 b(\034)11 b Fp(\))3088 2589 y Fi(\015)3088 2649 y(\015)3144 2615 y Fl(2)3183 2674 y Fv(.)404 2803 y(Ne)o(xt,)33 b(we)f(turn)g(to)g Fs(T)1163 2818 y Ff(T)1224 2803 y Fs(F)1287 2818 y Fr(R)1344 2803 y Fs(e)1389 2767 y Ff(\000)p Fr(it)p Fl(\()p Fr(H)1578 2776 y Fd(0)1614 2767 y Ff(\000)p Fr(E)1721 2776 y Fd(0)1755 2767 y Fl(\))1787 2803 y Fp(\010)1857 2818 y Ff(1)1932 2803 y Fp(\()p Fs(A;)17 b(\034)11 b Fp(\))p Fv(.)53 b(W)-8 b(e)33 b(note)f(that)f Fs(F)2867 2818 y Fr(R)2967 2803 y Fp(=)41 b Fs(F)3147 2818 y Fr(R)3221 2803 y Fs(P)3284 2818 y Fr(c)3351 2803 y Fv(and)257 2923 y(hence)371 3115 y Fs(F)434 3130 y Fr(R)509 3115 y Fs(e)554 3074 y Ff(\000)p Fr(it)p Fl(\()p Fr(H)743 3083 y Fd(0)778 3074 y Ff(\000)p Fr(E)885 3083 y Fd(0)919 3074 y Fl(\))1004 3115 y Fp(=)52 b Fs(e)1177 3074 y Ff(\000)p Fr(it)p Fl(\()p Fr(H)1366 3086 y Fc(f)1407 3074 y Ff(\000)p Fr(E)1514 3083 y Fd(0)1548 3074 y Fl(\))1596 3115 y Fs(F)1659 3130 y Fr(R)1734 3115 y Fs(P)1797 3130 y Fr(c)1848 3115 y Fs(e)1893 3074 y Ff(\000)p Fr(itH)2055 3086 y Fc(el)2166 3115 y Fp(=)g Fs(e)2339 3074 y Ff(\000)p Fr(it)p Fl(\()p Fr(H)2528 3086 y Fc(f)2569 3074 y Ff(\000)p Fr(E)2676 3083 y Fd(0)2710 3074 y Fl(\))2758 3115 y Fs(F)2821 3130 y Fr(R)2896 3115 y Fs(e)2941 3074 y Ff(\000)p Fr(it)p Fl(\()p Ff(\000)p Fl(\001\))3235 3115 y Fs(P)3298 3130 y Fr(c)3354 3115 y Fs(:)3238 3236 y Fv(\(II.36\))257 3431 y(Furthermore,)35 b(the)d(absolute)f(continuity)g(of)h(the)h(spectrum)e(of)i Fo(\000)p Fp(\001)g Fv(on)f Fs(L)2942 3394 y Fl(2)2982 3431 y Fp(\()p Fh(R)3086 3394 y Fl(3)3131 3431 y Fp(\))h Fv(implies)257 3551 y(that)1019 3743 y Fp(\(1)22 b Fo(\000)g Fs(F)1290 3758 y Fr(R)1348 3743 y Fp(\))p Fs(e)1431 3702 y Ff(\000)p Fr(it)p Fl(\()p Ff(\000)p Fl(\001\))1761 3743 y Fp(=)53 b Fg(1)1946 3758 y Ff(fj)p Fr(x)p Ff(j)p Fr()936 5017 y(>)936 5047 y(<)936 5226 y(>)936 5256 y(>)936 5286 y(:)1066 4887 y(Z)1166 4913 y Ff(1)1122 5113 y Fl(0)1257 5023 y Fs(ds)17 b(e)1416 4982 y Fr(is)p Fl(\()p Fr(H)1558 4991 y Fd(0)1592 4982 y Ff(\000)p Fr(E)1699 4991 y Fd(0)1734 4982 y Fl(\))1782 5023 y Fp([)p Fs(W)30 b(;)j(A)2064 5038 y Fr(s)2101 5023 y Fp(])17 b Fs(P)2208 5038 y Fr(d)2265 5023 y Fp(\010)2335 5038 y Fl(gs)2496 5023 y Fv(if)25 b Fp(0)j Fo(\024)g Fs(\034)39 b Fo(\024)28 b Fs(g)3001 4987 y Fr(\026)1066 5156 y Fi(Z)1166 5182 y Ff(1)1122 5381 y(\0001)1268 5291 y Fs(ds)17 b(e)1427 5250 y Fr(is)p Fl(\()p Fr(H)1569 5259 y Fd(0)1603 5250 y Ff(\000)p Fr(E)1710 5259 y Fd(0)1744 5250 y Fl(\))1792 5291 y Fp([)p Fs(W)31 b(;)i(A)2075 5306 y Fr(s)2112 5291 y Fp(])17 b Fs(P)2219 5306 y Fr(d)2276 5291 y Fp(\010)2346 5306 y Fl(gs)2496 5291 y Fv(if)25 b Fs(\034)39 b Fo(\025)28 b Fs(g)2819 5254 y Ff(\000)2896 5227 y Fd(1)p 2884 5239 55 3 v 2884 5281 a Fc(K)3238 5161 y Fv(\(II.39\))p eop %%Page: 21 22 21 21 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(21)257 573 y(which)25 b(we)g(complement)e(by)i(the)f(de\002nition)g (of)h(the)g(v)o(ector)406 958 y Fp(\010)476 917 y Fl(\012)476 983 y Ff(1)551 958 y Fp(\()p Fs(A;)17 b(\034)11 b Fp(\))45 b(:=)989 694 y Fi(8)989 784 y(>)989 813 y(>)989 843 y(<)989 1023 y(>)989 1053 y(>)989 1082 y(:)1119 684 y(Z)1218 710 y Ff(1)1174 909 y Fl(0)1310 819 y Fs(ds)17 b(e)1469 778 y Fr(is)p Fl(\()p Fr(H)1611 787 y Fd(0)1645 778 y Ff(\000)p Fr(E)1752 787 y Fd(0)1786 778 y Fl(\))1834 819 y Fp([)p Fs(W)31 b(;)i(A)2117 834 y Fr(s)2154 819 y Fp(])17 b Fs(P)2261 834 y Fr(d)2318 819 y Fp(\(0)k Fo(\010)i Fp(\012\))94 b Fv(if)25 b Fp(0)i Fo(\024)h Fs(\034)40 b Fo(\024)28 b Fs(g)3233 783 y Fr(\026)1119 952 y Fi(Z)1218 978 y Ff(1)1174 1178 y(\0001)1320 1088 y Fs(ds)17 b(e)1479 1047 y Fr(is)p Fl(\()p Fr(H)1621 1056 y Fd(0)1656 1047 y Ff(\000)p Fr(E)1763 1056 y Fd(0)1797 1047 y Fl(\))1845 1088 y Fp([)p Fs(W)30 b(;)k(A)2128 1103 y Fr(s)2164 1088 y Fp(])17 b Fs(P)2271 1103 y Fr(d)2328 1088 y Fp(\(0)22 b Fo(\010)h Fp(\012\))83 b Fv(if)25 b Fs(\034)39 b Fo(\025)28 b Fs(g)3051 1051 y Ff(\000)3127 1024 y Fd(1)p 3115 1036 55 3 v 3115 1077 a Fc(K)3238 1277 y Fv(\(II.40\))257 1468 y(whose)i(e)o(xistence)f(is)g(guaranteed)h (thanks)f(to)g(Lemma)g(IV)-13 b(.3.)46 b(Lemma)29 b(II.3)h(belo)n(w)f (sho)n(ws)257 1589 y(that)k(these)g(tw)o(o)f(v)o(ectors)g(dif)n(fer)h (by)g(at)g(most)f Fo(O)s Fp(\()p Fs(g)t Fp(\))p Fv(,)i(i.e.,)h(a)e (replacement)g(of)g(the)g(ground)257 1709 y(state)h Fp(\010)544 1724 y Fl(gs)646 1709 y Fv(of)g Fs(H)844 1724 y Fr(g)919 1709 y Fv(by)g(the)f(ground)h(state)g Fp(0)29 b Fo(\010)g Fp(\012)35 b Fv(of)f Fs(H)2229 1724 y Fl(0)2303 1709 y Fv(in)f(the)h(de\002nition)g(of)g Fp(\010)3174 1724 y Ff(1)3249 1709 y Fp(\()p Fs(A;)17 b(\034)11 b Fp(\))257 1829 y Fv(introduces)24 b(only)g(small)g(errors)h(which)g(are)g(ne)o (gligible,)e(as)i(we)g(shall)f(see.)257 2000 y Fk(Lemma)i(II.3.)49 b Fq(Assume)22 b(Hypotheses)h(1\2262,)g(and)g(let)g Fs(A)28 b Fp(=)f Fs(a)2395 1963 y Ff(\003)2435 2000 y Fp(\()p Fs(f)2521 2015 y Fl(1)2560 2000 y Fp(\))17 b Fs(a)2666 1963 y Ff(\003)2706 2000 y Fp(\()p Fs(f)2792 2015 y Fl(2)2831 2000 y Fp(\))g Fo(\001)g(\001)g(\001)d Fs(a)3069 1963 y Ff(\003)3109 2000 y Fp(\()p Fs(f)3195 2015 y Fr(N)3262 2000 y Fp(\))24 b Fq(with)257 2120 y Fs(f)305 2135 y Fl(1)345 2120 y Fs(;)17 b(:::;)g(f)562 2135 y Fr(N)657 2120 y Fo(2)28 b Fs(C)828 2084 y Ff(1)821 2145 y Fl(0)902 2120 y Fp(\()p Fh(R)1006 2084 y Fl(3)1074 2120 y Fo(n)22 b(f)p Fp(0)p Fo(g)p Fp(\))p Fq(.)30 b(Then)25 b(ther)l(e)g(e)n(xists)f (a)h(constant)f Fs(C)34 b(<)28 b Fo(1)p Fq(,)c(suc)o(h)g(that)1164 2234 y Fi(\015)1164 2294 y(\015)1220 2319 y Fp(\010)1290 2334 y Ff(1)1365 2319 y Fp(\()p Fs(A;)17 b(\034)11 b Fp(\))39 b Fo(\000)g Fp(\010)1836 2278 y Fl(\012)1836 2344 y Ff(1)1911 2319 y Fp(\()p Fs(A;)17 b(\034)11 b Fp(\))2157 2234 y Fi(\015)2157 2294 y(\015)2265 2319 y Fo(\024)53 b Fs(C)24 b(g)h(:)650 b Fv(\(II.41\))257 2518 y Fq(Pr)l(oof:)30 b Fv(W)-8 b(e)25 b(\002rst)g(note)f(that)h(due)g (to)f(Lemma)g(IV)-13 b(.3,)25 b(we)g(ha)n(v)o(e)g(the)f(estimate)269 2632 y Fi(\015)269 2692 y(\015)324 2717 y Fp(\010)394 2732 y Ff(1)469 2717 y Fp(\()p Fs(A;)17 b(\034)11 b Fp(\))39 b Fo(\000)h Fp(\010)941 2676 y Fl(\012)941 2742 y Ff(1)1016 2717 y Fp(\()p Fs(A;)17 b(\034)11 b Fp(\))1262 2632 y Fi(\015)1262 2692 y(\015)1345 2717 y Fo(\024)1816 b Fv(\(II.42\))352 2921 y Fo(\024)512 2806 y Fi(\015)512 2866 y(\015)512 2926 y(\015)568 2921 y Fp(\()p Fs(H)695 2870 y Fl(\(\003\))687 2948 y Fr(f)824 2921 y Fp(+)22 b(1\))1009 2880 y Fl(1+)1109 2852 y Fc(N)p 1109 2864 54 3 v 1121 2906 a Fd(2)1177 2840 y Fi(\000)1223 2921 y Fs(P)1286 2936 y Fr(d)1326 2921 y Fp(\010)1396 2936 y Fl(gs)1502 2921 y Fo(\000)40 b Fp(\(0)21 b Fo(\010)i Fp(\012\))1935 2840 y Fi(\001)1981 2806 y(\015)1981 2866 y(\015)1981 2926 y(\015)2059 2921 y Fo(\001)2108 2785 y Fi(Z)2164 3011 y Fe(R)2232 2921 y Fs(ds)2329 2836 y Fi(\015)2329 2896 y(\015)2401 2921 y Fp([)p Fs(W)30 b(;)k(A)2684 2936 y Fr(s)2721 2921 y Fp(])17 b(\()p Fs(H)2892 2870 y Fl(\(\003\))2884 2948 y Fr(f)3021 2921 y Fp(+)22 b(1\))3206 2880 y Ff(\000)p Fl(1)p Ff(\000)3361 2852 y Fc(N)p 3361 2864 V 3372 2906 a Fd(2)3428 2836 y Fi(\015)3428 2896 y(\015)352 3158 y Fo(\024)83 b Fs(C)582 3173 y Fl(1)638 3044 y Fi(\015)638 3104 y(\015)638 3163 y(\015)694 3158 y Fp(\()p Fs(H)821 3108 y Fl(\(\003\))813 3186 y Fr(f)950 3158 y Fp(+)22 b(1\))1135 3117 y Fl(1+)1235 3090 y Fc(N)p 1235 3102 V 1247 3143 a Fd(2)1303 3078 y Fi(\000)1348 3158 y Fs(P)1411 3173 y Fr(d)1452 3158 y Fp(\010)1522 3173 y Fl(gs)1628 3158 y Fo(\000)39 b Fp(\(0)22 b Fo(\010)h Fp(\012\))2061 3078 y Fi(\001)2107 3044 y(\015)2107 3104 y(\015)2107 3163 y(\015)2184 3158 y Fs(;)257 3378 y Fv(for)28 b(some)e(constant)h Fs(C)1068 3393 y Fl(1)1139 3378 y Fs(<)32 b Fo(1)p Fv(.)38 b(Additionally)25 b(using)h(Lemma)h(IV)-13 b(.2)27 b(and)g(the)g (decomposi-)257 3499 y(tion)d Fs(P)500 3514 y Fr(d)541 3499 y Fp(\010)611 3514 y Fl(gs)700 3499 y Fo(\000)f Fp(\(0)f Fo(\010)h Fp(\012\))28 b(=)1248 3418 y Fi(\002)1290 3499 y Fs(P)1353 3514 y Fr(d)1393 3499 y Fs(P)1456 3514 y Fl(\012)1511 3499 y Fp(\010)1581 3514 y Fl(gs)1671 3499 y Fo(\000)23 b Fp(\(0)e Fo(\010)i Fp(\012\))2087 3418 y Fi(\003)2151 3499 y Fp(+)f Fs(P)2312 3514 y Fr(d)2352 3499 y Fs(P)2429 3462 y Ff(?)2415 3524 y Fl(\012)2488 3499 y Fp(\010)2558 3514 y Fl(gs)2651 3499 y Fv(,)i(we)h(observ)o(e)g (that)356 3620 y Fi(\015)356 3680 y(\015)356 3740 y(\015)411 3735 y Fp(\()p Fs(H)538 3684 y Fl(\(\003\))530 3763 y Fr(f)668 3735 y Fp(+)d(1\))853 3694 y Fl(1+)p Fr(N)q(=)p Fl(2)1075 3654 y Fi(\000)1120 3735 y Fs(P)1183 3750 y Fr(d)1224 3735 y Fp(\010)1294 3750 y Fl(gs)1400 3735 y Fo(\000)39 b Fp(\(0)22 b Fo(\010)h Fp(\012\))1833 3654 y Fi(\001)1879 3620 y(\015)1879 3680 y(\015)1879 3740 y(\015)1987 3735 y Fo(\024)53 b(O)s Fp(\()p Fs(g)t Fp(\))21 b Fs(:)945 b Fa(\003)3238 3892 y Fv(\(II.43\))404 4083 y(The)27 b(\002nal)g(ingredient)f(for)h(the)g(proof)f(of)h(Theorem)g (I.1)g(is)f(Lemma)h(II.4,)g(belo)n(w)-6 b(.)36 b(Note)257 4204 y(that)26 b(due)h(to)f(the)h(de\002nitions)e(\(II.40\))i(we)g(may) f(restrict)g(ourselv)o(es)g(to)g(the)g(case)i Fs(\034)42 b Fp(=)31 b(0)26 b Fv(and)257 4324 y Fs(\034)40 b Fp(=)27 b Fo(1)p Fv(.)257 4494 y Fk(Lemma)f(II.4.)49 b Fq(Assume)41 b(Hypotheses)g(1\2262,)46 b(\002x)c Fs(N)5 b(;)17 b(m)59 b Fo(2)g Fh(N)e Fq(with)42 b Fs(m)59 b Fo(\024)h Fs(N)10 b Fq(,)46 b(and)c(let)257 4615 y Fs(')321 4630 y Fl(1)361 4615 y Fs(;)17 b(')469 4630 y Fl(2)508 4615 y Fs(;)g(:)g(:)g(:)f(')747 4630 y Fr(m)855 4615 y Fo(2)42 b Fs(C)1040 4579 y Ff(1)1033 4639 y Fl(0)1115 4615 y Fp(\()p Fh(R)1219 4579 y Fl(3)1292 4615 y Fo(n)28 b(f)p Fp(0)p Fo(g)p Fp(\))j Fq(be)i(an)f(orthonormal)f (system,)j Fo(h)p Fs(')2812 4630 y Fr(i)2839 4615 y Fo(j)p Fs(')2931 4630 y Fr(j)2967 4615 y Fo(i)42 b Fp(=)g Fs(\016)3209 4630 y Fr(i;j)3289 4615 y Fq(.)54 b(Let)257 4735 y Fs(A)28 b Fp(=)g Fs(a)513 4699 y Ff(\003)552 4735 y Fp(\()p Fs(')654 4750 y Fl(1)694 4735 y Fp(\))732 4699 y Fr(n)775 4708 y Fd(1)813 4735 y Fs(a)864 4699 y Ff(\003)904 4735 y Fp(\()p Fs(')1006 4750 y Fl(2)1045 4735 y Fp(\))1083 4699 y Fr(n)1126 4708 y Fd(2)1181 4735 y Fo(\001)17 b(\001)g(\001)e Fs(a)1365 4699 y Ff(\003)1405 4735 y Fp(\()p Fs(')1507 4750 y Fr(m)1573 4735 y Fp(\))1611 4699 y Fr(n)1654 4707 y Fc(m)1717 4735 y Fq(,)23 b(wher)l(e)h Fs(n)2086 4750 y Fr(j)2151 4735 y Fo(2)k Fh(N)38 b Fq(ar)l(e)24 b(suc)o(h)e(that)h Fs(n)2934 4750 y Fl(1)2990 4735 y Fp(+)16 b Fs(n)3140 4750 y Fl(2)3196 4735 y Fp(+)g Fs(:)h(:)g(:)f Fp(+)257 4855 y Fs(n)315 4870 y Fr(m)410 4855 y Fp(=)27 b Fs(N)10 b Fq(,)26 b(and)e(\002x)h(a)g(measur)o(able)e(set)i Fo(T)53 b(\022)28 b Fh(R)1922 4819 y Fl(3)1967 4855 y Fq(.)j(Then)537 4979 y Fi(\015)537 5039 y(\015)593 5064 y Fs(T)650 5079 y Ff(T)727 5064 y Fp(\010)797 5023 y Fl(\012)797 5089 y Ff(1)872 5064 y Fp(\()p Fs(A;)17 b(\034)11 b Fp(\))1118 4979 y Fi(\015)1118 5039 y(\015)1174 5006 y Fl(2)1296 5064 y Fp(=)83 b Fs(Q)1532 5023 y Fr(\034)1532 5089 y Ff(T)1593 5064 y Fp(\()p Fs(')p 1631 5099 64 4 v -1 w Fp(\))28 b(=)1402 b Fv(\(II.44\))1296 5300 y Fp(=)1455 5219 y Fi(\000)1501 5300 y Fs(n)1559 5315 y Fl(1)1598 5300 y Fp(!)17 b Fs(n)1700 5315 y Fl(2)1740 5300 y Fp(!)g Fo(\001)g(\001)g(\001)d Fs(n)1974 5315 y Fr(m)2041 5300 y Fp(!)2068 5219 y Fi(\001)2188 5175 y Fr(m)2147 5205 y Fi(X)2158 5415 y Fr(j)t Fl(=1)2307 5300 y Fs(n)2365 5315 y Fr(j)2435 5164 y Fi(Z)2491 5390 y Ff(T)2568 5300 y Fo(jh)p Fs(L)2701 5259 y Fr(p)2701 5324 y(\034)2760 5300 y Fs(;)34 b(')2885 5315 y Fr(j)2921 5300 y Fo(ij)2988 5259 y Fl(2)3027 5300 y Fs(d)3078 5259 y Fl(3)3117 5300 y Fs(p)22 b(;)p eop %%Page: 22 23 22 22 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(22)257 573 y Fq(with)27 b Fs(L)522 537 y Fr(p)522 597 y(\034)598 573 y Fp(=)33 b Fs(L)773 526 y Fr(p)773 597 y Fl(0)813 573 y Fq(,)28 b(for)e Fs(\034)44 b Fo(2)33 b Fp([0)p Fs(;)17 b(g)1365 537 y Fr(\026)1410 573 y Fp(])p Fq(,)28 b(and)f Fs(L)1733 537 y Fr(p)1733 597 y(\034)1809 573 y Fp(=)32 b Fs(L)1983 537 y Fr(p)1983 597 y Ff(1)2058 573 y Fq(,)c(for)f Fs(\034)44 b Fo(2)33 b Fp([)p Fs(g)2518 537 y Ff(\000)p Fl(1)p Fr(=K)2711 573 y Fs(;)17 b Fo(1)p Fp(\))p Fq(,)27 b(wher)l(e)h Fs(L)3278 526 y Fr(p)3278 597 y Fl(0)3346 573 y Fq(and)257 693 y Fs(L)323 657 y Fr(p)323 718 y Ff(1)423 693 y Fq(ar)l(e)d(de\002ned)g(in)f(Eqs.)h (\(I.38\)\226\(I.40\).)257 863 y(Pr)l(oof:)50 b Fv(W)-8 b(e)35 b(\002rst)g(write)g Fs(W)59 b Fp(=)46 b Fs(a)1483 827 y Ff(\003)1523 863 y Fp(\()p Fs(G)p Fp(\))29 b(+)h Fs(a)p Fp(\()p Fs(G)p Fp(\))35 b Fv(and)f(use)h(the)g(canonical)f (commutation)257 984 y(relations)24 b(to)h(obtain)f(that)296 1195 y Fp([)p Fs(W)m(;)17 b(A)529 1210 y Fr(s)567 1195 y Fp(])83 b(=)836 1114 y Fi(\002)877 1195 y Fs(a)928 1154 y Ff(\003)968 1195 y Fp(\()p Fs(G)p Fp(\))22 b(+)g Fs(a)p Fp(\()p Fs(G)p Fp(\))j Fs(;)41 b(a)1589 1154 y Ff(\003)1629 1195 y Fp(\()p Fs(e)1712 1154 y Ff(\000)p Fr(is!)1874 1195 y Fs(')1938 1210 y Fl(1)1977 1195 y Fp(\))2015 1154 y Fr(n)2058 1163 y Fd(1)2097 1195 y Fs(a)2148 1154 y Ff(\003)2187 1195 y Fp(\()p Fs(e)2270 1154 y Ff(\000)p Fr(is!)2433 1195 y Fs(')2497 1210 y Fl(2)2536 1195 y Fp(\))2574 1154 y Fr(n)2617 1163 y Fd(2)2672 1195 y Fo(\001)17 b(\001)g(\001)e Fs(a)2856 1154 y Ff(\003)2895 1195 y Fp(\()p Fs(e)2978 1154 y Ff(\000)p Fr(is!)3141 1195 y Fs(')3205 1210 y Fr(m)3271 1195 y Fp(\))3309 1154 y Fr(n)3352 1162 y Fc(m)3415 1114 y Fi(\003)677 1340 y Fp(=)836 1259 y Fi(\002)877 1340 y Fs(a)p Fp(\()p Fs(G)p Fp(\))25 b Fs(;)42 b(a)1226 1299 y Ff(\003)1265 1340 y Fp(\()p Fs(e)1348 1299 y Ff(\000)p Fr(is!)1510 1340 y Fs(')1574 1355 y Fl(1)1614 