Content-Type: multipart/mixed; boundary="-------------9905100237777" This is a multi-part message in MIME format. ---------------9905100237777 Content-Type: text/plain; name="99-158.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-158.keywords" Pauli-Fierz models ---------------9905100237777 Content-Type: application/postscript; name="infgrd.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="infgrd.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: infgrd.dvi %%Pages: 15 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSCommandLine: dvips infgrd.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 1999.05.10:0920 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def 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y(Applying)j(Lemma)j(A.2)g(in)e(the)h(App)s(endix)e(and)i(the)g (fact)i(that)f Fo(D)s Fr(\()p Fo(H)2624 1162 y Fi(1)p 2624 1174 31 4 v 2624 1216 a(2)2668 1225 y Fr(\))26 b(=)f Fo(D)s Fr(\()p Fo(H)3031 1141 y Fi(1)p 3031 1153 V 3031 1194 a(2)3014 1250 y Fm(0)3075 1225 y Fr(\))31 b(w)m(e)g(obtain)1155 1425 y(lim)1281 1439 y Fk(\034)8 b Fn(!)p Fm(+)p Fn(1)1520 1425 y Fr(\()p Fo(H)1631 1439 y Fk(\033)n(;\034)1753 1425 y Fp(\000)20 b Fo(\025)p Fr(\))1932 1392 y Fn(\000)p Fm(1)2052 1425 y Fr(=)25 b(\()p Fo(H)2259 1439 y Fk(\033)2326 1425 y Fp(\000)20 b Fo(\025)p Fr(\))2505 1392 y Fn(\000)p Fm(1)2599 1425 y Fo(;)1155 1609 y Fr(lim)1281 1623 y Fk(\033)r Fn(!)p Fm(0)1434 1609 y Fr(\()p Fo(H)1545 1623 y Fk(\033)1612 1609 y Fp(\000)g Fo(\025)p Fr(\))1791 1576 y Fn(\000)p Fm(1)1911 1609 y Fr(=)25 b(\()p Fo(H)i Fp(\000)20 b Fo(\025)p Fr(\))2324 1576 y Fn(\000)p Fm(1)2419 1609 y Fo(;)0 1517 y Fr(\(3.1\))0 1802 y(for)30 b Fo(\025)25 b Fp(2)g Fr(I)-15 b(R)p Fo(;)15 b(\025)26 b Fp(\034)f(\000)p Fr(1,)30 b(and)589 2011 y Fp(k)p Fr(\()q(\()p Fo(H)781 2025 y Fk(\033)n(;\034)902 2011 y Fp(\000)20 b Fo(z)t Fr(\))1074 1978 y Fn(\000)p Fm(1)1189 2011 y Fp(\000)g Fr(\()p Fo(H)1391 2025 y Fk(\033)1458 2011 y Fp(\000)g Fo(z)t Fr(\))1630 1978 y Fn(\000)p Fm(1)1725 2011 y Fr(\))q(\()p Fo(H)1872 2025 y Fm(0)1931 2011 y Fr(+)g(1\))2112 1948 y Fi(1)p 2113 1960 V 2113 2001 a(2)2157 2011 y Fp(k)26 b(2)f Fo(o)p Fr(\(1\))p Fp(j)p Fo(I)7 b(mz)t Fp(j)2696 1978 y Fn(\000)p Fm(1)2812 2011 y Fo(\034)36 b Fp(!)25 b Fr(+)p Fp(1)p Fo(;)589 2210 y Fp(k)p Fr(\()q(\()p Fo(H)781 2224 y Fk(\033)848 2210 y Fp(\000)20 b Fo(z)t Fr(\))1020 2177 y Fn(\000)p Fm(1)1135 2210 y Fp(\000)g Fr(\()p Fo(H)27 b Fp(\000)20 b Fo(z)t Fr(\))1536 2177 y Fn(\000)p Fm(1)1631 2210 y Fr(\)\()p Fo(H)1777 2224 y Fm(0)1837 2210 y Fr(+)g(1\))2018 2147 y Fi(1)p 2019 2159 V 2019 2201 a(2)2063 2210 y Fp(k)26 b(2)f Fo(o)p Fr(\(1\))p Fp(j)p Fo(I)7 b(mz)t Fp(j)2602 2177 y Fn(\000)p Fm(1)2718 2210 y Fo(\033)29 b Fp(!)c Fr(0)p Fo(;)0 2103 y Fr(\(3.2\))0 2409 y(for)30 b Fo(z)g Fp(2)p 323 2409 4 56 v 25 w Fr(C)o Fp(n)p Fr(I)-15 b(R.)0 2651 y Fe(3.3)112 b(Existence)37 b(of)g(ground)h(states)g(for)f(the)h (cut-o\013)f(Hamiltonians)0 2823 y Fr(Let)h(~)-52 b Fo(!)220 2837 y Fk(\033)292 2823 y Fr(:)25 b(I)-15 b(R)427 2787 y Fk(d)492 2823 y Fp(!)26 b Fr(I)-15 b(R)29 b(b)s(e)h(a)h(disp)s (ersion)c(relation)j(satisfying)1298 2954 