1340 y Fp(\))1652 1299 y Fr(n)1695 1308 y Fd(1)1733 1340 y Fs(a)1784 1299 y Ff(\003)1824 1340 y Fp(\()p Fs(e)1907 1299 y Ff(\000)p Fr(is!)2069 1340 y Fs(')2133 1355 y Fl(2)2172 1340 y Fp(\))2210 1299 y Fr(n)2253 1308 y Fd(2)2308 1340 y Fo(\001)17 b(\001)g(\001)e Fs(a)2492 1299 y Ff(\003)2532 1340 y Fp(\()p Fs(e)2615 1299 y Ff(\000)p Fr(is!)2777 1340 y Fs(')2841 1355 y Fr(m)2907 1340 y Fp(\))2945 1299 y Fr(n)2988 1307 y Fc(m)3051 1259 y Fi(\003)677 1571 y Fp(=)876 1447 y Fr(m)836 1477 y Fi(X)846 1687 y Fr(j)t Fl(=1)996 1571 y Fs(n)1054 1586 y Fr(j)1113 1431 y Fi(\032)1188 1491 y(\012)1235 1571 y Fs(G)1312 1487 y Fi(\014)1312 1546 y(\014)1345 1571 y Fs(e)1390 1530 y Ff(\000)p Fr(is!)1552 1571 y Fs(')1616 1586 y Fr(j)1652 1491 y Fi(\013)1722 1571 y Fo(\012)1854 1447 y Fr(m)1821 1477 y Fi(Y)1828 1687 y Fr(i)p Fl(=1)1965 1571 y Fs(a)2016 1530 y Ff(\003)2056 1571 y Fp(\()p Fs(e)2139 1530 y Ff(\000)p Fr(is!)2301 1571 y Fs(')2365 1586 y Fr(i)2393 1571 y Fp(\))2431 1530 y Fr(n)2474 1540 y Fc(i)2500 1530 y Ff(\000)p Fr(\016)2586 1540 y Fc(i;j)2664 1431 y Fi(\033)2761 1571 y Fs(;)450 b Fv(\(II.45\))257 1909 y(on)30 b Fp(dom[)p Fs(H)688 1858 y Fl(1+)p Fr(N)q(=)p Fl(2)680 1937 y Fr(f)909 1909 y Fp(])p Fv(,)i(where)e Fo(h)p Fs(G)p Fo(j)p Fs(e)1455 1873 y Ff(\000)p Fr(is!)1617 1909 y Fs(')1681 1924 y Fr(j)1717 1909 y Fo(i)38 b(2)g(B)s Fp(\()p Fo(H)2088 1924 y Fr(el)2147 1909 y Fp(\))30 b Fv(is)g(a)g(bounded)g(operator)g (acting)g(on)257 2029 y(the)25 b(electron)g(v)n(ariables)f(gi)n(v)o(en) f(by)942 2205 y Fi(\012)989 2286 y Fs(G)1066 2201 y Fi(\014)1066 2261 y(\014)1100 2286 y Fs(e)1145 2244 y Ff(\000)p Fr(is!)1307 2286 y Fs(')1371 2301 y Fr(j)1407 2205 y Fi(\013)1507 2286 y Fp(:=)1662 2150 y Fi(Z)1779 2286 y Fs(d)1830 2244 y Fl(3)1869 2286 y Fs(k)i(e)1990 2244 y Ff(\000)p Fr(is!)r Fl(\()p Fr(k)r Fl(\))2268 2286 y Fs(')2332 2301 y Fr(j)2368 2286 y Fp(\()p Fs(k)s Fp(\))17 b Fs(G)2592 2244 y Ff(\003)2631 2286 y Fp(\()p Fs(k)s Fp(\))22 b Fs(:)428 b Fv(\(II.46\))257 2538 y(Moreo)o(v)o(er)l(,)24 b(for)h(an)o(y)f Fs(s)k Fo(\025)g Fp(0)p Fv(,)533 2822 y Fs(e)578 2781 y Fr(is)p Fl(\()p Fr(H)720 2790 y Fd(0)755 2781 y Ff(\000)p Fr(E)862 2790 y Fd(0)897 2781 y Fl(\))928 2741 y Fi(\012)975 2822 y Fs(G)1052 2737 y Fi(\014)1052 2797 y(\014)1085 2822 y Fs(e)1130 2781 y Ff(\000)p Fr(is!)1293 2822 y Fs(')1357 2837 y Fr(j)1393 2741 y Fi(\013)1462 2822 y Fo(\012)1594 2697 y Fr(m)1562 2727 y Fi(Y)1568 2937 y Fr(i)p Fl(=1)1706 2822 y Fs(a)1757 2781 y Ff(\003)1796 2822 y Fp(\()p Fs(e)1879 2781 y Ff(\000)p Fr(is!)2041 2822 y Fs(')2105 2837 y Fr(i)2133 2822 y Fp(\))2171 2781 y Fr(n)2214 2791 y Fc(i)2241 2781 y Ff(\000)p Fr(\016)2327 2791 y Fc(i;j)2421 2822 y Fp(\(0)22 b Fo(\010)h Fp(\012\))616 3143 y(=)83 b Fs(e)820 3102 y Fr(is)p Fl(\()p Fr(H)962 3114 y Fc(el)1016 3102 y Ff(\000)p Fr(E)1123 3111 y Fd(0)1157 3102 y Fl(\))1189 3063 y Fi(\012)1236 3143 y Fs(G)1313 3059 y Fi(\014)1313 3118 y(\014)1346 3143 y Fs(e)1391 3102 y Ff(\000)p Fr(is!)1553 3143 y Fs(')1617 3158 y Fr(j)1654 3063 y Fi(\013)1723 3143 y Fo(\012)23 b Fs(e)1868 3102 y Fr(isH)1983 3114 y Fc(f)2075 3019 y Fr(m)2043 3049 y Fi(Y)2049 3259 y Fr(i)p Fl(=1)2187 3143 y Fs(a)2238 3102 y Ff(\003)2277 3143 y Fp(\()p Fs(e)2360 3102 y Ff(\000)p Fr(is!)2523 3143 y Fs(')2587 3158 y Fr(i)2615 3143 y Fp(\))2653 3102 y Fr(n)2696 3112 y Fc(i)2722 3102 y Ff(\000)p Fr(\016)2808 3112 y Fc(i;j)2902 3143 y Fp(\(0)f Fo(\010)h Fp(\012\))616 3465 y(=)83 b Fs(e)820 3424 y Fr(is)p Fl(\()p Ff(\000)p Fl(\001)p Ff(\000)p Fr(E)1125 3433 y Fd(0)1160 3424 y Fl(\))1191 3384 y Fi(\012)1238 3465 y Fs(P)1301 3480 y Fr(c)1352 3465 y Fs(G)17 b(P)1509 3480 y Fr(d)1549 3380 y Fi(\014)1549 3440 y(\014)1583 3465 y Fs(e)1628 3424 y Ff(\000)p Fr(is!)1790 3465 y Fs(')1854 3480 y Fr(j)1890 3384 y Fi(\013)1959 3465 y Fo(\012)2091 3340 y Fr(m)2059 3370 y Fi(Y)2065 3580 y Fr(i)p Fl(=1)2203 3465 y Fs(a)2254 3424 y Ff(\003)2293 3465 y Fp(\()p Fs(')2395 3480 y Fr(i)2423 3465 y Fp(\))2461 3424 y Fr(n)2504 3434 y Fc(i)2531 3424 y Ff(\000)p Fr(\016)2617 3434 y Fc(i;j)2711 3465 y Fp(\(0)22 b Fo(\010)h Fp(\012\))f Fs(:)616 3787 y Fp(=)775 3706 y Fi(\000)821 3787 y Fs( )884 3802 y Fr(j;s)991 3787 y Fo(\010)h Fp(0)1140 3706 y Fi(\001)1208 3787 y Fo(\012)1340 3662 y Fr(m)1307 3692 y Fi(Y)1314 3902 y Fr(i)p Fl(=1)1451 3787 y Fs(a)1502 3745 y Ff(\003)1542 3787 y Fp(\()p Fs(')1644 3802 y Fr(i)1672 3787 y Fp(\))1710 3745 y Fr(n)1753 3755 y Fc(i)1779 3745 y Ff(\000)p Fr(\016)1865 3755 y Fc(i;j)1943 3787 y Fp(\012)f Fs(;)1176 b Fv(\(II.47\))257 4090 y(where)21 b(the)e(F)o(ourier)h(transform)f(of)h Fs( )1547 4105 y Fr(j)1611 4090 y Fo(2)28 b Fs(L)1771 4054 y Fl(2)1811 4090 y Fp(\()p Fh(R)1915 4054 y Fl(3)1961 4090 y Fp(\))19 b Fv(with)g(respect)h(to)g(the)f(particle)h(coordinate) 257 4210 y Fs(x)26 b Fv(is)e(gi)n(v)o(en)f(by)949 4440 y Fp(^)929 4466 y Fs( )992 4481 y Fr(j;s)1077 4466 y Fp(\()p Fs(p)p Fp(\))53 b(=)1383 4331 y Fi(Z)1499 4466 y Fs(d)1550 4425 y Fl(3)1589 4466 y Fs(k)26 b(e)1711 4425 y Fr(is)p Fl(\()p Fr(p)1831 4402 y Fd(2)1865 4425 y Ff(\000)p Fr(E)1972 4434 y Fd(0)2006 4425 y Ff(\000)p Fr(!)r Fl(\()p Fr(k)r Fl(\)\))2263 4466 y Fp(^)-57 b Fs(\032)p Fp(\()p Fs(p;)17 b(k)s Fp(\))g Fs(')2609 4481 y Fr(j)2644 4466 y Fp(\()p Fs(k)s Fp(\))23 b Fs(:)414 b Fv(\(II.48\))257 4726 y(Here,)32 b(we)e(used)f(the)h(f)o(act)g(that)f Fs(P)1437 4741 y Fr(d)1494 4726 y Fs(G)p Fp(\()p Fs(k)s Fp(\))1701 4689 y Ff(\003)1740 4726 y Fs(P)1803 4741 y Fr(d)1880 4726 y Fp(=)37 b(0)30 b Fv(and)f(hence)h Fs(G)p Fp(\()p Fs(k)s Fp(\))2714 4689 y Ff(\003)2754 4726 y Fs(P)2817 4741 y Fr(d)2894 4726 y Fp(=)36 b Fs(P)3069 4741 y Fr(c)3121 4726 y Fs(G)p Fp(\()p Fs(k)s Fp(\))3328 4689 y Ff(\003)3367 4726 y Fs(P)3430 4741 y Fr(d)3470 4726 y Fv(,)257 4846 y(for)f(all)e Fs(k)48 b Fo(2)d Fh(R)817 4810 y Fl(3)862 4846 y Fv(.)58 b(Moreo)o(v)o(er)l(,)35 b Fs(P)1455 4861 y Fr(c)1506 4846 y Fs(G)p Fp(\()p Fs(k)s Fp(\))1713 4810 y Ff(\003)1753 4846 y Fs(P)1816 4861 y Fr(d)1856 4846 y Fp(\()1914 4820 y(~)1894 4846 y Fs( )e Fo(\010)c Fs(z)t Fp(\))45 b(=)g Fs(z)t(p)2447 4861 y Ff(#)2487 4846 y Fp(\()p Fs(k)s Fp(\))2617 4810 y Ff(\003)2685 4846 y Fo(\010)29 b Fp(0)p Fv(,)36 b(and)e(this)f(yields)257 4966 y(\(II.48\).)e(Eqs.)25 b(\(II.45\)\226\(II.48\))g(imply)e(that)439 5250 y Fp(\010)509 5209 y Fl(\012)509 5275 y Ff(1)584 5250 y Fp(\()p Fs(A;)17 b(\034)11 b Fp(\))53 b(=)1052 5125 y Fr(m)1011 5155 y Fi(X)1022 5365 y Fr(j)t Fl(=1)1171 5250 y Fs(n)1229 5265 y Fr(j)1283 5109 y Fi(\032)1374 5114 y(Z)1474 5141 y Ff(1)1429 5340 y(\000)p Fr(\034)1565 5250 y Fs(ds)1679 5169 y Fi(\000)1724 5250 y Fs( )1787 5265 y Fr(j;s)1894 5250 y Fo(\010)23 b Fp(0)2043 5169 y Fi(\001)2088 5109 y(\033)2185 5250 y Fo(\012)2317 5125 y Fr(m)2285 5155 y Fi(Y)2291 5365 y Fr(i)p Fl(=1)2429 5250 y Fs(a)2480 5209 y Ff(\003)2519 5250 y Fp(\()p Fs(')2621 5265 y Fr(i)2650 5250 y Fp(\))2688 5209 y Fr(n)2731 5219 y Fc(i)2757 5209 y Ff(\000)p Fr(\016)2843 5219 y Fc(i;j)2921 5250 y Fp(\012)f Fs(;)198 b Fv(\(II.49\))p eop %%Page: 23 24 23 23 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(23)257 573 y(P)o(assing)24 b(to)h(the)f(momentum)f(representation)h (for)h(the)g(particle)g(v)n(ariable)f(and)h(using)f(that)374 725 y Fi(\034)498 741 y Fr(m)466 771 y Fi(Y)472 981 y Fr(i)p Fl(=1)610 865 y Fs(a)661 824 y Ff(\003)700 865 y Fp(\()p Fs(')802 880 y Fr(i)830 865 y Fp(\))868 824 y Fr(n)911 834 y Fc(i)937 824 y Ff(\000)p Fr(\016)1023 834 y Fc(i;j)1118 865 y Fp(\012)1216 721 y Fi(\014)1216 781 y(\014)1216 841 y(\014)1216 900 y(\014)1354 741 y Fr(m)1321 771 y Fi(Y)1328 981 y Fr(i)p Fl(=1)1465 865 y Fs(a)1516 824 y Ff(\003)1556 865 y Fp(\()p Fs(')1658 880 y Fr(i)1686 865 y Fp(\))1724 824 y Fr(n)1767 834 y Fc(i)1793 824 y Ff(\000)p Fr(\016)1879 836 y Fc(i;`)1971 865 y Fp(\012)2041 725 y Fi(\035)2169 865 y Fp(=)52 b Fs(\016)2340 880 y Fr(j;`)2443 865 y Fo(\001)2526 741 y Fr(m)2493 771 y Fi(Y)2500 981 y Fr(i)p Fl(=1)2621 865 y Fp(\()p Fs(n)2717 880 y Fr(i)2767 865 y Fo(\000)23 b Fs(\016)2910 880 y Fr(i;j)2990 865 y Fp(\)!)f Fs(;)134 b Fv(\(II.50\))257 1167 y(we)25 b(hence)h(obtain)285 1354 y Fi(\015)285 1414 y(\015)341 1439 y Fs(T)398 1454 y Ff(T)475 1439 y Fp(\010)545 1398 y Fl(\012)545 1464 y Ff(1)620 1439 y Fp(\()p Fs(A;)17 b(\034)11 b Fp(\))866 1354 y Fi(\015)866 1414 y(\015)922 1381 y Fl(2)1044 1439 y Fp(=)1203 1295 y Fi(\015)1203 1354 y(\015)1203 1414 y(\015)1203 1474 y(\015)1316 1314 y Fr(m)1275 1344 y Fi(X)1286 1554 y Fr(j)t Fl(=1)1435 1439 y Fs(n)1493 1454 y Fr(j)1547 1328 y Fi(n)1630 1303 y(Z)1729 1330 y Ff(1)1685 1529 y(\000)p Fr(\034)1821 1439 y Fs(ds)1935 1358 y Fi(\000)1980 1439 y Fs(T)2037 1454 y Ff(T)2114 1439 y Fs( )2177 1454 y Fr(j;s)2284 1439 y Fo(\010)23 b Fp(0)2433 1358 y Fi(\001)2478 1328 y(o)2567 1439 y Fo(\012)2699 1314 y Fr(m)2666 1344 y Fi(Y)2673 1554 y Fr(i)p Fl(=1)2810 1439 y Fs(a)2861 1398 y Ff(\003)2901 1439 y Fp(\()p Fs(')3003 1454 y Fr(i)3031 1439 y Fp(\))3069 1398 y Fr(n)3112 1408 y Fc(i)3138 1398 y Ff(\000)p Fr(\016)3224 1408 y Fc(i;j)3302 1439 y Fp(\012)3372 1295 y Fi(\015)3372 1354 y(\015)3372 1414 y(\015)3372 1474 y(\015)3428 1321 y Fl(2)1044 1773 y Fp(=)83 b Fs(n)1261 1788 y Fl(1)1301 1773 y Fp(!)p Fs(n)1386 1788 y Fl(2)1425 1773 y Fp(!)17 b Fo(\001)g(\001)g(\001)e Fs(n)1660 1788 y Fr(m)1727 1773 y Fp(!)1828 1649 y Fr(m)1787 1679 y Fi(X)1798 1889 y Fr(j)t Fl(=1)1948 1773 y Fs(n)2006 1788 y Fr(j)2059 1638 y Fi(Z)2114 1863 y Ff(T)2191 1773 y Fs(d)2242 1732 y Fl(3)2281 1773 y Fs(p)2330 1629 y Fi(\014)2330 1689 y(\014)2330 1749 y(\014)2330 1808 y(\014)2380 1638 y(Z)2480 1664 y Ff(1)2435 1863 y(\000)p Fr(\034)2571 1773 y Fs(ds)2704 1747 y Fp(^)2685 1773 y Fs( )2748 1788 y Fr(j;s)2832 1773 y Fp(\()p Fs(p)p Fp(\))2957 1629 y Fi(\014)2957 1689 y(\014)2957 1749 y(\014)2957 1808 y(\014)2991 1655 y Fl(2)3052 1773 y Fs(:)159 b Fv(\(II.51\))257 2115 y(It)25 b(remains)f(to)g(e)n(v)n(aluate)g(the)g(inte)o(gral)1610 2035 y Fi(R)1677 2061 y Ff(1)1657 2150 y(\000)p Fr(\034)1772 2115 y Fs(ds)1905 2089 y Fp(^)1886 2115 y Fs( )1949 2130 y Fr(j;s)2034 2115 y Fp(\()p Fs(p)p Fp(\))g Fv(in)g(\(II.51\),)h(which) f(is)g(an)h(e)o(x)o(ercise)f(in)257 2235 y(distrib)n(ution)h(theory)-6 b(.)36 b(T)-8 b(o)26 b(this)g(end,)h(we)h(note)e(that)g(Hypothesis)g(3) g(implies)g(that)35 b Fp(^)-58 b Fs(\032)p Fp(\()p Fs(p;)34 b Fo(\001)17 b Fp(\))p Fs(')3460 2250 y Fr(j)257 2356 y Fv(is)32 b(tw)o(o)h(times)e(partially)h(dif)n(ferentiable)g(with)f (respect)i(to)f Fs(k)s Fv(,)j(and)e(that)f(its)f(deri)n(v)n(ati)n(v)o (es)f(of)257 2476 y(order)f Fo(\024)34 b Fp(2)28 b Fv(ha)n(v)o(e)g (bounded)g(support)f(a)o(w)o(ay)h(from)g Fs(k)37 b Fp(=)d(0)p Fv(.)41 b(Since)28 b(the)g(phase)h(in)3147 2450 y Fp(^)3127 2476 y Fs( )3190 2491 y Fr(j;s)3276 2476 y Fp(\()p Fs(p)p Fp(\))f Fv(is)257 2597 y(non-stationary)-6 b(,)31 b(a)o(w)o(ay)g(from)g Fs(k)42 b Fp(=)e(0)p Fv(,)32 b(tw)o(o)f(times)f(inte)o(gration)f(by)i (parts)g(back)g(and)h(forth)257 2717 y(yields)471 2849 y Fi(Z)571 2875 y Ff(1)526 3074 y Fl(1)662 2984 y Fs(ds)795 2958 y Fp(^)776 2984 y Fs( )839 2999 y Fr(j;s)923 2984 y Fp(\()p Fs(p)p Fp(\))80 b(=)1285 2849 y Fi(Z)1384 2875 y Ff(1)1340 3074 y Fl(1)1476 2984 y Fs(ds)1605 2849 y Fi(Z)1722 2984 y Fs(d)1773 2943 y Fl(3)1812 2984 y Fs(k)25 b(e)1933 2943 y Fr(is)p Fl(\()p Fr(p)2053 2920 y Fd(2)2087 2943 y Ff(\000)p Fr(E)2194 2952 y Fd(0)2229 2943 y Ff(\000)p Fr(!)r Fl(\()p Fr(k)r Fl(\)\))2486 2984 y Fp(^)-58 b Fs(\032)p Fp(\()p Fs(p;)17 b(k)s Fp(\))g Fs(')2831 2999 y Fr(j)2867 2984 y Fp(\()p Fs(k)s Fp(\))554 3245 y(=)713 3110 y Fi(Z)812 3136 y Ff(1)768 3335 y Fl(1)914 3178 y Fs(ds)p 914 3223 97 4 v 920 3314 a(s)966 3285 y Fl(2)1054 3110 y Fi(Z)1170 3245 y Fs(d)1221 3204 y Fl(3)1260 3245 y Fs(k)25 b(e)1381 3204 y Fr(is)p Fl(\()p Fr(p)1501 3181 y Fd(2)1536 3204 y Ff(\000)p Fr(E)1643 3213 y Fd(0)1677 3204 y Ff(\000)p Fr(!)r Fl(\()p Fr(k)r Fl(\)\))1925 3135 y Fi(\020)1985 3245 y Fs(i)p Fo(r)d(\001)2210 3178 y Fs(k)p 2183 3223 110 4 v 2183 3314 a Fo(j)p Fs(k)s Fo(j)2302 3135 y Fi(\021)2362 3157 y Fl(2)2401 3135 y Fi(h)2457 3245 y Fp(^)-58 b Fs(\032)p Fp(\()p Fs(p;)17 b(k)s Fp(\))g Fs(')2802 3260 y Fr(j)2838 3245 y Fp(\()p Fs(k)s Fp(\))2968 3135 y Fi(i)554 3510 y Fp(=)84 b(lim)713 3570 y Fr(")p Ff(!)p Fl(0)868 3374 y Fi(Z)968 3401 y Ff(1)923 3600 y Fl(1)1069 3442 y Fs(ds)p 1069 3487 97 4 v 1075 3578 a(s)1121 3549 y Fl(2)1209 3374 y Fi(Z)1325 3510 y Fs(d)1376 3469 y Fl(3)1415 3510 y Fs(k)25 b(e)1536 3469 y Fr(is)p Fl(\()p Fr(p)1656 3445 y Fd(2)1691 3469 y Ff(\000)p Fr(E)1798 3478 y Fd(0)1832 3469 y Ff(\000)p Fr(!)r Fl(\()p Fr(k)r Fl(\)+)p Fr(i")p Fl(\))2192 3399 y Fi(\020)2251 3510 y Fs(i)p Fo(r)e(\001)2477 3442 y Fs(k)p 2449 3487 110 4 v 2449 3578 a Fo(j)p Fs(k)s Fo(j)2569 3399 y Fi(\021)2628 3422 y Fl(2)2668 3399 y Fi(h)2724 3510 y Fp(^)-58 b Fs(\032)p Fp(\()p Fs(p;)17 b(k)s Fp(\))g Fs(')3069 3525 y Fr(j)3105 3510 y Fp(\()p Fs(k)s Fp(\))3235 3399 y Fi(i)554 3774 y Fp(=)84 b(lim)713 3834 y Fr(")p Ff(!)p Fl(0)868 3639 y Fi(Z)984 3774 y Fs(d)1035 3733 y Fl(3)1074 3774 y Fs(k)1162 3639 y Fi(Z)1261 3665 y Ff(1)1217 3864 y Fl(1)1353 3774 y Fs(ds)17 b(e)1512 3733 y Fr(is)p Fl(\()p Fr(p)1632 3710 y Fd(2)1665 3733 y Ff(\000)p Fr(E)1772 3742 y Fd(0)1806 3733 y Ff(\000)p Fr(!)r Fl(\()p Fr(k)r Fl(\)+)p Fr(i")p Fl(\))2175 3774 y Fp(^)-58 b Fs(\032)p Fp(\()p Fs(p;)17 b(k)s Fp(\))g Fs(')2520 3789 y Fr(j)2556 3774 y Fp(\()p Fs(k)s Fp(\))554 4056 y(=)84 b(lim)713 4116 y Fr(")p Ff(!)p Fl(0)868 3921 y Fi(Z)984 4056 y Fs(d)1035 4015 y Fl(3)1074 4056 