y Fj(8)1298 3029 y(>)1298 3054 y(>)1298 3079 y(>)1298 3104 y(<)1298 3253 y(>)1298 3278 y(>)1298 3303 y(>)1298 3328 y(:)1413 3019 y Fp(r)7 b Fr(~)-52 b Fo(!)1546 3033 y Fk(\033)1617 3019 y Fp(2)25 b Fo(L)1765 2986 y Fn(1)1840 3019 y Fr(\(I)-15 b(R)1960 2983 y Fk(d)2000 3019 y Fr(\))p Fo(;)1420 3202 y Fr(~)-52 b Fo(!)1470 3216 y Fk(\033)1516 3202 y Fr(\()p Fo(k)s Fr(\))27 b(=)e Fo(!)s Fr(\()p Fo(k)s Fr(\))31 b(if)e Fo(!)s Fr(\()p Fo(k)s Fr(\))d Fp(\025)f Fo(\033)n(;)1420 3386 y Fr(~)-52 b Fo(!)1470 3400 y Fk(\033)1516 3386 y Fr(\()p Fo(k)s Fr(\))27 b Fp(\025)e Fo(\033)s(=)p Fr(2)p Fo(:)0 3201 y Fr(\(3.3\))0 3611 y(Let)179 3588 y(~)155 3611 y Fo(H)231 3625 y Fk(\033)301 3611 y Fr(b)s(e)d(the)h(op)s(erator) g(asso)s(ciated)g(to)h(the)f(quadratic)f(form)h Fp(k)p Fo(K)2349 3548 y Fi(1)p 2349 3560 31 4 v 2349 3601 a(2)2393 3611 y Fo(u)p Fp(k)2490 3578 y Fm(2)2535 3611 y Fr(+)2611 3542 y Fj(R)2689 3611 y Fr(~)-52 b Fo(!)2739 3625 y Fk(\033)2785 3611 y Fr(\()p Fo(k)s Fr(\))p Fp(k)p Fo(a)p Fr(\()p Fo(k)s Fr(\))p Fo(u)p Fp(k)3215 3578 y Fm(2)3257 3611 y Fr(d)p Fo(k)8 b Fr(+)d Fo(V)3492 3625 y Fk(\033)3539 3611 y Fr(\()p Fo(u;)15 b(u)p Fr(\).)0 3796 y Fd(Lemma)33 b(3.2)46 b Fo(H)626 3810 y Fk(\033)705 3796 y Fq(admits)34 b(a)f(gr)-5 b(ound)34 b(state)f(in)f Fp(H)i Fq(if)e(and)i(only)f(if)2394 3773 y Fr(~)2370 3796 y Fo(H)2446 3810 y Fk(\033)2525 3796 y Fq(admits)h(a)f(gr)-5 b(ound)34 b(state)g(in)e Fp(H)q Fq(.)0 3980 y Fd(Pro)s(of.)63 b Fr(Let)36 b Fl(h)556 3994 y Fk(\033)636 3980 y Fr(:=)c Fo(L)826 3947 y Fm(2)866 3980 y Fr(\()p Fp(f)p Fo(k)s Fp(j)p Fo(!)s Fr(\()p Fo(k)s Fr(\))i Fo(<)f(\033)s 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Fr(\).)41 b(By)31 b([DG)q(,)f(Sect.)42 b(2.7],)32 b(the)e(op)s(erator)h Fo(U)10 b(H)3037 4498 y Fk(\033)3083 4484 y Fo(U)3155 4451 y Fn(\003)3225 4484 y Fr(is)29 b(equal)h(to)1192 4685 y(1)-20 b(l)1242 4707 y Fn(K\012)p Fm(\000\()p Fb(h)1458 4688 y Fa(?)1458 4724 y Fh(\033)1511 4707 y Fm(\))1563 4685 y Fp(\012)20 b Fr(d\000\()p Fo(!)1854 4699 y Fk(\033)n(;)p Fm(1)1951 4685 y Fr(\))h(+)e Fo(H)2180 4647 y Fm(2)2173 4707 y Fk(\033)2240 4685 y Fp(\012)h Fr(1)-20 b(l)2381 4703 y Fm(\000\()p Fb(h)2488 4711 y Fh(\033)2531 4703 y Fm(\))2562 4685 y Fo(;)0 4915 y Fr(where)47 b Fo(!)337 4929 y Fk(\033)n(;)p Fm(1)489 4915 y Fr(=)54 b Fo(!)671 4933 y Fn(j)p Fb(h)727 4941 y Fh(\033)820 4915 y Fr(and)47 b Fo(H)1097 4882 y Fm(2)1090 4937 y Fk(\033)1184 4915 y Fr(is)g(the)h(op)s(erator)g(asso)s(ciated)g(with)f(the)h(quadratic)f (form)g Fp(k)p Fo(K)3496 4852 y Fi(1)p 3496 4864 V 3496 4905 a(2)3540 4915 y Fo(u)p Fp(k)3637 4882 y Fm(2)3709 4915 y Fr(+)0 4973 y Fj(R)39 5069 y Fn(f)p Fk(!)r Fm(\()p Fk(k)r Fm(\))p Fn(\025)p Fk(\033)r Fn(g)366 5042 y Fo(!)423 5056 y Fk(\033)470 5042 y Fr(\()p Fo(k)s Fr(\))p Fp(k)p Fo(a)p Fr(\()p Fo(k)s Fr(\))p Fo(u)p Fp(k)900 5009 y Fm(2)942 5042 y Fr(d)p Fo(k)23 b Fr(+)d Fo(V)1207 5056 y Fk(\033)1254 5042 y Fr(\()p Fo(u;)15 b(u)p Fr(\).)42 b(Similarly)26 b Fo(U)2017 5019 y Fr(~)1993 5042 y Fo(H)2069 5056 y Fk(\033)2116 5042 y Fo(U)2188 5009 y Fn(\003)2257 5042 y Fr(is)k(equal)g(to)1192 5262 y(1)-20 b(l)1242 5284 y Fn(K\012)p Fm