y Fs(k)1155 3989 y Fo(\000)p Fs(i)17 b(e)1327 3953 y Fr(i)p Fl(\()p Fr(p)1414 3929 y Fd(2)1449 3953 y Ff(\000)p Fr(E)1556 3962 y Fd(0)1590 3953 y Ff(\000)p Fr(!)r Fl(\()p Fr(k)r Fl(\)\))1842 3989 y Fp(^)-58 b Fs(\032)p Fp(\()p Fs(p;)17 b(k)s Fp(\))p 1155 4034 952 4 v 1212 4125 a Fs(p)1261 4096 y Fl(2)1323 4125 y Fo(\000)22 b Fs(E)1494 4140 y Fl(0)1556 4125 y Fo(\000)h Fs(!)t Fp(\()p Fs(k)s Fp(\))e(+)h Fs(i")2133 4056 y(')2197 4071 y Fr(j)2233 4056 y Fp(\()p Fs(k)s Fp(\))g Fs(:)826 b Fv(\(II.52\))257 4326 y(Thus)25 b(in)f(the)h(case)g Fs(\034)39 b Fp(=)28 b(0)p Fv(:)417 4451 y Fi(Z)517 4477 y Ff(1)472 4677 y Fl(0)608 4587 y Fs(ds)741 4560 y Fp(^)722 4587 y Fs( )785 4602 y Fr(j;s)869 4587 y Fp(\()p Fs(p)p Fp(\))83 b(=)1236 4451 y Fi(Z)1336 4477 y Fl(1)1292 4677 y(0)1392 4587 y Fs(ds)1525 4560 y Fp(^)1506 4587 y Fs( )1569 4602 y Fr(j;s)1653 4587 y Fp(\()p Fs(p)p Fp(\))50 b(+)1954 4451 y Fi(Z)2053 4477 y Ff(1)2009 4677 y Fl(1)2145 4587 y Fs(ds)2278 4560 y Fp(^)2259 4587 y Fs( )2322 4602 y Fr(j;s)2406 4587 y Fp(\()p Fs(p)p Fp(\))707 b Fv(\(II.53\))1077 4852 y Fp(=)85 b(lim)1236 4912 y Fr(")p Ff(!)p Fl(0)1391 4716 y Fi(Z)1507 4852 y Fs(d)1558 4811 y Fl(3)1598 4852 y Fs(k)1897 4785 y Fo(\000)p Fs(i)26 b Fp(^)-58 b Fs(\032)p Fp(\()p Fs(p;)17 b(k)s Fp(\))p 1678 4829 837 4 v 1678 4920 a Fs(p)1727 4892 y Fl(2)1789 4920 y Fo(\000)22 b Fs(E)1960 4935 y Fl(0)2022 4920 y Fo(\000)h Fs(!)t Fp(\()p Fs(k)s Fp(\))e(+)h Fs(i")2547 4852 y(')2611 4867 y Fr(j)2648 4852 y Fp(\()p Fs(k)s Fp(\))27 b(=)2909 4771 y Fi(\012)2956 4852 y Fs(L)3022 4805 y Fr(p)3022 4876 y Fl(0)3079 4852 y Fs(;)33 b(')3203 4867 y Fr(j)3239 4771 y Fi(\013)3308 4852 y Fs(:)p eop %%Page: 24 25 24 24 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(24)257 573 y(F)o(or)25 b(the)g(case)g Fs(\034)39 b Fp(=)28 b Fo(1)p Fv(,)c(we)h(get)274 853 y Fp(lim)258 912 y Fr(t)p Ff(!1)441 717 y Fi(Z)541 743 y Fr(t)497 942 y Ff(\000)p Fr(t)598 853 y Fs(ds)731 826 y Fp(^)712 853 y Fs( )775 868 y Fr(j;s)859 853 y Fp(\()p Fs(p)p Fp(\))83 b(=)98 b(lim)1226 912 y Fr(t)p Ff(!1)1409 717 y Fi(Z)1509 743 y Fr(t)1464 942 y Ff(\000)p Fr(t)1566 853 y Fs(ds)1695 717 y Fi(Z)1812 853 y Fs(d)1863 811 y Fl(3)1902 853 y Fs(k)25 b(e)2023 811 y Fr(is)p Fl(\()p Fr(p)2143 788 y Fd(2)2177 811 y Ff(\000)p Fr(E)2284 820 y Fd(0)2319 811 y Ff(\000)p Fr(!)r Fl(\()p Fr(k)r Fl(\)\))2576 853 y Fp(^)-58 b Fs(\032)p Fp(\()p Fs(p;)17 b(k)s Fp(\))g Fs(')2921 868 y Fr(j)2957 853 y Fp(\()p Fs(k)s Fp(\))27 b(=)48 b Fv(\(II.54\))1067 1142 y Fp(=)98 b(lim)1226 1201 y Fr(t)p Ff(!1)1409 1006 y Fi(Z)1525 1142 y Fs(d)1576 1101 y Fl(3)1615 1142 y Fs(k)1702 1074 y(e)1747 1038 y Fr(it)p Fl(\()p Fr(p)1859 1015 y Fd(2)1894 1038 y Ff(\000)p Fr(E)2001 1047 y Fd(0)2035 1038 y Ff(\000)p Fr(!)r Fl(\()p Fr(k)r Fl(\)\))2283 1074 y Fo(\000)23 b Fs(e)2428 1038 y Fr(it)p Fl(\()p Fr(p)2540 1015 y Fd(2)2575 1038 y Ff(\000)p Fr(E)2682 1047 y Fd(0)2717 1038 y Ff(\000)p Fr(!)r Fl(\()p Fr(k)r Fl(\)\))p 1702 1119 1242 4 v 1949 1210 a Fs(i)p Fp(\()p Fs(p)2069 1181 y Fl(2)2130 1210 y Fo(\000)g Fs(E)2302 1225 y Fl(0)2364 1210 y Fo(\000)f Fs(!)t Fp(\()p Fs(k)s Fp(\)\))2984 1142 y(^)-58 b Fs(\032)p Fp(\()p Fs(p;)17 b(k)s Fp(\))g Fs(')3329 1157 y Fr(j)3365 1142 y Fp(\()p Fs(k)s Fp(\))1067 1414 y(=)98 b(lim)1226 1474 y Fr(t)p 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1677 y Fo(\000)23 b Fs(E)3171 1692 y Fl(0)3210 1677 y Fs(;)17 b Fp(\012\))p Fs(:)p Fa(\003)257 1943 y Fk(Pr)n(oof)26 b(of)g(Theor)n(em)i(I.1:)k Fv(The)26 b(proof)g(of)g(Theorem)g(I.1)f(consists)g(of)h(the)g(follo)n (wing)d(chain)257 2063 y(of)i(estimates,)583 2283 y Fs(Q)660 2242 y Fl(inf)660 2308 y Ff(T)750 2283 y Fp(\()p Fs(A)p Fp(\()p Fs(\034)11 b Fp(\)\010)1060 2298 y Fl(gs)1128 2283 y Fp(\))39 b(+)f Fo(O)s Fp(\()p Fs(g)1490 2242 y Fl(2+)p Fr(\026)1626 2283 y Fp(\))83 b(=)g Fs(Q)1983 2236 y Fl(sup)1983 2310 y Ff(T)2094 2283 y Fp(\()p Fs(A)p Fp(\()p Fs(\034)11 b Fp(\)\010)2404 2298 y Fl(gs)2472 2283 y Fp(\))39 b(+)f Fo(O)s Fp(\()p Fs(g)2834 2242 y Fl(2+)p Fr(\026)2970 2283 y Fp(\))1747 2446 y(=)83 b Fs(g)1957 2405 y Fl(2)1996 2361 y Fi(\015)1996 2421 y(\015)2051 2446 y Fs(T)2108 2461 y Ff(T)2185 2446 y Fp(\010)2255 2461 y Ff(1)2330 2446 y Fp(\()p Fs(A;)17 b(\034)11 b Fp(\))2576 2361 y Fi(\015)2576 2421 y(\015)2632 2387 y Fl(2)3238 2446 y Fv(\(II.55\))1747 2615 y Fp(=)83 b Fs(g)1957 2574 y Fl(2)1996 2530 y Fi(\015)1996 2590 y(\015)2051 2615 y Fs(T)2108 2630 y Ff(T)2185 2615 y Fp(\010)2255 2574 y Fl(\012)2255 2640 y Ff(1)2330 2615 y Fp(\()p Fs(A;)17 b(\034)11 b Fp(\))2576 2530 y Fi(\015)2576 2590 y(\015)2632 2557 y Fl(2)2710 2615 y Fp(+)39 b Fo(O)s Fp(\()p Fs(g)2996 2574 y Fl(2+)p Fr(\026)3132 2615 y Fp(\))68 b Fv(\(II.56\))1747 2760 y Fp(=)83 b Fs(Q)1983 2719 y Fr(\034)1983 2785 y Ff(T)2044 2760 y Fp(\()p Fs(')p 2082 2796 64 4 v Fp(\))38 b(+)h Fo(O)s Fp(\()p Fs(g)2508 2719 y Fl(2+)p Fr(\026)2644 2760 y Fp(\))22 b Fs(;)507 b Fv(\(II.57\))257 2980 y(where)29 b(Lemma)g(II.2,)g(II.3,)h(and)f(II.4)f(justify)g(Eq.)g(\(II.55\),)i (\(II.56\),)g(and)e(\(II.57\),)i(respecti-)257 3101 y(v)o(ely)-6 b(.)2971 b Fa(\003)p eop %%Page: 25 26 25 25 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(25)257 573 y Fu(III)143 b(The)35 b(Limit)g(of)g(Monochr)m(omatic)f (Light)257 796 y Fv(In)25 b(this)f(section)g(we)h(gi)n(v)o(e)f(a)h (proof)g(of)g(Theorem)f(I.2.)31 b(W)-8 b(e)25 b(remark,)g(that)f(the)h (functions)958 1012 y Fs(')1022 1027 y Fr(j;\016)1108 1012 y Fp(\()p Fs(r)m(;)17 b Fp(\012)1301 1027 y Fr(k)1344 1012 y Fp(\))53 b(:=)f Fs(!)1655 971 y Ff(\000)p Fl(1)1651 1038 y Fr(j)1765 1012 y Fs(\016)1812 971 y Ff(\000)p Fl(1)p Fr(=)p Fl(2)1994 1012 y Fs(\037)2055 1027 y Fr(j)2091 902 y Fi(\020)2161 945 y Fs(r)25 b Fo(\000)d Fs(!)2390 960 y Fr(j)p 2161 989 266 4 v 2270 1081 a Fs(\016)2437 902 y Fi(\021)2513 1012 y Fs(\024)2569 1027 y Fr(j)2605 1012 y Fp(\(\012)2713 1027 y Fr(k)2756 1012 y Fp(\))461 b Fv(\(III.1\))257 1217 y(de\002ned)28 b(in)e(\(I.45\))h(form)f(an)h (orthonormal)f(system)f(under)i(conditions)e(\(I.45\)\226\(I.47\))i (only)257 1338 y(in)f(the)h(limit)e Fs(\016)34 b Fo(!)c Fp(0)p Fv(.)36 b(For)27 b(suf)n(\002ciently)e(small)h Fs(\016)34 b(>)d Fp(0)p Fv(,)c(these)f(v)o(ectors)g(are)h(merely)f (orthog-)257 1458 y(onal.)31 b(Hence)25 b(in)f(the)h(pre)n(vious)f (calculations)f(we)j(ha)n(v)o(e)e(to)g(replace)i(\(II.50\))f(by)850 1571 y Fi(\034)974 1587 y Fr(m)941 1617 y Fi(Y)948 1827 y Fr(i)p Fl(=1)1085 1712 y Fs(a)1136 1671 y Ff(\003)1176 1712 y Fp(\()p Fs(')1278 1727 y Fr(i;\016)1359 1712 y Fp(\))1397 1671 y Fr(n)1440 1681 y Fc(i)1466 1671 y Ff(\000)p Fr(\016)1552 1681 y Fc(i;j)1647 1712 y Fp(\012)1745 1567 y Fi(\014)1745 1627 y(\014)1745 1687 y(\014)1745 1747 y(\014)1882 1587 y Fr(m)1850 1617 y Fi(Y)1856 1827 y Fr(i)p Fl(=1)1994 1712 y Fs(a)2045 1671 y Ff(\003)2084 1712 y Fp(\()p Fs(')2186 1727 y Fr(i;\016)2268 1712 y Fp(\))2306 1671 y Fr(n)2349 1681 y Fc(i)2375 1671 y Ff(\000)p Fr(\016)2461 1683 y Fc(i;`)2553 1712 y Fp(\012)2623 1571 y Fi(\035)3255 1712 y Fv(\(III.2\))1405 2033 y Fp(=)83 b Fs(\016)1607 2048 y Fr(j;`)1710 2033 y Fo(\001)1793 1909 y Fr(m)1760 1939 y Fi(Y)1767 2149 y Fr(i)p Fl(=1)1888 2033 y Fp(\()p Fs(n)1984 2048 y Fr(i)2034 2033 y Fo(\000)23 b Fs(\016)2177 2048 y Fr(i;j)2257 2033 y Fp(\)!)17 b Fo(k)p Fs(')2453 2048 y Fr(i)2481 2033 y Fo(k)2531 1992 y Fl(2\()p Fr(n)2636 2002 y Fc(i)2662 1992 y Ff(\000)p Fr(\016)2748 2002 y Fc(i;j)2822 1992 y Fl(\))2876 2033 y Fs(;)257 2297 y Fv(and)25 b(we)g(are)h(interested)e(in)g(the)h(limit) e Fs(\016)32 b Fo(!)27 b Fp(0)e Fv(of)g(the)f(quantity)803 2483 y Fs(Q)880 2447 y Fl(0)880 2508 y Ff(T)941 2483 y Fp(\()p Fs(A)1052 2498 y Fr(\016)1089 2483 y Fp(\010)1159 2498 y Fl(gs)1227 2483 y Fp(\))p 389 2527 1290 4 v 389 2618 a Fo(k)p Fs(')503 2633 y Fl(1)p Fr(;\016)596 2618 y Fo(k)646 2590 y Fl(2)p Fr(n)724 2599 y Fd(1)779 2618 y Fs(n)837 2633 y Fl(1)876 2618 y Fp(!)34 b Fo(\001)17 b(\001)g(\001)31 b(k)p Fs(')1200 2633 y Fr(m;\016)1320 2618 y Fo(k)1370 2590 y Fl(2)p Fr(n)1448 2598 y Fc(m)1527 2618 y Fs(n)1585 2633 y Fr(m)1652 2618 y Fp(!)1742 2550 y(=)1870 2414 y Fi(Z)1870 2698 y Ff(T)1986 2550 y Fs(d)2037 2509 y Fl(3)2076 2550 y Fs(p)2183 2426 y Fr(m)2142 2455 y Fi(X)2153 2665 y Fr(j)t Fl(=1)2409 2483 y Fs(n)2467 2498 y Fr(j)p 2312 2527 289 4 v 2312 2618 a Fo(k)p Fs(')2426 2633 y Fr(j;\016)2512 2618 y Fo(k)2562 2590 y Fl(2)2628 2550 y Fo(jh)p Fs(L)2761 2503 y Fr(p)2761 2575 y Fl(0)2817 2550 y Fs(;)i(')2941 2565 y Fr(j;\016)3027 2550 y Fo(ij)3094 2509 y Fl(2)3255 2550 y Fv(\(III.3\))257 2845 y(Since)h(the)f(right)g (side)g(of)g(\(III.3\))h(is)f(additi)n(v)o(e)e(in)i(the)g(functions)f Fs(')2657 2860 y Fr(j;\016)2743 2845 y Fv(,)j(we)f(may)f(consider)257 2965 y(just)28 b(a)h(single)e(photon)g(with)h(momentum)f(distrib)n (ution)f Fs(')2313 2980 y Fr(\016)2379 2965 y Fv(of)i(the)h(form)f (\(III.1\),)i(centered)257 3085 y(about)25 b Fs(!)565 3100 y Fl(0)629 3085 y Fv(and)f(get)257 3314 y Fo(h)p Fs(L)362 3266 y Fr(p)362 3338 y Fl(0)419 3314 y Fs(;)33 b(')543 3329 y Fr(\016)581 3314 y Fo(i)52 b Fp(=)801 3178 y Fi(Z)900 3204 y Ff(1)856 3404 y(\0001)1002 3314 y Fs(F)1065 3329 y Fr(p;\016)1158 3314 y Fp(\()p Fs(p)1245 3273 y Fl(2)1307 3314 y Fo(\000)22 b Fs(E)1478 3329 y Fl(0)1540 3314 y Fo(\000)h Fs(x)p Fp(\))1836 3246 y Fs(dx)p 1760 3291 258 4 v 1760 3382 a(x)f Fp(+)g Fs(i)p Fp(0)2080 3314 y(=)2208 3178 y Fi(Z)2308 3204 y Ff(1)2264 3404 y(\0001)2410 3314 y Fs(F)2473 3329 y Fr(p;\016)2566 3314 y Fp(\()p Fs(x)p Fp(\))3022 3246 y Fs(dx)p 2724 3291 702 4 v 2724 3382 a(p)2773 3353 y Fl(2)2835 3382 y Fo(\000)g Fs(E)3006 3397 y Fl(0)3068 3382 y Fo(\000)h Fs(x)f Fp(+)g Fs(i)p Fp(0)3457 3314 y Fs(;)3260 3503 y Fv(\(III.4\))257 3676 y(where)349 3882 y Fs(F)412 3897 y Fr(p;\016)505 3882 y Fp(\()p Fs(r)s Fp(\))53 b(=)f Fo(\000)p Fs(i)936 3746 y Fi(Z)992 3972 y Fe(S)1035 3953 y Fd(2)1084 3882 y Fs(r)1131 3841 y Fl(2)1187 3882 y Fs(')1251 3897 y Fr(\016)1288 3882 y Fp(\()p Fs(r)s(\033)t Fp(\))26 b(^)-58 b Fs(\032)p Fp(\()p Fs(p;)17 b(r)s(\033)t Fp(\))g Fs(d)1880 3841 y Fl(2)1918 3882 y Fs(\033)56 b Fp(=)d Fs(\016)2205 3841 y Ff(\000)p Fl(1)p Fr(=)p Fl(2)2386 3882 y Fs(\037)2447 3771 y Fi(\020)2517 3815 y Fs(r)25 b Fo(\000)d Fs(!)2746 3830 y Fl(0)p 2517 3859 269 4 v 2628 3950 a Fs(\016)2796 3771 y Fi(\021)2872 3882 y Fp(\002\()p Fs(p;)17 b(r)s Fp(\))91 b Fv(\(III.5\))257 4109 y(and)1053 4325 y Fp(\002\()p Fs(p;)17 b(r)s Fp(\))51 b(=)i Fo(\000)p Fs(i)1652 4189 y Fi(Z)1707 4415 y Fe(S)1750 4396 y Fd(2)1817 4258 y Fs(r)1864 4221 y Fl(2)p 1810 4302 101 4 v 1810 4393 a Fs(!)1871 4408 y Fl(0)1945 4325 y Fp(^)-58 b Fs(\032)q Fp(\()p Fs(p;)17 b(r)s(\033)t Fp(\))g Fs(\024)p Fp(\()p Fs(\033)t Fp(\))g Fs(d)2538 4284 y Fl(2)2575 4325 y Fs(\033)26 b(;)572 b Fv(\(III.6\))257 4559 y(for)23 b Fs(r)30 b Fo(\025)f Fp(0)p Fv(,)22 b(and)h Fp(\002\()p Fs(p;)17 b(r)s Fp(\))27 b Fo(\021)h Fs(F)1326 4574 y Fr(p;\016)1447 4559 y Fo(\021)g Fp(0)22 b Fv(on)g Fh(R)1811 4574 y Ff(\000)1876 4559 y Fv(.)30 b(W)-8 b(e)23 b(no)n(w)f(proceed)h(to)f(computing)f(the) h(limit)285 4730 y Fp(lim)257 4798 y Fr(\016)r Ff(!)p Fl(0)396 4779 y Fd(+)464 4595 y Fi(Z)520 4820 y Ff(T)597 4730 y Fs(d)648 4689 y Fl(3)687 4730 y Fs(p)17 b Fo(j)o(h)p Fs(L)885 4745 y Fr(p)925 4730 y Fs(;)g(')1033 4745 y Fr(\016)1070 4730 y Fo(ij)1137 4682 y Fl(2)1176 4730 y Fv(.)31 b(W)-8 b(e)25 b(write)547 4914 y Fi(Z)603 5140 y Ff(T)680 5050 y Fo(jh)p Fs(L)813 5003 y Fr(p)813 5074 y Fl(0)869 5050 y Fs(;)33 b(')993 5065 y Fr(\016)1031 5050 y Fo(ij)1098 5009 y Fl(2)1153 5050 y Fs(d)1204 5009 y Fl(3)1243 5050 y Fs(p)83 b Fp(=)1534 4914 y Fi(Z)1590 5140 y Ff(T)1666 4906 y Fi(\014)1666 4965 y(\014)1666 5025 y(\014)1666 5085 y(\014)1716 4914 y(Z)1816 4941 y Ff(1)1772 5140 y(\0001)1918 5050 y Fs(F)1981 5065 y Fr(p;\016)2074 5050 y Fp(\()p Fs(x)p Fp(\))2529 4983 y Fs(dx)p 2232 5027 702 4 v 2232 5118 a(p)2281 5090 y Fl(2)2342 5118 y Fo(\000)23 b Fs(E)2514 5133 y Fl(0)2576 5118 y Fo(\000)f Fs(x)h Fp(+)f Fs(i)p Fp(0)2943 4906 y Fi(\014)2943 4965 y(\014)2943 5025 y(\014)2943 5085 y(\014)2976 4932 y Fl(2)3016 5050 y Fs(d)3067 5009 y Fl(3)3106 5050 y Fs(p)100 b Fv(\(III.7\))1375 5341 y Fp(=)1534 5205 y Fi(Z)1590 5431 y Ff(T)1666 5197 y Fi(\014)1666 5256 y(\014)1666 5316 y(\014)1666 5376 y(\014)1716 5205 y(Z)1816 5232 y Ff(1)1772 5431 y(\0001)1928 5274 y Fs(\016)1975 5238 y Fl(1)p Fr(=)p Fl(2)2101 5274 y Fs(\037)p Fp(\()p Fs(x)p Fp(\))17 b(\002\()p Fs(p;)g(!)2578 5289 y Fl(0)2639 5274 y Fp(+)22 b Fs(\016)t(x)p Fp(\))p Fs(dx)p 1928 5318 1056 4 v 1971 5409 a(p)2020 5381 y Fl(2)2081 5409 y Fo(\000)h Fs(E)2253 5424 y Fl(0)2315 5409 y Fo(\000)f Fs(!)2475 5424 y Fl(0)2537 5409 y Fo(\000)g Fs(\016)t(x)h Fp(+)f Fs(i)p Fp(0)2994 5197 y Fi(\014)2994 5256 y(\014)2994 5316 y(\014)2994 5376 y(\014)3027 5223 y Fl(2)3066 5341 y Fs(d)3117 5300 y Fl(3)3156 5341 y Fs(p)p eop %%Page: 26 27 26 26 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(26)257 573 y(Here,)25 b(we)e(substitute)f(\(III.4\))i(in)f(the)h (inte)o(gral)e(\(III.5\))i(and)g(perform)f(a)h(change)g(of)f(v)n (ariables)257 693 y(in)i Fs(x)p Fv(.)31 b(W)-8 b(e)25 b(no)n(w)f(change)h(v)n(ariables)f(to)h(polar)f(coordinates)h(in)f(the) h Fs(p)p Fv(-inte)o(gral)f(to)g(get)383 823 y Fi(Z)439 1048 y Ff(T)516 958 y Fo(jh)p Fs(L)649 911 y Fr(p)649 983 y Fl(0)705 958 y Fs(;)33 b(')829 973 y Fr(\016)867 958 y Fo(ij)934 917 y Fl(2)989 958 y Fs(d)1040 917 y Fl(3)1079 958 y Fs(p)2127 b Fv(\(III.8\))381 1242 y Fp(=)550 1107 y Fi(Z)605 1332 y Fe(S)648 1313 y Fd(2)697 1242 y Fs(d)748 1201 y Fl(2)787 1242 y Fs(\033)863 1107 y Fi(Z)962 1133 y Ff(1)918 1332 y Fl(0)1054 1242 y Fs(dp)17 b Fg(1)1227 1257 y Ff(T)1286 1242 y Fp(\()p Fs(p\033)t Fp(\))p Fs(p)1519 1201 y Fl(2)1559 1098 y Fi(\014)1559 1158 y(\014)1559 1218 y(\014)1559 1277 y(\014)1608 1107 y(Z)1708 1133 y Ff(1)1664 1332 y(\0001)1820 1175 y Fs(\016)1867 1139 y Fl(1)p Fr(=)p Fl(2)1994 1175 y Fs(\037)p Fp(\()p Fs(x)p Fp(\))g(\002\()p Fs(p\033)n(;)33 b(!)2540 1190 y Fl(0)2602 1175 y Fp(+)22 b Fs(\016)t(x)p Fp(\))p Fs(dx)p 1820 1220 1127 4 v 1898 1311 a(p)1947 1282 y Fl(2)2009 1311 y Fo(\000)g Fs(E)2180 1326 y Fl(0)2242 1311 y Fo(\000)h Fs(!)2403 1326 y Fl(0)2464 1311 y Fo(\000)f Fs(\016)t(x)h Fp(+)f Fs(i)p Fp(0)2956 1098 y Fi(\014)2956 1158 y(\014)2956 1218 y(\014)2956 1277 y(\014)2989 1124 y Fl(2)381 1510 y Fp(=)559 1443 y(1)p 559 1488 49 4 v 559 1579 a(2)635 1375 y Fi(Z)690 1600 y Fe(S)733 1581 y Fd(2)782 1510 y Fs(d)833 1469 y Fl(2)873 1510 y Fs(\033)948 1375 y Fi(Z)1048 1401 y Ff(1)1004 1600 y(\000)p Fl(\()p Fr(E)1138 1609 y Fd(0)1172 1600 y Fl(+)p Fr(!)1271 1609 y Fd(0)1306 1600 y Fl(\))p Fr(=\016)1423 1510 y Fs(dy)d Fg(1)1597 1525 y Ff(T)1658 1430 y Fi(\000)1703 1420 y(p)p 1803 1420 551 4 v 90 x Fs(E)1875 1525 y Fl(0)1937 1510 y Fp(+)j Fs(!)2096 1525 y Fl(0)2157 1510 y Fp(+)g Fs(\016)t(y)e(\033)2429 1430 y Fi(\001)2491 1420 y(p)p 2591 1420 V 90 x Fs(!)2652 1525 y Fl(0)2713 1510 y Fp(+)i Fs(E)2883 1525 y Fl(0)2945 1510 y Fp(+)g Fs(\016)t(y)1022 1667 y Fi(\014)1022 1726 y(\014)1022 1786 y(\014)1022 1846 y(\014)1072 1676 y(Z)1171 1702 y Ff(1)1127 1901 y(\0001)1273 1811 y Fs(\037)p Fp(\()p Fs(x)p Fp(\))17 b(\002)1558 1730 y Fi(\000)1604 1721 y(p)p 1704 1721 V 90 x Fs(E)1776 1826 y Fl(0)1837 1811 y Fp(+)22 b Fs(!)1996 1826 y Fl(0)2058 1811 y Fp(+)g Fs(\016)t(y)d(\033)i(;)39 b(!)2473 1826 y Fl(0)2534 1811 y Fp(+)22 b Fs(\016)t(x)2734 1730 y Fi(\001)2969 1744 y Fs(dx)p 2807 1788 431 4 v 2807 1880 a(y)j Fo(\000)e Fs(x)f Fp(+)g Fs(i)p Fp(0)3247 1667 y Fi(\014)3247 1726 y(\014)3247 1786 y(\014)3247 1846 y(\014)3280 1693 y Fl(2)3342 1811 y Fs(:)257 2095 y Fv(In)e(the)g(last)f(step,)h(we)g (again)f(change)h(the)f(v)n(ariable,)h(passing)f(from)g Fs(p)h Fv(to)f Fs(y)31 b Fp(:=)d Fs(\016)2999 2059 y Ff(\000)p Fl(1)3093 2095 y Fp(\()p Fs(p)3180 2059 y Fl(2)3222 2095 y Fo(\000)s Fs(E)3374 2110 y Fl(0)3418 2095 y Fo(\000)257 2215 y Fs(!)318 2230 y Fl(0)358 2215 y Fp(\))p Fv(.)48 b(As)30 b Fs(\037)h Fv(has)f(support)g(in)g Fp([)p Fo(\000)p Fp(1)p Fs(;)17 b Fp(1])p Fv(,)32 b(we)f(observ)o(e)f(that)g(the)g Fs(x)p Fv(-inte)o(gral)h(is)f(non-singular)l(,)257 2336 y(for)25 b Fo(j)p Fs(y)t Fo(j)i Fs(>)g Fp(1)p Fv(,)e(and)f(we)i(obtain) e(the)g(estimate)466 2469 y Fi(\014)466 2528 y(\014)466 2588 y(\014)466 2648 y(\014)516 2477 y(Z)616 2504 y Ff(1)572 2703 y(\0001)718 2613 y Fs(\037)p Fp(\()p Fs(x)p Fp(\))17 b(\002)1003 2532 y Fi(\000)1048 2523 y(p)p 1148 2523 551 4 v 90 x Fs(E)1220 2628 y Fl(0)1282 2613 y Fp(+)22 b Fs(!)1441 2628 y Fl(0)1502 2613 y Fp(+)g Fs(\016)t(y)e(\033)g(;)39 b(!)1917 2628 y Fl(0)1979 2613 y Fp(+)22 b Fs(\016)t(x)2179 2532 y Fi(\001)2413 2546 y Fs(dx)p 2251 2590 431 4 v 2251 2681 a(y)j Fo(\000)e Fs(x)g Fp(+)f Fs(i)p Fp(0)2692 2469 y Fi(\014)2692 2528 y(\014)2692 2588 y(\014)2692 2648 y(\014)3255 2613 y Fv(\(III.9\))1022 2885 y Fo(\024)1207 2818 y(k)p Fs(\037)p Fo(k)1368 2834 y Fr(L)1416 2816 y Fd(1)p 1192 2862 278 4 v 1192 2953 a Fo(j)p Fs(y)t Fo(j)f(\000)i Fp(1)1513 2885 y(sup)1676 2775 y Fi(n)1742 2885 y Fo(j)p Fp(\002\()p Fs(p;)17 b(r)s Fp(\))p Fo(j)2114 2771 y Fi(\014)2114 2830 y(\014)2114 2890 y(\014)2172 2885 y Fs(p)28 b Fo(2)g Fh(R)2409 2844 y Fl(3)2471 2885 y Fs(;)44 b(r)30 b Fo(2)e Fp([0)17 b Fs(;)44 b(!)2935 2900 y Fl(0)2997 2885 y Fp(+)22 b(1])3171 2775 y Fi(o)3259 2885 y Fs(;)257 3161 y Fv(for)28 b(all)g Fo(j)p Fs(y)t Fo(j)k Fs(>)h Fp(1)p Fv(.)39 b(T)-8 b(o)28 b(estimate)f(the)g(inte)o (gral)g(in)g(\(III.9\))i(in)e(case)h(that)g Fo(j)p Fs(y)t Fo(j)j Fs(<)i Fp(2)p Fv(,)c(we)f(use)f(the)257 3281 y(representation)e (\(I.40\))g(of)g Fs(L)1270 3234 y Fr(p)1270 3306 y Fl(0)1334 3281 y Fv(and)g(the)g(identity)852 3420 y Fi(Z)951 3447 y Ff(1)907 3646 y(\0001)1063 3489 y Fs(f)11 b Fp(\()p Fs(x)p Fp(\))22 b Fo(\000)h Fs(f)11 b Fp(\()p Fo(\000)p Fs(x)p Fp(\))p 1063 3533 580 4 v 1301 3624 a(2)p Fs(x)1669 3556 y(dx)53 b Fp(=)f Fo(\000)2050 3420 y Fi(Z)2150 3447 y Ff(1)2106 3646 y(\0001)2252 3556 y Fs(f)2311 3515 y Ff(0)2334 3556 y Fp(\()p Fs(x)p Fp(\))39 b(ln)16 b Fo(j)p Fs(x)p Fo(j)h Fs(dx)k(;)321 b Fv(\(III.10\))257 3840 y(which)25 b(holds)f(for)h(all)f Fs(f)39 b Fo(2)28 b Fs(C)1290 3803 y Fl(1)1283 3864 y(0)1329 3840 y Fp(\()p Fh(R)5 b Fp(\))p Fv(.)37 b(This)24 b(yields)376 3975 y Fi(Z)476 4001 y Ff(1)431 4200 y(\0001)577 4110 y Fs(\037)p Fp(\()p Fs(x)p Fp(\))17 b(\002)862 4030 y Fi(\000)908 4020 y(p)p 1008 4020 551 4 v 90 x Fs(E)1080 4125 y Fl(0)1141 4110 y Fp(+)22 b Fs(!)1300 4125 y Fl(0)1362 4110 y Fp(+)g Fs(\016)t(y)d(\033)i(;)39 b(!)1777 4125 y Fl(0)1838 4110 y Fp(+)22 b Fs(\016)t(x)2038 4030 y Fi(\001)2273 4043 y Fs(dx)p 2111 4087 431 4 v 2111 4179 a(y)j Fo(\000)e Fs(x)f Fp(+)g Fs(i)p Fp(0)3205 4110 y Fv(\(III.11\))459 4378 y Fp(=)83 b Fo(\000)712 4243 y Fi(Z)812 4269 y Ff(1)767 4468 y(\0001)914 4378 y Fs(\037)p Fp(\()p Fs(y)25 b Fo(\000)d Fs(z)t Fp(\))17 b(\002)1365 4298 y Fi(\000)1411 4288 y(p)p 1511 4288 551 4 v 90 x Fs(E)1583 4393 y Fl(0)1645 4378 y Fp(+)22 b Fs(!)1804 4393 y Fl(0)1865 4378 y Fp(+)g Fs(\016)t(y)e(\033)g(;)39 b(!)2280 4393 y Fl(0)2342 4378 y Fp(+)22 b Fs(\016)t Fp(\()p Fs(y)j Fo(\000)d Fs(z)t Fp(\))2784 4298 y Fi(\001)2933 4311 y Fs(dz)p 2857 4355 252 4 v 2857 4447 a(z)27 b Fp(+)22 b Fs(i)p Fp(0)459 4606 y(=)83 b Fs(i\031)21 b(\037)p Fp(\()p Fs(y)t Fp(\))c(\002)1009 4526 y Fi(\000)1053 4516 y(p)p 1153 4516 551 4 v 90 x Fs(E)1225 4621 y Fl(0)1286 4606 y Fp(+)22 b Fs(!)1445 4621 y Fl(0)1507 4606 y Fp(+)g Fs(\016)t(y)d(\033)i(;)39 b(!)1922 4621 y Fl(0)1983 4606 y Fp(+)22 b Fs(\016)t(y)2180 4526 y Fi(\001)618 4819 y Fo(\000)712 4683 y Fi(Z)812 4710 y Ff(1)767 4909 y(\0001)914 4679 y Fi(\032)1026 4752 y Fs(d)p 998 4796 107 4 v 998 4887 a(dx)1114 4708 y Fi(h)1161 4819 y Fs(\037)p Fp(\()p Fs(x)p Fp(\))17 b(\002)1446 4738 y Fi(\000)1492 4729 y(p)p 1592 4729 551 4 v 90 x Fs(E)1664 4834 y Fl(0)1725 4819 y Fp(+)22 b Fs(!)1884 4834 y Fl(0)1946 4819 y Fp(+)g Fs(\016)t(y)d(\033)i(;)39 b(!)2361 4834 y Fl(0)2422 4819 y Fp(+)22 b Fs(\016)t(x)2622 4738 y Fi(\001)2668 4708 y(i)2715 4679 y(\033)2823 4819 y Fp(ln)16 b Fo(j)p Fs(y)25 b Fo(\000)e Fs(x)p Fo(j)17 b Fs(dx)22 b(:)p eop %%Page: 27 28 27 27 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(27)257 573 y(Thus)25 b(we)g(obtain)f(the)g(follo)n(wing)f(estimate)h (on)h(the)f(magnitude)g(of)h(this)e(inte)o(gral,)331 702 y Fi(\014)331 762 y(\014)331 822 y(\014)331 882 y(\014)381 711 y(Z)481 737 y Ff(1)437 937 y(\0001)583 847 y Fs(\037)p Fp(\()p Fs(x)p Fp(\))17 b(\002)868 766 y Fi(\000)913 757 y(p)p 1013 757 551 4 v 90 x Fs(E)1085 862 y Fl(0)1147 847 y Fp(+)22 b Fs(!)1306 862 y Fl(0)1367 847 y Fp(+)g Fs(\016)t(y)e(\033)g(;)39 b(!)1782 862 y Fl(0)1844 847 y Fp(+)22 b Fs(\016)t(x)2044 766 y Fi(\001)2278 779 y Fs(dx)p 2116 824 431 4 v 2116 915 a(y)j Fo(\000)e Fs(x)g Fp(+)f Fs(i)p Fp(0)2557 702 y Fi(\014)2557 762 y(\014)2557 822 y(\014)2557 882 y(\014)3205 847 y Fv(\(III.12\))414 1091 y Fo(\024)84 b Fs(\031)20 b Fo(k)p Fs(\037)p Fo(k)811 1106 y Fr(L)859 1087 y Fb(1)962 1091 y Fp(sup)1125 981 y Fi(n)1191 1091 y Fo(j)p Fp(\002\()p Fs(p;)d(r)s Fp(\))p Fo(j)1563 977 y Fi(\014)1563 1037 y(\014)1563 1096 y(\014)1621 1091 y Fs(p)28 b Fo(2)g Fh(R)1858 1050 y Fl(3)1920 1091 y Fs(;)44 b(r)30 b Fo(2)e Fp([0)17 b Fs(;)44 b(!)2384 1106 y Fl(0)2446 1091 y Fp(+)22 b(2])2620 981 y Fi(o)603 1327 y Fp(+)54 b(sup)723 1413 y Ff(j)p Fr(y)r Ff(j\024)p Fl(2)906 1217 y Fi(n)989 1192 y(Z)1089 1218 y Ff(1)1044 1417 y(\0001)1191 1247 y Fi(\000)1236 1327 y Fo(j)p Fs(\037)1325 1286 y Ff(0)1348 1327 y Fp(\()p Fs(x)p Fp(\))p Fo(j)22 b Fp(+)g Fo(j)p Fs(\037)p Fp(\()p Fs(x)p Fp(\))p Fo(j)1875 1247 y Fi(\001)1937 1243 y(\014)1937 1302 y(\014)1987 1327 y Fp(ln)16 b Fo(j)p Fs(y)25 b Fo(\000)e Fs(x)p Fo(j)2369 1243 y Fi(\014)2369 1302 y(\014)2419 1327 y Fs(dx)2525 1217 y Fi(o)1047 1587 y Fo(\001)17 b Fp(sup)1255 1477 y Fi(n)1322 1587 y Fo(j)p Fp(\002\()p Fs(p;)g(r)s Fp(\))p Fo(j)j Fp(+)i Fo(j)p Fs(@)1867 1602 y Fr(r)1905 1587 y Fp(\002\()p Fs(p;)17 b(r)s Fp(\))p Fo(j)2249 1473 y Fi(\014)2249 1533 y(\014)2249 1592 y(\014)2307 1587 y Fs(p)28 b Fo(2)g Fh(R)2544 1546 y Fl(3)2606 1587 y Fs(;)44 b(r)30 b Fo(2)f Fp([0)17 b Fs(;)43 b(!)3070 1602 y Fl(0)3132 1587 y Fp(+)22 b(2])3306 1477 y Fi(o)3394 1587 y Fs(;)257 1835 y Fv(for)h(all)f Fo(j)p Fs(y)t Fo(j)k Fs(<)i Fp(2)p Fv(.)i(Putting)21 b(together)h(the)h(estimates)e(\(III.9\))i(and)g (\(III.12\),)g(we)g(conclude)f(that)330 1968 y Fi(\014)330 2027 y(\014)330 2087 y(\014)330 2147 y(\014)380 1977 y(Z)479 2003 y Ff(1)435 2202 y(\0001)581 2112 y Fs(\037)p Fp(\()p Fs(x)p Fp(\))17 b(\002)866 2031 y Fi(\000)912 2022 y(p)p 1011 2022 551 4 v 1011 2112 a Fs(E)1083 2127 y Fl(0)1145 2112 y Fp(+)22 b Fs(!)1304 2127 y Fl(0)1366 2112 y Fp(+)g Fs(\016)t(y)d(\033)i(;)39 b(!)1781 2127 y Fl(0)1842 2112 y Fp(+)22 b Fs(\016)t(x)2042 2031 y Fi(\001)2277 2045 y Fs(dx)p 2115 2089 431 4 v 2115 2181 a(y)j Fo(\000)e Fs(x)f Fp(+)g Fs(i)p Fp(0)2555 1968 y Fi(\014)2555 2027 y(\014)2555 2087 y(\014)2555 2147 y(\014)2641 2112 y Fo(\024)2880 2045 y Fs(C)p 2781 2089 276 4 v 2781 2181 a Fo(j)p Fs(y)t Fo(j)f Fp(+)h(1)3089 2112 y Fs(;)89 b Fv(\(III.13\))257 2385 y(for)30 b(some)f(constant)g Fs(C)44 b(<)37 b Fo(1)29 b Fv(depending)g(on)g Fs(\037)p Fv(,)i Fp(\002)p Fv(,)g(and)f(their)f(deri)n(v)n(ati)n(v)o(es.)43 b(Hence,)31 b(we)257 2505 y(can)26 b(apply)e(Lebesgue')-5 b(s)24 b(Dominated)g(Con)l(v)o(er)n(gence)h(Theorem)f(to)h(\(III.8\))g (which)g(gi)n(v)o(es)460 2796 y Fp(lim)432 2863 y Fr(\016)r Ff(!)p Fl(0)571 2844 y Fd(+)639 2660 y Fi(Z)695 2886 y Ff(T)755 2796 y Fo(h)p Fs(L)860 2749 y Fr(p)860 2820 y Fl(0)916 2796 y Fs(;)34 b(')1041 2811 y Fr(\016)1079 2796 y Fo(ij)1146 2755 y Fl(2)1201 2796 y Fs(d)1252 2755 y Fl(3)1291 2796 y Fs(p)53 b Fp(=)f(\002)1597 2811 y Fl(0)1670 2660 y Fi(Z)1769 2687 y Ff(1)1725 2886 y(\0001)1871 2796 y Fs(h)p Fp(\()p Fs(y)t Fp(\))2072 2651 y Fi(\014)2072 2711 y(\014)2072 2771 y(\014)2072 2831 y(\014)2121 2660 y(Z)2220 2687 y Ff(1)2176 2886 y(\0001)2390 2728 y Fs(\037)p Fp(\()p Fs(x)p Fp(\))17 b Fs(dx)p 2332 2773 431 4 v 2332 2864 a(y)26 b Fo(\000)c Fs(x)h Fp(+)f Fs(i)p Fp(0)2773 2651 y Fi(\014)2773 2711 y(\014)2773 2771 y(\014)2773 2831 y(\014)2806 2678 y Fl(2)2862 2796 y Fs(dy)j(;)192 b Fv(\(III.14\))257 3069 y(where)26 b Fp(\002)602 3084 y Fl(0)666 3069 y Fv(is)e(de\002ned)h(in)g(\(I.56\),)g(and)1169 3289 y Fs(h)p Fp(\()p Fs(y)t Fp(\))52 b(=)100 b(lim)1533 3351 y Fr(\016)r Ff(!)p Fl(+)p Ff(1)1780 3289 y Fg(1)1836 3304 y Fl([)p Ff(\000)p Fl(\()p Fr(E)1990 3313 y Fd(0)2024 3304 y Fl(+)p Fr(!)2123 3313 y Fd(0)2157 3304 y Fl(\))p Fr(=\016)20 b(;)15 b Ff(1)p Fl(\))2407 3289 y Fp(\()p Fs(y)t Fp(\))21 b Fs(:)622 b Fv(\(III.15\))257 3544 y(Hence,)26 b(we)f(obtain)f(the)g(claim)h(\(I.55\),)587 3875 y Fp(lim)585 3938 y Fr(\016)r Ff(!)p Fl(0)741 3740 y Fi(Z)741 4023 y Ff(T)858 3875 y Fo(jh)p Fs(L)991 3828 y Fr(p)991 3900 y Fl(0)1047 3875 y Fs(;)33 b(')1171 3890 y Fr(\016)1209 3875 y Fo(ij)1276 3834 y Fl(2)1331 3875 y Fs(d)1382 3834 y Fl(3)1421 3875 y Fs(p)53 b Fp(=)1651 3671 y Fi(8)1651 3761 y(<)1651 3940 y(:)1877 3754 y Fp(0)178 b Fv(if)83 b Fs(E)2320 3769 y Fl(0)2382 3754 y Fp(+)22 b Fs(!)2541 3769 y Fl(0)2607 3754 y Fs(<)28 b Fp(0)22 b Fs(;)1791 3874 y Fp(\002)1867 3889 y Fl(0)1929 3874 y Fs(I)1972 3889 y Fl(0)2104 3874 y Fv(if)83 b Fs(E)2320 3889 y Fl(0)2382 3874 y Fp(+)22 b Fs(!)2541 3889 y Fl(0)2607 3874 y Fp(=)28 b(0)22 b Fs(;)1781 3995 y Fp(\002)1857 4010 y Fl(0)1919 3995 y Fs(I)1962 4010 y Fl(+)2104 3995 y Fv(if)83 b Fs(E)2320 4010 y Fl(0)2382 3995 y Fp(+)22 b Fs(!)2541 4010 y Fl(0)2607 3995 y Fs(>)28 b Fp(0)22 b Fs(;)3205 3875 y Fv(\(III.16\))257 4209 y(with)i Fs(I)502 4224 y Fl(0)567 4209 y Fv(and)h Fs(I)779 4224 y Fl(+)863 4209 y Fv(as)f(de\002ned)i(in)e(\(I.57\))h (and)g(\(I.58\).)1360 b Fa(\003)p eop %%Page: 28 29 28 28 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(28)257 573 y Fu(IV)143 b(T)-13 b(echnical)34 b(Estimates)257 796 y Fv(In)25 b(this)f(section)g(we)h(deri)n(v)o(e)f(the)h(estimates)e (\(II.6\),)i(\(II.8\),)h(and)e(\(II.11\),)h(which)g(is)f(the)g(basic) 257 917 y(input)g(for)h(the)g(asymptotics)e(of)i(the)f(time)g(e)n(v)n (olution)f(asserted)i(in)f(Theorem)h(II.1.)404 1037 y(Before)h(we)f (turn)f(to)h(these)f(bounds,)g(we)h(deri)n(v)o(e)f(a)h(preparatory)g (lemma.)257 1207 y Fk(Lemma)h(IV)-14 b(.1.)48 b Fq(Suppose)36 b(that)g Fs(\013)51 b Fo(\025)f Fp(1)p Fq(,)40 b Fp(\003)49 b Fs(<)h Fo(1)p Fq(,)39 b(and)e(let)f Fs(G)50 b Fo(2)g Fs(L)2808 1171 y Fl(2)2848 1207 y Fp([)p Fh(R)2941 1171 y Fl(3)2986 1207 y Fp(;)17 b Fo(B)s Fp(\()p Fo(H)q Fp(\)])38 b Fq(with)257 1328 y Fs(C)i Fp(:=)504 1247 y Fi(R)587 1328 y Fs(d)638 1292 y Fl(3)677 1328 y Fs(k)731 1247 y Fi(\000)776 1328 y Fp(1)25 b(+)f Fs(!)t Fp(\()p Fs(k)s Fp(\))1145 1292 y Ff(\000)p Fl(1)1238 1247 y Fi(\001)1301 1328 y Fo(k)p Fs(G)p Fp(\()p Fs(k)s Fp(\))p Fo(k)1608 1292 y Fl(2)1696 1328 y Fs(<)33 b Fo(1)p Fq(.)40 b(Denote)27 b Fs(a)2336 1292 y Ff(\003)2376 1328 y Fp(\()p Fs(G)p Fp(\))33 b(:=)2698 1247 y Fi(R)2781 1328 y Fs(d)2832 1292 y Fl(3)2871 1328 y Fs(k)20 b(G)p Fp(\()p Fs(k)s Fp(\))k Fo(\012)h Fs(a)3326 1292 y Ff(\003)3365 1328 y Fp(\()p Fs(k)s Fp(\))257 1448 y Fq(and)g Fs(a)p Fp(\()p Fs(G)p Fp(\))i(:=)794 1368 y Fi(R)877 1448 y Fs(d)928 1412 y Fl(3)967 1448 y Fs(k)20 b(G)p Fp(\()p Fs(k)s Fp(\))1245 1412 y Ff(\003)1306 1448 y Fo(\012)j Fs(a)p Fp(\()p Fs(k)s Fp(\))p Fq(.)31 b(Then)813 1602 y Fi(\015)813 1662 y(\015)869 1687 y Fp(\()p Fs(H)996 1636 y Fl(\(\003\))988 1715 y Fr(f)1125 1687 y Fp(+)22 b(1\))1310 1646 y Fr(\013)1376 1687 y Fs(a)p Fp(\()p Fs(G)p Fp(\))17 b(\()p Fs(H)1716 1702 y Fr(f)1783 1687 y Fp(+)22 b(1\))1968 1646 y Ff(\000)p Fr(\013)p Ff(\000)p Fl(1)p Fr(=)p Fl(2)2233 1602 y Fi(\015)2233 1662 y(\015)2371 1687 y Fo(\024)84 b Fs(C)29 b(;)604 b Fv(\(IV)-13 b(.1\))774 1777 y Fi(\015)774 1837 y(\015)829 1862 y Fp(\()p Fs(H)956 1811 y Fl(\(\003\))948 1890 y Fr(f)1086 1862 y Fp(+)22 b(1\))1271 1821 y Fr(\013)1337 1862 y Fs(a)1388 1821 y Ff(\003)1427 1862 y Fp(\()p Fs(G)p Fp(\))17 b(\()p Fs(H)1716 1877 y Fr(f)1783 1862 y Fp(+)22 b(1\))1968 1821 y Ff(\000)p Fr(\013)p Ff(\000)p Fl(1)p Fr(=)p Fl(2)2233 1777 y Fi(\015)2233 1837 y(\015)2371 1862 y Fo(\024)84 b Fs(C)23 b Fp(\(2)f(+)g(4\003\))2987 1821 y Fr(\013)3058 1862 y Fs(;)177 b Fv(\(IV)-13 b(.2\))690 1952 y Fi(\015)690 2011 y(\015)745 2036 y Fp(\()p Fs(H)872 1985 y Fl(\(\003\))864 2064 y Fr(f)1002 2036 y Fp(+)22 b(1\))1187 1995 y Fr(\013)1253 2036 y Fs(a)p Fp(\()p Fs(G)1419 2051 y Fl(\003)1472 2036 y Fp(\))17 b(\()p Fs(H)1654 1985 y Fl(\(\003\))1646 2064 y Fr(f)1783 2036 y Fp(+)22 b(1\))1968 1995 y Ff(\000)p Fr(\013)p Ff(\000)p Fl(1)p Fr(=)p Fl(2)2233 1952 y Fi(\015)2233 2011 y(\015)2371 2036 y Fo(\024)84 b Fs(C)29 b(;)604 b Fv(\(IV)-13 b(.3\))651 2126 y Fi(\015)651 2186 y(\015)706 2211 y Fp(\()p Fs(H)833 2160 y Fl(\(\003\))825 2239 y Fr(f)963 2211 y Fp(+)22 b(1\))1148 2170 y Fr(\013)1213 2211 y Fs(a)1264 2170 y Ff(\003)1304 2211 y Fp(\()p Fs(G)1419 2226 y Fl(\003)1472 2211 y Fp(\))17 b(\()p Fs(H)1654 2160 y Fl(\(\003\))1646 2239 y Fr(f)1783 2211 y Fp(+)22 b(1\))1968 2170 y Ff(\000)p Fr(\013)p Ff(\000)p Fl(1)p Fr(=)p Fl(2)2233 2126 y Fi(\015)2233 2186 y(\015)2371 2211 y Fo(\024)84 b Fs(C)23 b Fp(\(2)f(+)g(4\003\)) 2987 2170 y Fr(\013)3058 2211 y Fs(;)177 b Fv(\(IV)-13 b(.4\))257 2431 y Fq(wher)l(e)33 b Fs(G)606 2446 y Fl(\003)659 2431 y Fp(\()p Fs(k)s Fp(\))41 b(:=)g Fg(1)1030 2446 y Ff(f)p Fr(!)r Fl(\()p Fr(k)r Fl(\))p Fr(<)p Fl(\003)p Ff(g)1365 2431 y Fs(G)p Fp(\()p Fs(k)s Fp(\))p Fq(,)33 b(and)f Fg(1)1868 2446 y Ff(f)p Fr(!)r Fl(\()p Fr(k)r Fl(\))p Fr(<)p Fl(\003)p Ff(g)2218 2431 y Fq(denotes)f(the)h(c)o(har)o (acteristic)d(func-)257 2551 y(tion)24 b(of)h Fo(f)p Fs(k)30 b Fo(2)e Fh(R)831 2515 y Fl(3)904 2551 y Fo(j)g Fs(!)t Fp(\()p Fs(k)s Fp(\))f Fs(<)g Fp(\003)p Fo(g)p Fq(.)257 2746 y(Pr)l(oof)o(.)47 b Fv(First,)25 b(we)g(introduce)f(a)h (more)g(compact)f(notation,)828 2960 y Fs(G)905 2975 y Fl(1)1027 2960 y Fp(:=)83 b Fs(G;)269 b(H)1667 2975 y Fl(1)1789 2960 y Fp(:=)83 b Fs(H)2056 2975 y Fr(f)2102 2960 y Fs(;)286 b(!)2476 2975 y Fl(1)2598 2960 y Fp(:=)83 b Fs(!)t(;)828 3097 y(G)905 3112 y Fl(2)1027 3097 y Fp(:=)g Fs(G)1290 3112 y Fl(\003)1343 3097 y Fs(;)216 b(H)1667 3112 y Fl(2)1789 3097 y Fp(:=)83 b Fs(H)2064 3046 y Fl(\(\003\))2056 3125 y Fr(f)2172 3097 y Fs(;)216 b(!)2476 3112 y Fl(2)2598 3097 y Fp(:=)83 b Fs(!)2845 3112 y Fl(\003)2898 3097 y Fs(;)3262 3032 y Fv(\(IV)-13 b(.5\))257 3314 y(and)25 b(observ)o(e)f(that)h(Eqs.)f(\(IV)-13 b(.1\)\226\(IV)g(.4\))25 b(are)h(equi)n(v)n(alent)d(to)746 3459 y Fi(\015)746 3519 y(\015)802 3544 y Fp(\()p Fs(H)929 3493 y Fl(\(\003\))921 3572 y Fr(f)1058 3544 y Fp(+)f(1\))1243 3503 y Fr(\013)1309 3544 y Fs(a)p Fp(\()p Fs(G)1475 3559 y Fr(j)1512 3544 y Fp(\))17 b(\()p Fs(H)1686 3559 y Fr(j)1744 3544 y Fp(+)22 b(1\))1929 3503 y Ff(\000)p Fr(\013)p Ff(\000)p Fl(1)p Fr(=)p Fl(2)2193 3459 y Fi(\015)2193 3519 y(\015)2332 3544 y Fo(\024)83 b Fs(C)29 b(;)644 b Fv(\(IV)-13 b(.6\))707 3634 y Fi(\015)707 3694 y(\015)762 3718 y Fp(\()p Fs(H)889 3668 y Fl(\(\003\))881 3746 y Fr(f)1019 3718 y Fp(+)22 b(1\))1204 3677 y Fr(\013)1270 3718 y Fs(a)1321 3677 y Ff(\003)1360 3718 y Fp(\()p Fs(G)1475 3733 y Fr(j)1512 3718 y Fp(\))17 b(\()p Fs(H)1686 3733 y Fr(j)1744 3718 y Fp(+)22 b(1\))1929 3677 y Ff(\000)p Fr(\013)p Ff(\000)p Fl(1)p Fr(=)p Fl(2)2193 3634 y Fi(\015)2193 3694 y(\015)2332 3718 y Fo(\024)83 b Fs(C)24 b Fp(\(2)e(+)g(4\003\))2948 3677 y Fr(\013)3019 3718 y Fs(;)216 b Fv(\(IV)-13 b(.7\))257 3938 y(with)22 b Fs(j)33 b Fp(=)28 b(1)p Fs(;)17 b Fp(2)p Fv(.)29 b(W)-8 b(e)23 b(apply)e(the)h(operator)g(on)g(the)g(left)g (side)g(of)g(\(IV)-13 b(.6\))23 b(to)e(a)i(normalized)e(v)o(ec-)257 4059 y(tor)26 b Fs( )35 b Fo(2)c(H)c Fv(and)f(obtain)g(the)g(desired)g (estimate)f(by)h(means)g(of)g(the)g(pull-through)f(formula,)257 4189 y(the)g(Cauchy-Schw)o(arz)h(inequality)-6 b(,)23 b(and)i Fs(H)1808 4139 y Fl(\(\003\))1800 4217 y Fr(f)1937 4189 y Fp(+)d(1)28 b Fo(\024)g Fs(H)2298 4204 y Fr(j)2356 4189 y Fp(+)22 b(1)g(+)g Fs(!)2684 4204 y Fr(j)2721 4189 y Fv(,)370 4354 y Fi(\015)370 4414 y(\015)425 4439 y Fp(\()p Fs(H)552 4388 y Fl(\(\003\))544 4467 y Fr(f)682 4439 y Fp(+)g(1\))867 4398 y Fr(\013)933 4439 y Fs(a)p Fp(\()p Fs(G)1099 4454 y Fr(j)1135 4439 y Fp(\))17 b(\()p Fs(H)1309 4454 y Fr(j)1367 4439 y Fp(+)22 b(1\))1552 4398 y Ff(\000)p Fr(\013)p Ff(\000)p Fl(1)p Fr(=)p Fl(2)1834 4439 y Fs( )1901 4354 y Fi(\015)1901 4414 y(\015)3262 4439 y Fv(\(IV)-13 b(.8\))453 4693 y Fo(\024)613 4557 y Fi(Z)730 4693 y Fs(d)781 4652 y Fl(3)820 4693 y Fs(k)30 b Fo(k)p Fs(G)1028 4708 y Fr(j)1065 4693 y Fp(\()p Fs(k)s Fp(\))p Fo(k)1262 4548 y Fi(\015)1262 4608 y(\015)1262 4668 y(\015)1262 4728 y(\015)1461 4613 y Fs(H)1550 4562 y Fl(\(\003\))1542 4641 y Fr(f)1679 4613 y Fp(+)22 b(1)p 1326 4670 634 4 v 1326 4761 a Fs(H)1407 4776 y Fr(j)1466 4761 y Fp(+)g(1)g(+)g Fs(!)1794 4776 y Fr(j)1830 4761 y Fp(\()p Fs(k)s Fp(\))1970 4548 y Fi(\015)1970 4608 y(\015)1970 4668 y(\015)1970 4728 y(\015)2025 4575 y Fr(\013)2091 4578 y Fi(\015)2091 4638 y(\015)2091 4698 y(\015)2147 4693 y Fs(a)p Fp(\()p Fs(k)s Fp(\))17 b(\()p Fs(H)2464 4708 y Fr(j)2522 4693 y Fp(+)22 b(1\))2707 4652 y Ff(\000)p Fl(1)p Fr(=)p Fl(2)2888 4693 y Fs( )2955 4578 y Fi(\015)2955 4638 y(\015)2955 4698 y(\015)453 4990 y Fo(\024)613 4850 y Fi(\022)703 4855 y(Z)846 4923 y Fo(k)p Fs(G)973 4938 y Fr(j)1009 4923 y Fp(\()p Fs(k)s Fp(\))p Fo(k)1189 4887 y Fl(2)1245 4923 y Fs(d)1296 4887 y Fl(3)1335 4923 y Fs(k)p 846 4967 544 4 v 1004 5059 a(!)1065 5074 y Fr(j)1101 5059 y Fp(\()p Fs(k)s Fp(\))1399 4850 y Fi(\023)1473 4872 y Fl(1)p Fr(=)p Fl(2)1605 4990 y Fo(\001)1655 4846 y Fi(\015)1655 4906 y(\015)1655 4965 y(\015)1655 5025 y(\015)1804 4923 y Fs(H)1885 4938 y Fr(j)p 1720 4967 287 4 v 1720 5059 a Fs(H)1801 5074 y Fr(j)1859 5059 y Fp(+)g(1)2016 4846 y Fi(\015)2016 4906 y(\015)2016 4965 y(\015)2016 5025 y(\015)2072 4872 y Fl(1)p Fr(=)p Fl(2)2234 4990 y Fo(\024)2364 4850 y Fi(\022)2454 4855 y(Z)2597 4923 y Fo(k)p Fs(G)2724 4938 y Fr(j)2760 4923 y Fp(\()p Fs(k)s Fp(\))p Fo(k)2940 4887 y Fl(2)2996 4923 y Fs(d)3047 4887 y Fl(3)3086 4923 y Fs(k)p 2597 4967 544 4 v 2755 5059 a(!)2816 5074 y Fr(j)2852 5059 y Fp(\()p Fs(k)s Fp(\))3150 4850 y Fi(\023)3224 4872 y Fl(1)p Fr(=)p Fl(2)3356 4990 y Fs(:)p eop %%Page: 29 30 29 29 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(29)257 573 y(for)32 b(an)o(y)g Fs( )44 b Fo(2)d(H)q Fv(,)34 b Fo(k)p Fs( )t Fo(k)40 b Fp(=)g(1)p Fv(.)51 b(T)-8 b(o)32 b(deri)n(v)o(e)f(\(IV)-13 b(.7\),)34 b(we)e(use)f(the)h (canonical)f(commutation)257 693 y(relations,)288 824 y Fi(\015)288 884 y(\015)343 909 y Fp(\()p Fs(H)470 858 y Fl(\(\003\))462 936 y Fr(f)600 909 y Fp(+)22 b(1\))785 867 y Fr(\013)850 909 y Fs(a)901 867 y Ff(\003)941 909 y Fp(\()p Fs(G)1056 924 y Fr(j)1092 909 y Fp(\))17 b(\()p Fs(H)1266 924 y Fr(j)1324 909 y Fp(+)22 b(1\))1509 867 y Ff(\000)p Fr(\013)p Ff(\000)p Fl(1)p Fr(=)p Fl(2)1791 909 y Fs( )1858 824 y Fi(\015)1858 884 y(\015)1913 850 y Fl(2)3262 909 y Fv(\(IV)-13 b(.9\))372 1118 y Fp(=)531 982 y Fi(Z)647 1118 y Fs(d)698 1077 y Fl(3)738 1118 y Fs(k)19 b(d)859 1077 y Fl(3)901 1092 y Fp(~)898 1118 y Fs(k)969 1008 y Fi(D)1030 1118 y Fs( )1114 1004 y Fi(\014)1114 1063 y(\014)1114 1123 y(\014)1174 1118 y Fs(G)1251 1133 y Fr(j)1288 1118 y Fp(\()p Fs(k)s Fp(\))1418 1077 y Ff(\003)1457 1118 y Fs(G)1534 1133 y Fr(j)1571 1118 y Fp(\()1612 1092 y(~)1609 1118 y Fs(k)s Fp(\))j Fo(\012)g Fp(\()p Fs(H)1941 1133 y Fr(j)2000 1118 y Fp(+)g(1\))2185 1077 y Ff(\000)p Fr(\013)p Ff(\000)p Fl(1)p Fr(=)p Fl(2)2466 1118 y Fs(a)p Fp(\()p Fs(k)s Fp(\))2664 1037 y Fi(\000)2710 1118 y Fs(H)2799 1067 y Fl(\(\003\))2791 1146 y Fr(f)2928 1118 y Fp(+)g(1)3075 1037 y Fi(\001)3121 1060 y Fl(2)p Fr(\013)3228 1118 y Fs(a)3279 1077 y Ff(\003)3318 1118 y Fp(\()3359 1092 y(~)3356 1118 y Fs(k)s Fp(\))1358 1352 y(\()p Fs(H)1477 1367 y Fr(j)1535 1352 y Fp(+)g(1\))1720 1311 y Ff(\000)p Fr(\013)p Ff(\000)p Fl(1)p Fr(=)p Fl(2)2002 1352 y Fs( )2069 1241 y Fi(E)372 1594 y Fp(=)531 1458 y Fi(Z)647 1594 y Fs(d)698 1553 y Fl(3)738 1594 y Fs(k)d(d)859 1553 y Fl(3)901 1568 y Fp(~)898 1594 y Fs(k)969 1454 y Fi(\034)1044 1594 y Fs(a)p Fp(\()1136 1568 y(~)1133 1594 y Fs(k)s Fp(\))e(\()p Fs(H)1361 1609 y Fr(j)1419 1594 y Fp(+)22 b(1\))1604 1553 y Ff(\000)p Fl(1)p Fr(=)p Fl(2)1769 1594 y Fs( )1836 1450 y Fi(\014)1836 1509 y(\014)1836 1569 y(\014)1836 1629 y(\014)1896 1594 y Fs(G)1973 1609 y Fr(j)2010 1594 y Fp(\()p Fs(k)s Fp(\))2140 1553 y Ff(\003)2179 1594 y Fs(G)2256 1609 y Fr(j)2293 1594 y Fp(\()2334 1568 y(~)2331 1594 y Fs(k)s Fp(\))767 1903 y Fo(\012)844 1762 y Fi(\024)1029 1742 y(\000)1074 1823 y Fs(H)1163 1772 y Fl(\(\003\))1155 1851 y Fr(f)1293 1823 y Fp(+)g(1)g(+)g Fs(!)1621 1838 y Fl(\003)1674 1823 y Fp(\()p Fs(k)s Fp(\))g(+)g Fs(!)1985 1838 y Fl(\003)2038 1823 y Fp(\()2079 1797 y(~)2076 1823 y Fs(k)s Fp(\))2168 1742 y Fi(\001)2213 1765 y Fl(2)p 907 1880 1467 4 v 907 1903 a Fi(\000)953 1984 y Fs(H)1034 1999 y Fr(j)1093 1984 y Fp(+)g(1)g(+)g Fs(!)1421 1999 y Fr(j)1457 1984 y Fp(\()p Fs(k)s Fp(\))1587 1903 y Fi(\001)16 b(\000)1695 1984 y Fs(H)1776 1999 y Fr(j)1834 1984 y Fp(+)22 b(1)g(+)g Fs(!)2162 1999 y Fr(j)2198 1984 y Fp(\()2239 1958 y(~)2236 1984 y Fs(k)s Fp(\))2328 1903 y Fi(\001)2384 1762 y(\025)2437 1784 y Fr(\013)2508 1903 y Fs(a)p Fp(\()p Fs(k)s Fp(\))17 b(\()p Fs(H)2825 1918 y Fr(j)2883 1903 y Fp(+)22 b(1\))3068 1862 y Ff(\000)p Fl(1)p Fr(=)p Fl(2)3250 1903 y Fs( )3317 1762 y Fi(\035)531 2188 y Fp(+)651 2052 y Fi(Z)768 2188 y Fs(d)819 2147 y Fl(3)858 2188 y Fs(k)928 2077 y Fi(D)989 2188 y Fs( )1056 2073 y Fi(\014)1056 2133 y(\014)1056 2193 y(\014)1117 2188 y Fs(G)1194 2203 y Fr(j)1231 2188 y Fp(\()p Fs(k)s Fp(\))1361 2147 y Ff(\003)1400 2188 y Fs(G)1477 2203 y Fr(j)1513 2188 y Fp(\()p Fs(k)s Fp(\))h Fo(\012)f Fp(\()p Fs(H)1884 2203 y Fr(j)1943 2188 y Fp(+)g(1\))2128 2147 y Ff(\000)p Fl(2)p Fr(\013)p Ff(\000)p Fl(1)2374 2107 y Fi(\000)2419 2188 y Fs(H)2508 2137 y Fl(\(\003\))2500 2216 y Fr(f)2638 2188 y Fp(+)g(1)g(+)g Fs(!)2966 2203 y Fl(\003)3019 2188 y Fp(\()p Fs(k)s Fp(\))3149 2107 y Fi(\001)3194 2129 y Fl(2)p Fr(\013)3296 2188 y Fs( )3363 2077 y Fi(E)371 2487 y Fo(\024)531 2343 y Fi(\015)531 2402 y(\015)531 2462 y(\015)531 2522 y(\015)597 2408 y Fs(H)686 2357 y Fl(\(\003\))678 2435 y Fr(f)815 2408 y Fp(+)g(1)g(+)g(2\003)p 597 2464 603 4 v 754 2555 a Fs(H)835 2570 y Fr(j)894 2555 y Fp(+)g(1)1209 2343 y Fi(\015)1209 2402 y(\015)1209 2462 y(\015)1209 2522 y(\015)1264 2369 y Fl(2)p Fr(\013)1365 2347 y Fi(\022)1455 2351 y(Z)1571 2487 y Fs(d)1622 2446 y Fl(3)1662 2487 y Fs(k)30 b Fo(k)p Fs(G)1870 2502 y Fr(j)1907 2487 y Fp(\()p Fs(k)s Fp(\))p Fo(k)2104 2372 y Fi(\015)2104 2432 y(\015)2104 2492 y(\015)2158 2487 y Fs(a)p Fp(\()p Fs(k)s Fp(\))17 b(\()p Fs(H)2475 2502 y Fr(j)2533 2487 y Fp(+)22 b(1\))2718 2446 y Ff(\000)p Fl(1)p Fr(=)p Fl(2)2900 2487 y Fs( )2967 2372 y Fi(\015)2967 2432 y(\015)2967 2492 y(\015)3022 2347 y(\023)3095 2369 y Fl(2)531 2761 y Fp(+)635 2646 y Fi(\015)635 2706 y(\015)635 2766 y(\015)690 2761 y Fp(\()p Fs(H)809 2776 y Fr(j)868 2761 y Fp(+)g(1\))1053 2720 y Ff(\000)p Fl(2)p Fr(\013)p Ff(\000)p Fl(1)1299 2680 y Fi(\000)1344 2761 y Fs(H)1433 2710 y Fl(\(\003\))1425 2789 y Fr(f)1563 2761 y Fp(+)g(1)g(+)g Fs(!)1891 2776 y Fl(\003)1944 2761 y Fp(\()p Fs(k)s Fp(\))2074 2680 y Fi(\001)2119 2703 y Fl(2)p Fr(\013)2204 2646 y Fi(\015)2204 2706 y(\015)2204 2766 y(\015)2282 2620 y(\022)2372 2625 y(Z)2488 2761 y Fs(d)2539 2720 y Fl(3)2578 2761 y Fs(k)e Fo(k)p Fs(G)2776 2776 y Fr(j)2812 2761 y Fp(\()p Fs(k)s Fp(\))p Fo(k)2992 2720 y Fl(2)3031 2620 y Fi(\023)371 3033 y Fo(\024)83 b Fs(C)24 b Fp(\(1)e(+)g(2\003\))987 2992 y Fl(2)p Fr(\013)1087 2893 y Fi(\022)1161 3033 y Fp(1)44 b(+)1374 2898 y Fi(Z)1490 3033 y Fs(d)1541 2992 y Fl(3)1580 3033 y Fs(k)20 b(!)1712 3048 y Fr(j)1748 3033 y Fp(\()p Fs(k)s Fp(\))1895 2919 y Fi(\015)1895 2978 y(\015)1895 3038 y(\015)1950 3033 y Fs(a)p Fp(\()p Fs(k)s Fp(\))d(\()p Fs(H)2267 3048 y Fr(j)2325 3033 y Fp(+)22 b(1\))2510 2992 y Ff(\000)p Fl(1)p Fr(=)p Fl(2)2692 3033 y Fs( )2759 2919 y Fi(\015)2759 2978 y(\015)2759 3038 y(\015)2814 2945 y Fl(2)2853 2893 y Fi(\023)371 3306 y Fo(\024)83 b Fs(C)24 b Fp(\(1)e(+)g(2\003\))987 3264 y Fl(2)p Fr(\013)1087 3165 y Fi(\022)1161 3306 y Fp(1)44 b(+)1374 3191 y Fi(\015)1374 3251 y(\015)1374 3311 y(\015)1524 3238 y Fs(H)1605 3253 y Fr(j)p 1439 3283 287 4 v 1439 3374 a Fs(H)1520 3389 y Fr(j)1579 3374 y Fp(+)22 b(1)1735 3191 y Fi(\015)1735 3251 y(\015)1735 3311 y(\015)1791 3165 y(\023)1917 3306 y Fo(\024)53 b Fs(C)23 b Fp(\(2)f(+)g(4\003\))2502 3264 y Fl(2)p Fr(\013)2609 3306 y Fs(:)257 3585 y Fv(Note)j(that)f(in)h(the)f(last)h(step)f(we)h (used)g Fs(H)1678 3600 y Fr(j)1742 3585 y Fp(=)1845 3505 y Fi(R)1928 3585 y Fs(d)1979 3549 y Fl(3)2018 3585 y Fs(k)20 b(!)2150 3600 y Fr(j)2186 3585 y Fp(\()p Fs(k)s Fp(\))d Fs(a)2384 3549 y Ff(\003)2423 3585 y Fp(\()p Fs(k)s Fp(\))p Fs(a)p Fp(\()p Fs(k)s Fp(\))p Fv(.)659 b Fa(\003)404 3706 y Fv(No)n(w)23 b(we)g(are)i(in)e(position)f(to)h (establish)f(Eq.)h(\(II.6\))h(which)f(sho)n(ws)g(that)g(the)g (interacting)257 3826 y(atom-photon)30 b(ground)g(state)h Fp(\010)1395 3841 y Fl(gs)1494 3826 y Fv(is)g(well-localized)f(in)h (ener)n(gy)-6 b(,)32 b(e)n(v)o(en)f(for)g(the)g(noninter)n(-)257 3946 y(acting)25 b(Hamiltonian.)257 4117 y Fk(Lemma)h(IV)-14 b(.2.)48 b Fq(F)-10 b(or)24 b(any)h Fs(\013)j Fo(\025)h Fp(1)p Fq(,)24 b Fp(\003)k Fs(<)f Fo(1)p Fq(,)e(and)f(suf)n (\002ciently)g(small)f Fs(g)31 b(>)d Fp(0)p Fq(,)d(we)g(have)536 4257 y Fi(\015)536 4316 y(\015)592 4341 y Fp(\()p Fs(H)719 4290 y Fl(\(\003\))711 4369 y Fr(f)848 4341 y Fp(+)d Fg(1)p Fp(\))1040 4300 y Fr(\013)1106 4341 y Fs(P)1169 4356 y Fr(c)1220 4341 y Fp(\010)1290 4356 y Fl(gs)1358 4257 y Fi(\015)1358 4316 y(\015)1458 4341 y Fp(+)1578 4257 y Fi(\015)1578 4316 y(\015)1633 4341 y Fp(\()p Fs(H)1760 4290 y Fl(\(\003\))1752 4369 y Fr(f)1890 4341 y Fp(+)g Fg(1)p Fp(\))2082 4300 y Fr(\013)2148 4341 y Fs(P)2225 4300 y Ff(?)2211 4366 y Fl(\012)2300 4341 y Fp(\010)2370 4356 y Fl(gs)2438 4257 y Fi(\015)2438 4316 y(\015)2546 4341 y Fo(\024)53 b(O)s Fp(\()p Fs(g)t Fp(\))21 b Fs(:)279 b Fv(\(IV)-13 b(.10\))257 4557 y Fq(Pr)l(oof)o(.)47 b Fv(The)39 b(proof)g(is)g(similar)f(to)g(the)h(one)g(for)g([2,)k(Thm.)38 b(X].)i(W)-8 b(e)39 b(\002rst)g(note)g(that)f(the)257 4677 y(asserted)25 b(bound)f(\(IV)-13 b(.10\))25 b(is)g(implied)e(by)h (\(I.25\))h(and)g(the)g(follo)n(wing)e(estimate,)1041 4818 y Fi(\015)1041 4878 y(\015)1096 4903 y Fp(\()p Fs(H)1223 4852 y Fl(\(\003\))1215 4930 y Fr(f)1352 4903 y Fp(+)f Fg(1)p Fp(\))1544 4862 y Fr(\013)1611 4903 y Fg(1)1667 4918 y Ff(f)p Fr(H)1760 4927 y Fd(0)1794 4918 y Ff(\025)p Fr(E)1901 4927 y Fd(0)1936 4918 y Fr(=)p Fl(3)p Ff(g)2062 4903 y Fp(\010)2132 4918 y Fl(gs)2200 4818 y Fi(\015)2200 4878 y(\015)2308 4903 y Fo(\024)53 b(O)s Fp(\()p Fs(g)t Fp(\))21 b Fs(;)517 b Fv(\(IV)-13 b(.11\))257 5147 y(because)26 b(on)e Fp(Ran)16 b Fg(1)969 5163 y Ff(f)p Fr(H)1062 5172 y Fd(0)1097 5163 y Fr()30 b Fp(1)c Fv(is)g(the)g(de)o(gree)g(of)h(dif)n (ferentiability)d(in)257 4899 y(Hypothesis)g(1.)30 b(Denoting)1004 5145 y Fp(\(1)22 b(+)g Fo(j)p Fs(x)p Fo(j)p Fp(\))1360 5104 y Fl(3)p Fr(=)p Fl(2)1522 5145 y Fp(:=)1678 5005 y Fi(\022)1793 5085 y Fp(\(1)f(+)i Fo(j)p Fs(x)p Fo(j)p Fp(\))2149 5049 y Fl(3)p Fr(=)p Fl(2)2341 5085 y Fp(0)2001 5206 y(0)291 b(1)2432 5005 y Fi(\023)2527 5145 y Fo(\012)23 b Fg(1)f Fs(;)480 b Fv(\(IV)-13 b(.15\))257 5388 y(we)25 b(pro)o(v)o(e)f(the)h(follo)n(wing)e(lemma.)p eop %%Page: 31 32 31 31 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(31)257 573 y Fk(Lemma)26 b(IV)-14 b(.3.)48 b Fq(F)-10 b(or)28 b(any)h Fs(N)46 b Fo(2)35 b Fh(N)44 b Fq(and)28 b Fs(K)43 b(>)35 b Fp(1)p Fq(,)29 b(ther)l(e)g(e)n(xists)f(a)h (constant)f Fs(C)42 b(<)35 b Fo(1)28 b Fq(suc)o(h)257 693 y(that)461 829 y Fi(\015)461 888 y(\015)516 913 y Fp(\(1)22 b(+)g Fo(j)p Fs(x)p Fo(j)p Fp(\))872 872 y Fl(3)p Fr(=)p Fl(2)998 913 y Fp([)p Fs(W)31 b(;)i(A)1281 928 y Fr(s)1318 913 y Fp(])17 b(\()p Fs(H)1489 862 y Fl(\(\003\))1481 941 y Fr(f)1618 913 y Fp(+)22 b Fg(1)p Fp(\))1810 872 y Ff(\000)p Fl(1)p Ff(\000)p Fl(\()p Fr(N)q(=)p Fl(2\))2142 829 y Fi(\015)2142 888 y(\015)2250 913 y Fo(\024)53 b Fs(C)24 b Fp(\(1)d(+)h Fo(j)p Fs(s)p Fo(j)p Fp(\))2820 872 y Ff(\000)p Fr(K)2965 913 y Fs(;)220 b Fv(\(IV)-13 b(.16\))257 1133 y Fq(for)25 b(all)f Fs(s)j Fo(2)h Fh(R)5 b Fq(.)257 1338 y(Pr)l(oof)o(.)47 b Fv(First)22 b(observ)o(e)f(that)h(on)g Fp(dom)o(\()p Fs(H)1692 1287 y Fr(N)q(=)p Fl(2)1684 1365 y Fr(f)1824 1338 y Fp(\))p Fv(,)g(we)g(ha)n(v)o(e)g Fs(e)2299 1301 y Ff(\000)p Fr(isH)2469 1310 y Fd(0)2508 1338 y Fs(a)2559 1301 y Ff(\003)2598 1338 y Fp(\()p Fs(f)11 b Fp(\))p Fs(e)2778 1301 y Fr(isH)2893 1310 y Fd(0)2959 1338 y Fp(=)28 b Fs(a)3114 1301 y Ff(\003)3153 1338 y Fp(\()p Fs(e)3236 1301 y Ff(\000)p Fr(is!)3399 1338 y Fs(f)11 b Fp(\))257 1458 y Fv(and)25 b(hence)917 1678 y Fs(A)990 1693 y Fr(s)1080 1678 y Fp(=)52 b Fs(a)1259 1637 y Ff(\003)1299 1678 y Fp(\()p Fs(e)1382 1637 y Ff(\000)p Fr(is!)1544 1678 y Fs(f)1592 1693 y Fl(1)1632 1678 y Fp(\))17 b Fs(a)1738 1637 y Ff(\003)1777 1678 y Fp(\()p Fs(e)1860 1637 y Ff(\000)p Fr(is!)2022 1678 y Fs(f)2070 1693 y Fl(2)2110 1678 y Fp(\))g Fo(\001)g(\001)g(\001)d Fs(a)2348 1637 y Ff(\003)2388 1678 y Fp(\()p Fs(e)2471 1637 y Ff(\000)p Fr(is!)2633 1678 y Fs(f)2681 1693 y Fr(N)2748 1678 y Fp(\))22 b Fs(:)377 b Fv(\(IV)-13 b(.17\))257 1898 y(Writing)28 b Fs(W)48 b Fp(=)35 b Fs(a)898 1862 y Ff(\003)938 1898 y Fp(\()p Fs(G)p Fp(\))24 b(+)h Fs(a)p Fp(\()p Fs(G)p Fp(\))k Fv(and)f(using)g(the)g(canonical)h(commutation)d (relations,)j(we)257 2018 y(thus)24 b(ha)n(v)o(e)413 2238 y Fp([)p Fs(W)30 b(;)k(A)696 2253 y Fr(s)732 2238 y Fp(])84 b(=)1001 2158 y Fi(\002)1043 2238 y Fs(a)1094 2197 y Ff(\003)1134 2238 y Fp(\()p Fs(G)p Fp(\))22 b(+)g Fs(a)p Fp(\()p Fs(G)p Fp(\))j Fs(;)41 b(a)1755 2197 y Ff(\003)1795 2238 y Fp(\()p Fs(e)1878 2197 y Ff(\000)p Fr(is!)2040 2238 y Fs(f)2088 2253 y Fl(1)2127 2238 y Fp(\))17 b Fs(a)2233 2197 y Ff(\003)2273 2238 y Fp(\()p Fs(e)2356 2197 y Ff(\000)p Fr(is!)2518 2238 y Fs(f)2566 2253 y Fl(2)2605 2238 y Fp(\))g Fo(\001)g(\001)g(\001)e Fs(a)2844 2197 y Ff(\003)2883 2238 y Fp(\()p Fs(e)2966 2197 y Ff(\000)p Fr(is!)3128 2238 y Fs(f)3176 2253 y Fr(N)3244 2238 y Fp(\))3299 2158 y Fi(\003)843 2384 y Fp(=)1001 2303 y Fi(\002)1043 2384 y Fs(a)p Fp(\()p Fs(G)p Fp(\))25 b Fs(;)42 b(a)1392 2343 y Ff(\003)1431 2384 y Fp(\()p Fs(e)1514 2343 y Ff(\000)p Fr(is!)1676 2384 y Fs(f)1724 2399 y Fl(1)1764 2384 y Fp(\))17 b Fs(a)1870 2343 y Ff(\003)1909 2384 y Fp(\()p Fs(e)1992 2343 y Ff(\000)p Fr(is!)2154 2384 y Fs(f)2202 2399 y Fl(2)2242 2384 y Fp(\))g Fo(\001)g(\001)g(\001)d Fs(a)2480 2343 y Ff(\003)2520 2384 y Fp(\()p Fs(e)2603 2343 y Ff(\000)p Fr(is!)2765 2384 y Fs(f)2813 2399 y Fr(N)2880 2384 y Fp(\))2935 2303 y Fi(\003)3212 2384 y Fv(\(IV)-13 b(.18\))843 2632 y Fp(=)1042 2507 y Fr(N)1001 2537 y Fi(X)1012 2747 y Fr(j)t Fl(=1)1162 2491 y Fi(\032)1237 2551 y(\012)1284 2632 y Fs(G)1361 2547 y Fi(\014)1361 2607 y(\014)1394 2632 y Fs(e)1439 2590 y Ff(\000)p Fr(is!)1601 2632 y Fs(f)1649 2647 y Fr(j)1686 2551 y Fi(\013)1755 2632 y Fo(\012)1892 2507 y Fr(N)1860 2537 y Fi(Y)1864 2736 y Fc(i)p Fd(=1)p Fc(;)1875 2792 y(i)p Fb(6)p Fd(=)p Fc(j)2009 2632 y Fs(a)2060 2590 y Ff(\003)2100 2632 y Fp(\()p Fs(e)2183 2590 y Ff(\000)p Fr(is!)2345 2632 y Fs(f)2393 2647 y Fr(i)2421 2632 y Fp(\))2459 2491 y Fi(\033)2556 2632 y Fs(;)257 3000 y Fv(where)33 b Fo(h)p Fs(G)p Fo(j)p Fs(e)722 2964 y Ff(\000)p Fr(is!)884 3000 y Fs(f)932 3015 y Fr(j)968 3000 y Fo(i)42 b(2)h(B)s Fp(\()p Fo(H)1348 3015 y Fr(el)1407 3000 y Fp(\))32 b Fv(is)g(a)h(bounded)f(operator)g(acting)h(on)f(the)g (electron)h(v)n(ari-)257 3120 y(ables)25 b(gi)n(v)o(en)e(by)958 3305 y Fi(\012)1005 3386 y Fs(G)1082 3301 y Fi(\014)1082 3361 y(\014)1115 3386 y Fs(e)1160 3344 y Ff(\000)p Fr(is!)1323 3386 y Fs(f)1371 3401 y Fr(j)1407 3305 y Fi(\013)1507 3386 y Fp(:=)1662 3250 y Fi(Z)1779 3386 y Fs(d)1830 3344 y Fl(3)1869 3386 y Fs(k)i(e)1990 3344 y Ff(\000)p Fr(is!)r Fl(\()p Fr(k)r Fl(\))2268 3386 y Fs(f)2316 3401 y Fr(j)2352 3386 y Fp(\()p Fs(k)s Fp(\))17 b Fs(G)2576 3344 y Ff(\003)2615 3386 y Fp(\()p Fs(k)s Fp(\))22 b Fs(:)418 b Fv(\(IV)-13 b(.19\))257 3654 y(Note)40 b(that)f Fs(k)58 b Fo(7!)c Fs(f)991 3669 y Fr(j)1028 3654 y Fp(\()p Fs(k)s Fp(\))p Fs(G)1235 3618 y Ff(\003)1274 3654 y Fp(\()p Fs(k)s Fp(\))40 b Fv(is)f Fs(K)62 b(>)55 b Fp(1)39 b Fv(times)g(continuously)e(dif)n (ferentiable,)42 b(and,)257 3774 y(thanks)e(to)g(the)g(support)g (properties)g(of)g Fs(f)1788 3789 y Fr(j)1825 3774 y Fv(,)k(there)d(e)o(xists)e Fs(r)2444 3789 y Fr(j)2537 3774 y Fs(>)56 b Fp(0)40 b Fv(such)g(that)g Fs(f)3219 3789 y Fr(j)3256 3774 y Fs(G)3333 3738 y Ff(\003)3429 3774 y Fo(2)257 3895 y Fs(C)334 3859 y Fr(K)403 3895 y Fp([)p Fs(B)504 3910 y Fl(\003)590 3895 y Fo(n)33 b Fs(B)747 3910 y Fr(r)779 3920 y Fc(j)815 3895 y Fp(;)17 b Fo(B)s Fp(\()p Fo(H)1049 3910 y Fr(el)1108 3895 y Fp(\)])40 b Fv(has)f(compact)g(support)f(a)o(w)o(ay)h(from)g(zero.)75 b(Thus,)42 b(thanks)d(to)257 4015 y(Hypothesis)32 b(1,)k Fs(K)41 b Fv(times)32 b(\()p Fo(B)s Fp(\()p Fo(H)1446 4030 y Fr(el)1505 4015 y Fp(\))p Fv(-v)n(alued\))h(inte)o(gration)f(by) h(parts)h(yields)e(the)i(standard)257 4136 y(estimate)24 b(for)h(oscillatory)f(inte)o(grals,)618 4265 y Fi(\015)618 4325 y(\015)618 4385 y(\015)673 4380 y Fp(\(1)e(+)g Fo(j)p Fs(x)p Fo(j)p Fp(\))1029 4338 y Fl(3)p Fr(=)p Fl(2)1156 4299 y Fi(\012)1203 4380 y Fs(G)1280 4295 y Fi(\014)1280 4355 y(\014)1313 4380 y Fs(e)1358 4338 y Ff(\000)p Fr(is!)1520 4380 y Fs(f)1568 4395 y Fr(j)1605 4299 y Fi(\013)1668 4265 y(\015)1668 4325 y(\015)1668 4385 y(\015)3212 4380 y Fv(\(IV)-13 b(.20\))701 4613 y Fo(\024)84 b(j)p Fs(s)p Fo(j)964 4572 y Ff(\000)p Fr(K)1119 4478 y Fi(Z)1236 4613 y Fs(d)1287 4572 y Fl(3)1326 4613 y Fs(k)1396 4499 y Fi(\015)1396 4558 y(\015)1396 4618 y(\015)1452 4613 y Fp(\(1)22 b(+)g Fo(j)p Fs(x)p Fo(j)p Fp(\))1808 4572 y Fl(3)p Fr(=)p Fl(2)1934 4532 y Fi(\000)1980 4613 y Fo(r)2063 4628 y Fr(k)2127 4613 y Fo(\001)g Fp(\()p Fs(k)s(=)p Fo(j)p Fs(k)s Fo(j)p Fp(\))2466 4532 y Fi(\001)2511 4555 y Fr(K)2602 4613 y Fs(f)2650 4628 y Fr(j)2686 4613 y Fp(\()p Fs(k)s Fp(\))17 b Fs(G)2910 4572 y Ff(\003)2949 4613 y Fp(\()p Fs(k)s Fp(\))3079 4499 y Fi(\015)3079 4558 y(\015)3079 4618 y(\015)701 4819 y Fo(\024)84 b(O)944 4738 y Fi(\000)989 4819 y Fp(\(1)22 b(+)g Fo(j)p Fs(s)p Fo(j)p Fp(\))1336 4778 y Ff(\000)p Fr(K)1459 4738 y Fi(\001)1527 4819 y Fs(;)1658 b Fv(\(IV)-13 b(.21\))257 5039 y(pro)o(vided)28 b Fo(j)p Fs(s)p Fo(j)34 b(\025)i Fp(1)p Fv(.)42 b(F)o(or)29 b Fo(j)p Fs(s)p Fo(j)34 b(\024)i Fp(1)p Fv(,)29 b(Estimate)f(\(IV)-13 b(.20\))29 b(is)f(tri)n(vial)g(and)h(hence)g(holds)f(for)h(all)257 5159 y Fs(s)k Fo(2)g Fh(R)5 b Fv(.)45 b(Inserting)27 b(\(IV)-13 b(.20\))27 b(into)g(\(IV)-13 b(.18\))28 b(and)f(undoing)g (the)g(free)h(time)f(e)n(v)n(olution)f(\(which)p eop %%Page: 32 33 32 32 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(32)257 574 y(is)25 b(possible)e(because)i Fs(H)1121 589 y Fl(0)1186 574 y Fv(commutes)e(with)h Fs(H)1911 523 y Fl(\(\003\))1903 602 y Fr(f)2019 574 y Fv(\),)h(we)g(obtain)498 739 y Fi(\015)498 798 y(\015)570 823 y Fp(\(1)d(+)g Fo(j)p Fs(x)p Fo(j)p Fp(\))926 782 y Fl(3)p Fr(=)p Fl(2)1053 823 y Fp([)p Fs(W)30 b(;)j(A)1335 838 y Fr(s)1372 823 y Fp(])17 b(\()p Fs(H)1543 772 y Fl(\(\003\))1535 851 y Fr(f)1672 823 y Fp(+)22 b Fg(1)p Fp(\))1864 782 y Ff(\000)p Fl(1)p Ff(\000)p Fl(\()p Fr(N)q(=)p Fl(2\))2196 739 y Fi(\015)2196 798 y(\015)3212 823 y Fv(\(IV)-13 b(.22\))582 1077 y Fo(\024)83 b(O)824 996 y Fi(\000)870 1077 y Fp(\(1)22 b(+)g Fo(j)p Fs(s)p Fo(j)p Fp(\))1217 1036 y Ff(\000)p Fr(K)1339 996 y Fi(\001)1458 953 y Fr(N)1418 982 y Fi(X)1429 1192 y Fr(j)t Fl(=1)1578 933 y Fi(\015)1578 992 y(\015)1578 1052 y(\015)1578 1112 y(\015)1634 937 y(\022)1761 953 y Fr(N)1729 982 y Fi(Y)1734 1181 y Fc(i)p Fd(=1)p Fc(;)1744 1238 y(i)p Fb(6)p Fd(=)p Fc(j)1878 1077 y Fs(a)1929 1036 y Ff(\003)1969 1077 y Fp(\()p Fs(e)2052 1036 y Ff(\000)p Fr(is!)2214 1077 y Fs(f)2262 1092 y Fr(i)2290 1077 y Fp(\))2328 937 y Fi(\023)2418 1077 y Fp(\()p Fs(H)2545 1026 y Fl(\(\003\))2537 1105 y Fr(f)2675 1077 y Fp(+)g Fg(1)p Fp(\))2867 1036 y Ff(\000)p Fl(1)p Ff(\000)p Fl(\()p Fr(N)q(=)p Fl(2\))3199 933 y Fi(\015)3199 992 y(\015)3199 1052 y(\015)3199 1112 y(\015)582 1471 y Fp(=)84 b Fo(O)824 1390 y Fi(\000)870 1471 y Fp(\(1)22 b(+)g Fo(j)p Fs(s)p Fo(j)p Fp(\))1217 1429 y Ff(\000)p Fr(K)1339 1390 y Fi(\001)1458 1346 y Fr(N)1418 1376 y Fi(X)1429 1586 y Fr(j)t Fl(=1)1578 1326 y Fi(\015)1578 1386 y(\015)1578 1446 y(\015)1578 1505 y(\015)1634 1330 y(\022)1761 1346 y Fr(N)1729 1376 y Fi(Y)1734 1575 y Fc(i)p Fd(=1)p Fc(;)1744 1631 y(i)p Fb(6)p Fd(=)p Fc(j)1878 1471 y Fs(a)1929 1429 y Ff(\003)1969 1471 y Fp(\()p Fs(f)2055 1486 y Fr(i)2083 1471 y Fp(\))2121 1330 y Fi(\023)2211 1471 y Fp(\()p Fs(H)2338 1420 y Fl(\(\003\))2330 1498 y Fr(f)2468 1471 y Fp(+)g Fg(1)p Fp(\))2660 1429 y Ff(\000)p Fl(1)p Ff(\000)p Fl(\()p Fr(N)q(=)p Fl(2\))2992 1326 y Fi(\015)2992 1386 y(\015)2992 1446 y(\015)2992 1505 y(\015)3069 1471 y Fs(:)257 1851 y Fv(T)-8 b(o)24 b(estimate)g(a)g(product)g(of)g Fs(N)31 b Fo(\000)20 b Fp(1)k Fv(creation)g(operators)g Fs(a)2315 1815 y Ff(\003)2355 1851 y Fp(\()2414 1825 y(~)2393 1851 y Fs(f)2441 1866 y Fl(1)2480 1851 y Fp(\))p Fs(a)2569 1815 y Ff(\003)2609 1851 y Fp(\()2668 1825 y(~)2647 1851 y Fs(f)2695 1866 y Fl(2)2734 1851 y Fp(\))17 b Fo(\001)g(\001)g(\001)e Fs(a)2973 1815 y Ff(\003)3012 1851 y Fp(\()3072 1825 y(~)3050 1851 y Fs(f)3098 1866 y Fr(N)7 b Ff(\000)p Fl(1)3256 1851 y Fp(\))24 b Fv(with)279 1946 y Fp(~)257 1972 y Fs(f)305 1987 y Fl(1)345 1972 y Fs(;)410 1946 y Fp(~)389 1972 y Fs(f)437 1987 y Fl(2)476 1972 y Fs(;)17 b(:)g(:)g(:)33 b(;)732 1946 y Fp(~)712 1972 y Fs(f)760 1987 y Fr(N)871 1972 y Fo(2)45 b Fs(C)1059 1936 y Ff(1)1052 1996 y Fl(0)1133 1972 y Fp(\()p Fs(B)1245 1987 y Fl(\003)1320 1972 y Fo(n)22 b(f)p Fp(0)p Fo(g)p Fp(\))p Fv(,)i(we)h(use)f(Estimate)g(\(IV)-13 b(.4\))25 b Fs(N)32 b Fo(\000)22 b Fp(1)j Fv(times)f(and)g(deri)n(v)o (e)372 2140 y Fi(\015)372 2200 y(\015)372 2260 y(\015)372 2319 y(\015)427 2144 y(\022)517 2160 y Fr(N)7 b Ff(\000)p Fl(1)530 2190 y Fi(Y)527 2399 y Fr(n)p Fl(=1)687 2285 y Fs(a)738 2243 y Ff(\003)778 2285 y Fp(\()837 2258 y(~)816 2285 y Fs(f)864 2300 y Fr(n)911 2285 y Fp(\))949 2144 y Fi(\023)1038 2285 y Fp(\()p Fs(H)1165 2234 y Fl(\(\003\))1157 2312 y Fr(f)1295 2285 y Fp(+)22 b Fg(1)p Fp(\))1487 2243 y Ff(\000)p Fl(\()p Fr(N)7 b Ff(\000)p Fl(1\))p Fr(=)p Fl(2\))1852 2140 y Fi(\015)1852 2200 y(\015)1852 2260 y(\015)1852 2319 y(\015)3212 2285 y Fv(\(IV)-13 b(.23\))456 2622 y Fp(=)615 2478 y Fi(\015)615 2537 y(\015)615 2597 y(\015)615 2657 y(\015)687 2498 y Fr(N)7 b Ff(\000)p Fl(1)700 2528 y Fi(Y)697 2737 y Fr(n)p Fl(=1)857 2482 y Fi(\022)931 2622 y Fp(\()p Fs(H)1058 2571 y Fl(\(\003\))1050 2650 y Fr(f)1187 2622 y Fp(+)22 b Fg(1)p Fp(\))1379 2581 y Fl(\()p Fr(n)p Ff(\000)p Fl(1\))p Fr(=)p Fl(2)1642 2622 y Fs(a)1693 2581 y Ff(\003)1732 2622 y Fp(\()1791 2596 y(~)1770 2622 y Fs(f)1818 2637 y Fr(n)1865 2622 y Fp(\))17 b(\()p Fs(H)2047 2571 y Fl(\(\003\))2039 2650 y Fr(f)2176 2622 y Fp(+)22 b Fg(1)p Fp(\))2368 2581 y Ff(\000)p Fr(n=)p Fl(2)2541 2482 y Fi(\023)2614 2478 y(\015)2614 2537 y(\015)2614 2597 y(\015)2614 2657 y(\015)455 2960 y Fo(\024)615 2835 y Fr(N)7 b Ff(\000)p Fl(1)628 2865 y Fi(Y)626 3074 y Fr(n)p Fl(=1)785 2845 y Fi(\015)785 2905 y(\015)785 2965 y(\015)841 2960 y Fp(\()p Fs(H)968 2909 y Fl(\(\003\))960 2988 y Fr(f)1097 2960 y Fp(+)22 b Fg(1)p Fp(\))1289 2919 y Fl(\()p Fr(n)p Ff(\000)p Fl(1\))p Fr(=)p Fl(2)1552 2960 y Fs(a)1603 2919 y Ff(\003)1642 2960 y Fp(\()1702 2933 y(~)1680 2960 y Fs(f)1728 2975 y Fr(n)1775 2960 y Fp(\))17 b(\()p Fs(H)1957 2909 y Fl(\(\003\))1949 2988 y Fr(f)2086 2960 y Fp(+)22 b Fg(1)p Fp(\))2278 2919 y Ff(\000)p Fr(n=)p Fl(2)2451 2845 y Fi(\015)2451 2905 y(\015)2451 2965 y(\015)2559 2960 y Fo(\024)53 b(O)2771 2879 y Fi(\000)2817 2960 y Fp(\(1)21 b(+)h(\003\))3129 2919 y Fl(\()p Fr(N)3219 2895 y Fd(2)3254 2919 y Fl(\))3286 2879 y Fi(\001)3354 2960 y Fs(:)257 3260 y Fv(Inserting)i(this)g(bound) g(into)g(\(IV)-13 b(.22\))25 b(yields)f(\(IV)-13 b(.16\),)25 b(since)g Fp(\003)i Fs(<)h Fo(1)c Fv(is)h(bounded.)241 b Fa(\003)404 3381 y Fv(W)-8 b(e)34 b(\002nally)f(turn)g(to)g(pro)o (ving)f(\(II.11\).)57 b(Note)33 b(that)g(the)g(follo)n(wing)f(Lemma)h (uses)g Fs(H)3450 3396 y Fr(f)257 3518 y Fv(rather)26 b(than)e Fs(H)800 3467 y Fl(\(\003\))792 3545 y Fr(f)908 3518 y Fv(.)257 3688 y Fk(Lemma)i(IV)-14 b(.4.)48 b Fq(Ther)l(e)40 b(e)n(xists)d(a)h(constant)f Fs(C)60 b(<)52 b Fo(1)38 b Fq(suc)o(h)g(that,)i(for)e(any)g Fs(r)55 b Fo(2)e Fh(R)d Fq(and)257 3808 y Fs(s)28 b Fo(2)g Fh(R)491 3767 y Fl(+)491 3833 y(0)556 3808 y Fq(,)416 3943 y Fi(\015)416 4003 y(\015)488 4028 y Fs(W)i(e)655 3987 y Ff(\000)p Fr(ir)r(H)826 3996 y Fd(0)882 4028 y Fs(P)945 4043 y Fr(c)996 4028 y Fp([)p Fs(W)g(;)k(A)1279 4043 y Fr(s)1316 4028 y Fp(])17 b(\()p Fs(H)1479 4043 y Fr(f)1545 4028 y Fp(+)22 b Fg(1)p Fp(\))1737 3987 y Ff(\000)p Fl(2)p Ff(\000)p Fl(\()p Fr(N)q(=)p Fl(2\))2069 3943 y Fi(\015)2069 4003 y(\015)2177 4028 y Fo(\024)53 b Fs(C)2401 3947 y Fi(\000)2447 4028 y Fp(\(1)21 b(+)h Fo(j)p Fs(r)s Fo(j)p Fp(\))2794 3987 y Ff(\000)p Fl(3)p Fr(=)p Fl(2)2958 3947 y Fi(\001)3026 4028 y Fs(:)159 b Fv(\(IV)-13 b(.24\))257 4258 y Fq(Pr)l(oof)o(.)47 b Fv(The)30 b(proof)g(is)g(based)g(on)g(the)g(decay)h(of)f(the)g(Schw)o (artz)h(k)o(ernel)f Fs(e)2870 4222 y Ff(\000)p Fr(ir)r Fl(\()p Ff(\000)p Fl(\001\))3156 4258 y Fp(\()p Fs(x;)17 b(y)t Fp(\))36 b(=)257 4379 y(\(4)p Fs(\031)t(ir)s Fp(\))521 4342 y Ff(\000)p Fl(3)p Fr(=)p Fl(2)719 4379 y Fp(exp)q([)p Fs(i)p Fp(\()p Fs(x)29 b Fo(\000)g Fs(y)t Fp(\))1246 4342 y Fl(2)1285 4379 y Fp(\(4)p Fs(r)s Fp(\))1457 4342 y Ff(\000)p Fl(1)1550 4379 y Fp(])34 b Fv(of)f(the)h(propagator)e Fs(e)2392 4342 y Ff(\000)p Fr(ir)r Fl(\()p Ff(\000)p Fl(\001\))2711 4379 y Fv(of)i(the)f(free)i(particle.)257 4499 y(More)25 b(precisely)-6 b(,)24 b(de\002ning)h Fs(C)1323 4514 y Fl(1)1389 4499 y Fp(:=)j(\(4)p Fs(\031)t Fp(\))1704 4463 y Ff(\000)p Fl(1)1815 4419 y Fi(R)1881 4499 y Fp(\(1)22 b(+)g Fo(j)p Fs(x)p Fo(j)p Fp(\))2237 4463 y Fl(2)p Fr(\015)2316 4499 y Fs(d)2367 4463 y Fl(3)2406 4499 y Fs(x)28 b(<)g Fo(1)p Fv(,)c(we)i(observ)o(e)e(that)693 4644 y Fi(\014)693 4703 y(\014)726 4648 y(\012)773 4728 y Fp(\(1)e(+)g Fo(j)p Fs(x)p Fo(j)p Fp(\))1129 4687 y Ff(\000)p Fr(\015)1228 4728 y Fs(')1320 4644 y Fi(\014)1320 4703 y(\014)1378 4728 y Fs(e)1423 4687 y Ff(\000)p Fr(ir)r Fl(\()p Ff(\000)p Fl(\001\))1725 4728 y Fp(\(1)g(+)g Fo(j)p Fs(x)p Fo(j)p Fp(\))2081 4687 y Ff(\000)p Fr(\015)2180 4728 y Fs( )2247 4648 y Fi(\013)2294 4644 y(\014)2294 4703 y(\014)3212 4728 y Fv(\(IV)-13 b(.25\))777 4958 y Fp(=)1101 4891 y(1)p 947 4935 357 4 v 947 5027 a(\(4)p Fs(\031)20 b(r)s Fp(\))1194 4998 y Fl(3)p Fr(=)p Fl(2)1314 4814 y Fi(\014)1314 4874 y(\014)1314 4933 y(\014)1314 4993 y(\014)1363 4823 y(Z)1480 4958 y Fs(d)1531 4917 y Fl(3)1570 4958 y Fs(x)d(d)1693 4917 y Fl(3)1732 4958 y Fs(y)p 1810 4804 196 4 v 1810 4891 a(')p Fp(\()p Fs(x)p Fp(\))22 b Fs(e)2072 4855 y Fr(i)p Fl(\()p Fr(x)p Ff(\000)p Fr(y)r Fl(\))2282 4831 y Fd(2)2318 4855 y Fr(=)p Fl(\(4)p Fr(r)r Fl(\))2503 4891 y Fs( )t Fp(\()p Fs(y)t Fp(\))p 1810 4935 888 4 v 1842 5027 a(\(1)f(+)h Fo(j)p Fs(x)p Fo(j)p Fp(\))2197 4998 y Fr(\015)2269 5027 y Fp(\(1)g(+)g Fo(j)p Fs(y)t Fo(j)p Fp(\))2622 4998 y Fr(\015)2707 4814 y Fi(\014)2707 4874 y(\014)2707 4933 y(\014)2707 4993 y(\014)776 5233 y Fo(\024)947 5165 y(k)p Fs(')p Fo(k)27 b(k)p Fs( )t Fo(k)p 947 5210 358 4 v 947 5301 a Fp(\(4)p Fs(\031)20 b(r)s Fp(\))1194 5272 y Fl(3)p Fr(=)p Fl(2)1337 5092 y Fi(\022)1427 5097 y(Z)1698 5165 y Fs(d)1749 5129 y Fl(3)1788 5165 y Fs(x)p 1553 5210 436 4 v 1553 5301 a Fp(\(1)i(+)g Fo(j)p Fs(x)p Fo(j)p Fp(\))1909 5272 y Fl(2)p Fr(\015)1998 5092 y Fi(\023)2152 5233 y Fo(\024)80 b Fs(C)2379 5248 y Fl(1)2435 5233 y Fo(k)p Fs(')p Fo(k)17 b(k)p Fs( )t Fo(k)g Fs(r)2847 5191 y Ff(\000)p Fl(3)p Fr(=)p Fl(2)3032 5233 y Fs(;)p eop %%Page: 33 34 33 33 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(33)257 573 y(for)25 b(an)o(y)g Fs(';)17 b( )31 b Fo(2)d Fs(L)928 537 y Fl(1)990 573 y Fo(\\)23 b Fs(L)1145 537 y Fl(2)1184 573 y Fp(\()p Fh(R)1288 537 y Fl(3)1334 573 y Fp(\))p Fv(,)h(since)h Fs(\015)33 b(>)27 b Fp(3)p Fs(=)p Fp(2)p Fv(.)j(This)24 b(estimate)g(yields)407 708 y Fi(\015)407 768 y(\015)462 793 y Fp(\(1)e(+)g Fo(j)p Fs(x)p Fo(j)p Fp(\))818 752 y Ff(\000)p Fr(\015)934 793 y Fs(e)979 752 y Ff(\000)p Fr(ir)r(H)1150 761 y Fd(0)1205 793 y Fs(P)1268 808 y Fr(c)1319 793 y Fp(\(1)g(+)g Fo(j)p Fs(x)p Fo(j)p Fp(\))1675 752 y Ff(\000)p Fr(\015)1774 708 y Fi(\015)1774 768 y(\015)3212 793 y Fv(\(IV)-13 b(.26\))490 1007 y Fp(=)649 863 y Fi(\015)649 922 y(\015)649 982 y(\015)649 1042 y(\015)704 867 y(\022)819 947 y Fp(\(1)22 b(+)g Fo(j)p Fs(x)p Fo(j)p Fp(\))1175 911 y Ff(\000)p Fr(\015)1291 947 y Fs(e)1336 911 y Ff(\000)p Fr(ir)r Fl(\()p Ff(\000)p Fl(\001\))1638 947 y Fp(\(1)g(+)g Fo(j)p Fs(x)p Fo(j)p Fp(\))1994 911 y Ff(\000)p Fr(\015)2176 947 y Fp(0)1432 1067 y(0)695 b(0)2266 867 y Fi(\023)2362 1007 y Fo(\012)22 b Fs(e)2506 966 y Ff(\000)p Fr(ir)r(H)2677 978 y Fc(f)2721 863 y Fi(\015)2721 922 y(\015)2721 982 y(\015)2721 1042 y(\015)2829 1007 y Fo(\024)53 b Fs(C)3029 1022 y Fl(1)3085 1007 y Fs(r)3132 966 y Ff(\000)p Fl(3)p Fr(=)p Fl(2)3319 1007 y Fs(:)257 1279 y Fv(Inserting)24 b(\(IV)-13 b(.26\))25 b(into)f(\(IV)-13 b(.24\))25 b(and)g(using)f (that)g Fs(H)2097 1294 y Fr(f)2167 1279 y Fv(and)h Fs(H)2417 1294 y Fl(0)2481 1279 y Fv(commute,)f(we)h(obtain)370 1415 y Fi(\015)370 1475 y(\015)442 1499 y Fs(W)31 b(e)610 1458 y Ff(\000)p Fr(ir)r(H)781 1467 y Fd(0)836 1499 y Fs(P)899 1514 y Fr(c)950 1499 y Fp([)p Fs(W)f(;)k(A)1233 1514 y Fr(s)1270 1499 y Fp(])17 b(\()p Fs(H)1433 1514 y Fr(f)1499 1499 y Fp(+)22 b Fg(1)p Fp(\))1691 1458 y Ff(\000)p Fl(2)p Ff(\000)p Fl(\()p Fr(N)q(=)p Fl(2\))2023 1415 y Fi(\015)2023 1475 y(\015)3212 1499 y Fv(\(IV)-13 b(.27\))453 1658 y Fo(\024)614 1574 y Fi(\015)614 1634 y(\015)669 1658 y Fs(W)30 b Fp(\(1)22 b(+)g Fo(j)p Fs(x)p Fo(j)p Fp(\))1147 1617 y Fr(\015)1214 1658 y Fo(\012)g Fp(\()p Fs(H)1432 1673 y Fr(f)1499 1658 y Fp(+)g(1\))1684 1617 y Ff(\000)p Fl(1)p Fr(=)p Fl(2)1849 1574 y Fi(\015)1849 1634 y(\015)1932 1574 y(\015)1932 1634 y(\015)1987 1658 y Fp(\(1)g(+)g Fo(j)p Fs(x)p Fo(j)p Fp(\))2343 1617 y Ff(\000)p Fr(\015)2459 1658 y Fs(e)2504 1617 y Ff(\000)p Fr(ir)r(H)2675 1626 y Fd(0)2730 1658 y Fs(P)2793 1673 y Fr(c)2844 1658 y Fp(\(1)g(+)g Fo(j)p Fs(x)p Fo(j)p Fp(\))3200 1617 y Ff(\000)p Fr(\015)3299 1574 y Fi(\015)3299 1634 y(\015)1086 1733 y(\015)1086 1793 y(\015)1142 1817 y Fp(\(1)f(+)h Fo(j)p Fs(x)p Fo(j)p Fp(\))1497 1776 y Fr(\015)1564 1817 y Fo(\012)g Fp(\()p Fs(H)1782 1832 y Fr(f)1850 1817 y Fp(+)g(1\))2035 1776 y Fl(1)p Fr(=)p Fl(2)2161 1817 y Fp([)p Fs(W)30 b(;)k(A)2444 1832 y Fr(s)2481 1817 y Fp(])17 b(\()p Fs(H)2644 1832 y Fr(f)2711 1817 y Fp(+)22 b(1\))2896 1776 y Ff(\000)p Fl(2)p Ff(\000)p Fl(\()p Fr(N)q(=)p Fl(2\))3227 1733 y Fi(\015)3227 1793 y(\015)453 1976 y Fo(\024)84 b Fs(C)684 1991 y Fl(1)740 1976 y Fs(r)787 1935 y Ff(\000)p Fl(3)p Fr(=)p Fl(2)979 1892 y Fi(\015)979 1952 y(\015)1034 1976 y Fs(W)30 b Fp(\(1)22 b(+)g Fo(j)p Fs(x)p Fo(j)p Fp(\))1512 1935 y Fr(\015)1579 1976 y Fo(\012)g Fp(\()p Fs(H)1797 1991 y Fr(f)1865 1976 y Fp(+)g(1\))2050 1935 y Ff(\000)p Fl(1)p Fr(=)p Fl(2)2214 1892 y Fi(\015)2214 1952 y(\015)1086 2051 y(\015)1086 2111 y(\015)1142 2136 y Fp(\(1)f(+)h Fo(j)p Fs(x)p Fo(j)p Fp(\))1497 2094 y Fr(\015)1564 2136 y Fo(\012)g Fp(\()p Fs(H)1782 2151 y Fr(f)1850 2136 y Fp(+)g(1\))2035 2094 y Fl(1)p Fr(=)p Fl(2)2161 2136 y Fp([)p Fs(W)30 b(;)k(A)2444 2151 y Fr(s)2481 2136 y Fp(])17 b(\()p Fs(H)2644 2151 y Fr(f)2711 2136 y Fp(+)22 b(1\))2896 2094 y Ff(\000)p Fl(2)p Ff(\000)p Fl(\()p Fr(N)q(=)p Fl(2\))3227 2051 y Fi(\015)3227 2111 y(\015)3305 2136 y Fs(:)257 2364 y Fv(Ne)o(xt,)e(we)f(use)g(\(IV)-13 b(.18\))20 b(and)f(the)g(pull-through)e(formula)i(to)f(commute)g Fp(\()p Fs(H)2846 2379 y Fr(f)2892 2364 y Fp(+)q(1\))3056 2328 y Fl(1)p Fr(=)p Fl(2)3185 2364 y Fv(through)257 2485 y Fp([)p Fs(W)31 b(;)i(A)540 2500 y Fr(s)577 2485 y Fp(])p Fv(,)25 b(observing)f(that)g Fp(supp)q Fs(f)1495 2500 y Fr(j)1559 2485 y Fo(\022)29 b Fs(B)1739 2500 y Fl(\003)1792 2485 y Fp(\(0\))p Fv(.)h(This)24 b(yields)448 2620 y Fi(\015)448 2680 y(\015)448 2740 y(\015)504 2735 y Fp(\(1)d(+)h Fo(j)p Fs(x)p Fo(j)p Fp(\))859 2694 y Fr(\015)926 2735 y Fo(\012)g Fp(\()p Fs(H)1144 2750 y Fr(f)1212 2735 y Fp(+)g(1\))1397 2694 y Fl(1)p Fr(=)p Fl(2)1523 2735 y Fp([)p Fs(W)30 b(;)k(A)1806 2750 y Fr(s)1842 2735 y Fp(])17 b(\()p Fs(H)2005 2750 y Fr(f)2072 2735 y Fp(+)22 b(1\))2257 2694 y Ff(\000)p Fl(2)p Ff(\000)p Fl(\()p 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Fr(f)2246 3160 y Fp(+)22 b(1\))2431 3119 y Ff(\000)p Fl(3)p Fr(=)p Fl(2)p Ff(\000)p Fl(\()p Fr(N)q(=)p Fl(2\))2850 3046 y Fi(\015)2850 3105 y(\015)2850 3165 y(\015)2958 3160 y Fo(\024)53 b Fs(C)3158 3175 y Fl(2)3220 3160 y Fs(;)257 3401 y Fv(for)31 b(some)g(constant)f Fs(C)1079 3416 y Fl(2)1157 3401 y Fs(<)38 b Fo(1)p Fv(,)32 b(uniformly)d(in)i Fs(s)39 b Fo(2)g Fh(R)5 b Fv(.)54 b(Here)32 b(we)f(additionally)e(inserted)257 3522 y(\(IV)-13 b(.16\))25 b(to)g(deri)n(v)o(e)f(the)h(last)f(inequality)-6 b(.)29 b(Finally)-6 b(,)24 b(there)h(e)o(xists)e(a)i(constant)f Fs(C)2992 3537 y Fl(3)3059 3522 y Fs(<)k Fo(1)p Fv(,)d(such)257 3642 y(that)1052 3748 y Fi(\015)1052 3807 y(\015)1052 3867 y(\015)1107 3862 y Fs(W)31 b Fp(\(1)22 b(+)g Fo(j)p Fs(x)p Fo(j)p Fp(\))1586 3821 y Fr(\015)1652 3862 y Fo(\012)h Fp(\()p Fs(H)1871 3877 y Fr(f)1938 3862 y Fp(+)f(1\))2123 3821 y Ff(\000)p Fl(1)p Fr(=)p Fl(2)2287 3748 y Fi(\015)2287 3807 y(\015)2287 3867 y(\015)2395 3862 y Fo(\024)53 b Fs(C)2595 3877 y Fl(3)2657 3862 y Fs(;)528 b Fv(\(IV)-13 b(.29\))257 4110 y(as)32 b(follo)n(ws)e(from)h(writing)g Fs(W)f Fp(\(1)d(+)g Fo(j)p Fs(x)p Fo(j)p Fp(\))1733 4073 y Fr(\015)1817 4110 y Fp(=)40 b Fs(a)1984 4073 y Ff(\003)2023 4110 y Fp([)p Fs(G)17 b Fp(\(1)27 b(+)g Fo(j)p Fs(x)p Fo(j)p Fp(\))2510 4073 y Fr(\015)2554 4110 y Fp(])g(+)g Fs(a)p Fp([)p Fs(G)17 b Fp(\(1)27 b(+)g Fo(j)p Fs(x)p Fo(j)p Fp(\))3249 4073 y Fr(\015)3293 4110 y Fp(])k Fv(and)257 4230 y(applying)24 b(Lemma)g(IV)-13 b(.1.)2266 b Fa(\003)p eop %%Page: 34 35 34 34 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(34)257 573 y Fu(A)143 b(Elimination)37 b(of)i(Matrix)f(Elements)f (by)h(a)h(Bogoliubo)o(v)504 756 y(T)-11 b(ransf)l(ormation)257 979 y Fv(Consider)21 b Fs(H)719 994 y Fr(el)805 979 y Fp(=)27 b Fo(\000)p Fp(\001)6 b Fo(\000)g Fs(V)24 b Fp(\()p Fs(x)p Fp(\))p Fv(,)d(a)g(Schr)8 b(\250)-41 b(odinger)21 b(operator)f(on)h Fh(R)2524 943 y Fl(3)2591 979 y Fv(ha)n(ving)e(a)i (single,)g(simple)257 1099 y(ne)o(gati)n(v)o(e)j(eigen)l(v)n(alue,)h Fs(e)1136 1114 y Fl(0)1206 1099 y Fs(<)30 b Fp(0)p Fv(,)c(associated)g (to)f(a)i(normalized)e(v)o(ector)g Fs(')2833 1114 y Fl(0)2873 1099 y Fv(.)34 b(This)25 b(Hamilto-)257 1220 y(nian)j(represents)h(an)g (atom)e(which)h(we)h(couple)f(to)g(the)h(photonic)e(\002eld)i (described)f(in)g(Sub-)257 1340 y(sect.)g(I.1.2)g(with)f(an)h (interaction)g(described)f(in)h(Subsect.)g(I.1.3.)40 b(More)28 b(precisely)-6 b(,)28 b(we)g(as-)257 1461 y(sume)c(the)f (interaction)g Fs(W)38 b Fv(to)23 b(be)h(of)g(the)g(form)g(\(I.11\),)g (with)f Fs(G)k Fo(2)h Fs(L)2645 1424 y Fl(2)2685 1461 y Fp([)p Fh(R)2778 1424 y Fl(3)2824 1461 y Fp(;)17 b Fo(B)s Fp(\()p Fo(H)3058 1476 y Fr(el)3117 1461 y Fp(\)])24 b Fv(being)f(a)257 1581 y(square-inte)o(grable)e(function)g(with)g(v)n (alues)g(in)g(the)h(bounded)f(operators)g(on)h Fo(H)3002 1596 y Fr(el)3060 1581 y Fv(,)h(satisfying)1115 1710 y Fi(Z)1232 1846 y Fs(d)1283 1805 y Fl(3)1322 1846 y Fs(k)1392 1735 y Fi(n)1459 1846 y Fp(1)f(+)g Fs(!)t Fp(\()p Fs(k)s Fp(\))1823 1805 y Ff(\000)p Fl(2)1916 1735 y Fi(o)1999 1761 y(\015)1999 1821 y(\015)2055 1846 y Fs(G)p Fp(\()p Fs(k)s Fp(\))2262 1761 y Fi(\015)2262 1821 y(\015)2317 1788 y Fl(2)2409 1846 y Fo(\024)53 b Fp(1)22 b Fs(:)645 b Fv(\(A.1\))257 2108 y(Note)25 b(that)f(this)g(assumption)f(is)h (stronger)h(than)f(\(I.15\).)31 b(W)-8 b(e)25 b(no)n(w)f(sho)n(w)257 2278 y Fk(Theor)n(em)j(A.1.)49 b Fq(Assume)37 b(\(A.1\).)67 b(F)-10 b(or)36 b Fs(g)54 b(>)c Fp(0)37 b Fq(small)f(enough,)k(ther)l (e)d(e)n(xists)f(a)h(unitary)257 2398 y(tr)o(ansform)23 b Fs(U)740 2413 y Fr(g)808 2398 y Fo(2)28 b(B)s Fp(\()p Fo(H)1092 2413 y Fr(el)1173 2398 y Fo(\012)23 b(F)10 b Fp(\))24 b Fq(r)l(eal)h(analytic)f(in)g Fs(g)29 b Fq(suc)o(h)24 b(that)873 2609 y Fi(e)851 2634 y Fs(H)932 2649 y Fr(g)1025 2634 y Fp(:=)52 b Fs(U)1246 2649 y Fr(g)1303 2634 y Fs(H)1384 2649 y Fr(g)1441 2634 y Fs(U)1517 2593 y Ff(\003)1507 2659 y Fr(g)1610 2634 y Fp(=)1735 2609 y Fi(e)1713 2634 y Fs(H)1794 2649 y Fr(el)1875 2634 y Fo(\012)22 b Fg(1)2030 2649 y Fr(f)2115 2634 y Fp(+)38 b Fg(1)2285 2649 y Fr(el)2366 2634 y Fo(\012)23 b Fs(H)2547 2649 y Fr(f)2631 2634 y Fp(+)38 b Fs(g)2799 2609 y Fi(f)2796 2634 y Fs(W)394 b Fv(\(A.2\))257 2854 y Fq(wher)l(e)403 3057 y Fo(\017)523 3032 y Fi(e)501 3057 y Fs(H)582 3072 y Fr(el)668 3057 y Fp(=)28 b Fs(H)853 3072 y Fr(el)934 3057 y Fp(+)22 b Fs(g)1083 3021 y Fl(2)1121 3057 y Fp(\001)p Fs(V)g Fq(,)j(and)f Fp(\001)p Fs(V)47 b Fq(is)25 b(a)f(bounded)g(self-adjoint) f(oper)o(ator;)403 3261 y Fo(\017)48 b Fq(if)36 b Fs(W)50 b Fq(acts)36 b(as)f(a)i(multiplication)c(oper)o(ator)h(in)i(the)g (electr)l(on)f(variable)o(,)k Fp(\001)p Fs(V)58 b Fq(is)35 b(a)501 3381 y(potential,)23 b(i.e)o(.,)h(a)h(multiplication)d(oper)o (ator;)403 3585 y Fo(\017)523 3559 y Fi(e)501 3585 y Fs(H)582 3600 y Fr(el)666 3585 y Fq(has)i(a)g(single)o(,)g(simple)g(ne) l(gative)g(eig)o(en)l(value)o(,)h Fs(e)2354 3600 y Fr(g)2422 3585 y Fs(<)i Fp(0)p Fq(,)e(associated)e(to)i(the)f(one-)501 3705 y(dimensional)f(eig)o(enpr)l(ojector)g Fp(\005)1692 3720 y Fr(g)1732 3705 y Fq(;)403 3908 y Fo(\017)48 b Fq(we)26 b(have)1366 4128 y Fp(\(\005)1477 4143 y Fr(g)1539 4128 y Fo(\012)d Fg(1)1695 4143 y Fr(f)1740 4128 y Fp(\))1809 4103 y Fi(f)1806 4128 y Fs(W)41 b Fp(\(\005)2050 4143 y Fr(g)2113 4128 y Fo(\012)22 b Fg(1)2268 4143 y Fr(f)2314 4128 y Fp(\))52 b(=)h(0)21 b Fs(:)652 b Fv(\(A.3\))257 4390 y Fq(All)25 b(the)f(quantities)f(intr)l(oduced)g(abo)o(ve)i(ar)l (e)g(r)l(eal-analytic)f(in)g Fs(g)t Fq(.)404 4560 y Fv(Theorem)36 b(A.1)g(pro)o(v)o(es)f(that,)k(starting)c(from)h(a)h(Schr)8 b(\250)-41 b(odinger)37 b(operator)f(on)g Fh(R)3236 4524 y Fl(3)3318 4560 y Fv(with)257 4680 y(a)31 b(single,)f(simple)f(ne)o (gati)n(v)o(e)f(bound)h(state,)i(using)e(a)h(unitary)f(transform,)i (one)f(can)h(al)o(w)o(ays)257 4801 y(pass)25 b(to)f(an)h(interaction)f (ful\002lling)g(Assumption)e(\(I.12\).)257 4996 y Fq(Pr)l(oof)o(.)47 b Fv(Pick)25 b Fs(h)j Fo(2)g Fs(L)998 4960 y Fl(2)1038 4996 y Fp(\()p Fh(R)1142 4960 y Fl(3)1187 4996 y Fp(\))p Fv(,)d(and)g(consider)f(the)h(Bogoliubo)o(v)e(transform)1315 5216 y Fs(U)10 b Fp(\()p Fs(h)p Fp(\))53 b(=)g Fg(1)22 b Fo(\012)h Fs(e)1928 5175 y Fr(ig)r Fl([)p Fr(a)2045 5151 y Fb(\003)2081 5175 y Fl(\()p Fr(h)p Fl(\)+)p Fr(a)p Fl(\()p Fr(h)p Fl(\)])2410 5216 y Fs(:)845 b Fv(\(A.4\))p eop %%Page: 35 36 35 35 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(35)257 573 y(A)25 b(standard)g(computation)e(gi)n(v)o(es)823 802 y Fs(U)10 b Fp(\()p Fs(h)p Fp(\))25 b Fs(H)1137 817 y Fr(g)1202 802 y Fs(U)10 b Fp(\()p Fs(h)p Fp(\))1410 761 y Ff(\003)1503 802 y Fp(=)1653 777 y Fi(e)1631 802 y Fs(H)1712 817 y Fr(el)1793 802 y Fo(\012)23 b Fg(1)1949 817 y Fr(f)2044 802 y Fp(+)50 b Fg(1)2226 817 y Fr(el)2307 802 y Fo(\012)22 b Fs(H)2487 817 y Fr(f)2582 802 y Fp(+)50 b Fs(g)2762 777 y Fi(f)2759 802 y Fs(W)35 b(;)369 b Fv(\(A.5\))257 1022 y(where)1068 1217 y Fi(e)1046 1242 y Fs(H)1127 1257 y Fr(el)1282 1242 y Fp(=)96 b Fs(H)1535 1257 y Fr(el)1644 1242 y Fp(+)49 b Fs(g)1820 1201 y Fl(2)1876 1242 y Fp(\001)p Fs(V)22 b Fp(\()p Fs(h)p Fp(\))g Fs(;)1065 b Fv(\(A.6\))894 1436 y Fp(\001)p Fs(V)21 b Fp(\()p Fs(h)p Fp(\))83 b(:=)1454 1300 y Fi(Z)1571 1436 y Fs(d)1622 1395 y Fl(3)1661 1436 y Fs(k)1715 1325 y Fi(n)1781 1436 y Fs(h)p Fp(\()p Fs(k)s Fp(\))p Fs(G)p Fp(\()p Fs(k)s Fp(\))22 b(+)p 2294 1349 187 4 v 22 w Fs(h)p Fp(\()p Fs(k)s Fp(\))p Fs(G)2557 1395 y Ff(\003)2597 1436 y Fp(\()p Fs(k)s Fp(\))2727 1325 y Fi(o)2815 1436 y Fs(;)440 b Fv(\(A.7\))1083 1630 y Fi(f)1080 1655 y Fs(W)110 b Fp(=)96 b Fs(W)64 b Fp(+)50 b Fg(1)22 b Fo(\012)1913 1574 y Fi(\010)1971 1655 y Fs(a)2022 1614 y Ff(\003)2062 1655 y Fp(\()p Fs(!)t(h)p Fp(\))g(+)g Fs(a)p Fp(\()p Fs(!)t(h)p Fp(\))2627 1574 y Fi(\011)2706 1655 y Fs(:)549 b Fv(\(A.8\))257 1875 y(W)-8 b(e)28 b(remark)g(that)f Fp(\001)p Fs(V)55 b Fp(:)33 b Fs(L)1221 1839 y Fl(2)1261 1875 y Fp(\()p Fh(R)1365 1839 y Fl(3)1410 1875 y Fp(\))g Fo(!)g(B)s Fp(\()p Fo(H)1804 1890 y Fr(el)1863 1875 y Fp(\))28 b Fv(is)f(a)h(bounded)f(linear)g(operator)h(with)f(norm)257 2005 y(bounded)38 b(by)g(one,)j(thanks)c(to)h(\(A.1\).)71 b(Hence,)2053 1980 y Fi(e)2031 2005 y Fs(H)2112 2020 y Fr(el)2208 2005 y Fv(has)38 b(a)h(single,)h(simple)d(eigen)l(v)n (alue)261 2126 y Fp(~)-53 b Fs(e)q Fp(\()p Fs(h)p Fp(\))27 b(=)h Fs(e)611 2141 y Fl(0)656 2126 y Fp(+)6 b Fo(O)s Fp(\()p Fs(g)909 2089 y Fl(2)947 2126 y Fp(\))21 b Fv(in)f(a)g (vicinity)f(of)i Fs(e)1642 2141 y Fl(0)1681 2126 y Fv(,)g(and)g(we)f (denote)g(the)h(corresponding)e(normalized)257 2246 y(eigen)l(v)o (ector)25 b(by)f Fs(')930 2261 y Fr(g)970 2246 y Fp(\()p Fs(h)p Fp(\))p Fv(.)404 2366 y(T)-8 b(o)37 b(pro)o(v)o(e)f(Theorem)h (A.1,)j(we)d(only)g(need)g(to)g(sho)n(w)f(that,)k(for)e Fs(g)54 b(>)c Fp(0)37 b Fv(suf)n(\002ciently)257 2487 y(small,)24 b(we)h(can)g(construct)f Fs(h)1273 2502 y Fr(g)1341 2487 y Fo(2)k Fs(L)1501 2451 y Fl(2)1541 2487 y Fp(\()p Fh(R)1645 2451 y Fl(3)1691 2487 y Fp(\))c Fv(such)h(that)408 2720 y 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y Fp(\()p Fs(h)p Fp(\))52 b(:=)1341 3520 y(1)p 1295 3564 141 4 v 1295 3655 a(2)p Fs(i\031)1462 3452 y Fi(Z)1518 3677 y Fr(\015)1554 3686 y Fd(0)1609 3506 y Fi(\000)1655 3587 y Fs(z)27 b Fo(\000)22 b Fs(H)1907 3602 y Fr(el)1988 3587 y Fo(\000)h Fs(g)2139 3546 y Fl(2)2177 3587 y Fp(\001)p Fs(V)f Fp(\()p Fs(h)p Fp(\))2469 3506 y Fi(\001)2515 3529 y Ff(\000)p Fl(1)2626 3587 y Fs(')2690 3602 y Fl(0)2746 3587 y Fs(dz)k(;)337 b Fv(\(A.10\))257 3873 y(where)23 b Fs(\015)574 3888 y Fl(0)634 3873 y Fv(is)f(a)g(small)e(circle)i(of)g(center)g Fs(e)1684 3888 y Fl(0)1746 3873 y Fv(and)g(radius)f Fo(j)p Fs(e)2250 3888 y Fl(0)2289 3873 y Fs(=)p Fp(2)p Fo(j)p Fv(.)29 b(The)22 b(function)f Fs(')3064 3888 y Fr(g)3104 3873 y Fp(\()p Fs(h)p Fp(\))h Fv(is)f(real)257 3993 y(analytic)32 b(in)f Fs(g)k Fv(and)d(continuous)e(in)i Fs(h)p Fv(,)h(for)g Fs(h)f Fv(in)f(the)h(unit)f(ball)g(in)g Fs(L)2708 3957 y Fl(2)2748 3993 y Fp(\()p Fh(R)2852 3957 y Fl(3)2898 3993 y Fp(\))p Fv(;)k(real)d(analytic)257 4113 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4656 y Fp(\()p Fs(H)2329 4671 y Fr(g)2368 4656 y Fp(\))p Fv(,)36 b(which)e(yields)e(\(B.1\).)58 b(As)34 b(a)257 4776 y(consequence)26 b(of)g(\(B.1\))g(we)g(may)f (assume)g Fp(\011)k(=)g Fg(1)2058 4791 y Fr(c)2093 4776 y Fp(\()p Fs(H)2212 4791 y Fr(g)2251 4776 y Fp(\)\011)d Fv(in)f(the)h(proof)f(of)h(\(B.2\).)33 b(These)257 4897 y(v)o(ectors)24 b(out)h(of)g(the)f(continuous)f(subspace)i(ha)n(v)o(e)g (the)f(property)h(that)1298 5117 y Fp(lim)1296 5180 y Fr(\016)r Ff(&)p Fl(0)1469 5117 y Fg(1)1525 5132 y Fl(\()p Fr(E)t Ff(\000)p Fr(\016)o(;E)t Fl(+)p Fr(\016)r Fl(\))1888 5117 y Fp(\()p Fs(H)2007 5132 y Fr(g)2047 5117 y Fp(\)\011)52 b(=)h(0)22 b Fs(;)848 b Fv(\(B.4\))p eop %%Page: 37 38 37 37 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(37)257 573 y(for)36 b(an)o(y)e Fs(E)52 b Fo(2)47 b Fh(R)5 b Fv(.)67 b(As)35 b Fs(H)1208 588 y Fr(g)1283 573 y Fv(is)f(bounded)g(from)h(belo)n(w)f(and)h(in)f(vie)n(w)h(of)g (\(B.4\),)j(it)c(hence)257 693 y(\(choosing)24 b Fs(E)34 b Fp(:=)28 b(0)p 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b(The)23 b(measure)257 2441 y(in)f(\(B.6\))g(is)g(then)f(de\002ned)i(as)f Fs(d\026)p Fp(\()p Fs(z)t Fp(\))27 b(:=)h(\(2)p Fs(\031)t(i)p Fp(\))1899 2404 y Ff(\000)p Fl(1)2005 2414 y Fp(\026)1993 2441 y Fs(@)18 b Fp(~)-61 b Fs(\037)p Fp(\()p Fs(z)t Fp(\))17 b Fs(dz)f Fo(^)11 b Fs(d)6 b Fp(\026)-55 b Fs(z)6 b Fv(.)30 b(By)22 b(means)f(of)h(\(B.6\),)h(the)257 2561 y(second)29 b(resolv)o(ent)f(equation,)h(and)g(the)g(f)o(act)g(that)f Fs(F)2096 2576 y Fr(R)2171 2561 y Fs(\037)p Fp([)p Fs(H)2340 2576 y Fl(0)2379 2561 y Fp(])36 b(=)f Fs(F)2616 2576 y Fr(R)2690 2561 y Fs(P)2753 2576 y Fr(c)2805 2561 y Fs(\037)p Fp([)p Fs(H)2974 2576 y Fl(0)3013 2561 y Fp(])h(=)f(0)p Fv(,)29 b(since)257 2681 y Fs(H)338 2696 y Fl(0)405 2681 y Fo(\025)f Fp(0)d Fv(on)g Fp(Ran)16 b Fs(P)963 2696 y Fr(c)998 2681 y Fv(,)24 b(we)i(thus)d(ha)n(v)o(e)681 2919 y Fs(F)744 2934 y Fr(R)819 2919 y Fs(\037)p Fp([)p Fs(H)988 2934 y Fr(g)1028 2919 y Fp(])83 b(=)g Fs(F)1360 2934 y Fr(R)1435 2808 y Fi(\020)1494 2919 y Fs(\037)p 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b(Math.)h(Phys.)p Fv(,)f(207\(2\):249\226290,)f(1999.)307 2453 y([5])49 b(V)-13 b(.)26 b(Bach,)g(J.)f(Fr)8 b(\250)-41 b(ohlich,)25 b(and)g(I.)h(M.)e(Sigal.)37 b(Return)26 b(to)f(equilibrium.)35 b Fq(J)n(.)25 b(Math.)g(Phys.)p Fv(,)472 2573 y(41\(6\):3985\2264060,)e (June)h(2000.)307 2776 y([6])49 b(J.)36 b(Derezinski)h(and)f(C.)h(G)6 b(\264)-39 b(erard.)74 b Fq(Scattering)35 b(theory)h(of)g(classical)f (and)i(quantum)472 2897 y(N-particle)25 b(systems)p Fv(.)35 b(T)-7 b(e)o(xt)24 b(and)h(Monographs)e(in)h(Physics.)g(Springer)l(,)h (1997.)307 3100 y([7])49 b(D.)25 b(Eastham.)35 b Fq(Atomic)24 b(Physics)g(of)h(Laser)o(s)p Fv(.)35 b(T)-8 b(aylor)25 b(and)f(Francis,)h(London,)f(1986.)307 3304 y([8])49 b(A.)38 b(Einstein.)1055 3282 y(\250)1036 3304 y(Uber)g(einen)f(die)h (Erzeugung)f(und)g(V)-11 b(erw)o(andlung)37 b(des)h(Lichts)f(be-)472 3424 y(tref)n(fenden)d(heuristischen)f(Gesichtspunkt.)64 b Fq(Annalen)33 b(der)h(Physik)p Fv(,)i(17:132\226148,)472 3544 y(1905.)307 3748 y([9])49 b(J.)36 b(Fr)8 b(\250)-41 b(ohlich,)39 b(M.)d(Griesemer)l(,)j(and)e(B.)g(Schlein.)72 b(Asymptotic)35 b(completeness)g(for)472 3868 y(Rayleigh)25 b(scattering.)35 b Fq(Pr)l(eprint)p Fv(,)24 b(2001.)257 4072 y([10])49 b(Ch.)26 b(G)6 b(\264)-39 b(erard.)41 b(Asymptotic)24 b(completeness)h(for)h(the)g(spin-boson)e(model)h(with) g(a)i(par)n(-)472 4192 y(ticle)e(number)f(cutof)n(f.)35 b Fq(Re)o(v)-7 b(.)24 b(Math.)g(Phys.)p Fv(,)h(8:549\226589,)e(1996.) 257 4395 y([11])49 b(M.)65 b(Griesemer)l(,)75 b(E.)66 b(Lieb,)75 b(and)65 b(M.)g(Loss.)165 b(Ground)65 b(states)f(in)h (nonrela-)472 4516 y(ti)n(vistic)60 b(quantum)g(electrodynamics.)153 b Fq(In)l(vent.)61 b(math.)p Fv(,)70 b(June)61 b(2001.)153 b(DOI)472 4636 y(10.1007/s002220100159.)257 4840 y([12])49 b(H.)25 b(Hak)o(en.)36 b Fq(Light)p Fv(,)24 b(v)n(olume)g(1.)35 b(North-Holland,)24 b(1981.)257 5043 y([13])49 b(D.)25 b(Haliday)f(and)h(R.)g(Resnick.)36 b Fq(Physics)24 b(\(P)-8 b(art)24 b(I)i(and)e(II\))p Fv(.)36 b(W)l(ile)o(y)-6 b(,)24 b(1978.)257 5246 y([14])49 b(W)-9 b(.)29 b(Hall)o(w)o(achs.)50 b(Ueber)29 b(den)g(Ein\003uss)g(des)g(Lichtes)f(auf)i(electrostatisch)e (geladene)472 5367 y(K)8 b(\250)-41 b(orper)-5 b(.)36 b Fq(Annalen)24 b(der)g(Physik)p Fv(,)h(33:301,)e(1888.)p eop %%Page: 39 40 39 39 bop 257 266 a Fw(BKZ-Aug)25 b(31,)g(2001)2360 b Fv(39)257 573 y([15])49 b(H.)26 b(Hertz.)38 b(Ueber)26 b(einen)g(Ein\003uss)e(des)i(ultra)n(violetten)e(Lichtes)h(auf)h(die)f (electrische)472 693 y(Entladung.)34 b Fq(Annalen)24 b(der)h(Physik)p Fv(,)g(31:983,)e(1887.)257 897 y([16])49 b(P)-11 b(.)30 b(Lenard.)947 875 y(\250)928 897 y(Uber)f(die)g (lichtelektrische)f(Wirkung.)49 b Fq(Annalen)28 b(der)i(Physik)p Fv(,)f(8:149\226)472 1017 y(198,)24 b(1902.)257 1220 y([17])49 b(A.)25 b(Melin)f(and)g(J.)h(Sj)8 b(\250)-41 b(ostrand.)35 b Fq(F)-10 b(ourier)23 b(Inte)l(gr)o(al)h(Oper)o(ator)o (s)f(with)h(Comple)n(x-valued)472 1341 y(Phase)h(Functions)p Fv(,)e(v)n(olume)h(459)g(of)h Fq(L.)g(N.)g(M.)g Fv(.)36 b(Springer)l(,)25 b(Heidelber)n(g,)f(1976.)257 1544 y([18])49 b(R.)27 b(Millikan.)37 b(A)26 b(direct)g(photoelectric)f(determination) f(of)j(Planck')-5 b(s)25 b(h.)39 b Fq(Phys.)26 b(Re)o(v)-7 b(.)p Fv(,)472 1665 y(7:355\226388,)23 b(1916.)257 1868 y([19])49 b(R.)41 b(Millikan.)85 b(Einstein')-5 b(s)39 b(photoelectric)h(equation)g(and)h(contact)f(electromoti)n(v)o(e)472 1988 y(force.)d Fq(Phys.)24 b(Re)o(v)-7 b(.)p Fv(,)24 b(7:18\22632,)g(1916.)257 2192 y([20])49 b(B.)35 b(Schlein.)68 b(Nichtrelati)n(vistische)32 b(Elektronensysteme)h(in)h(einem)h (quantisierten)472 2312 y(elektromagnetischen)24 b(Feld.)35 b Fq(diploma)24 b(thesis,)g(ETH-Z)2472 2313 y(\250)2464 2312 y(uric)o(h)p Fv(,)g(August)g(1999.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF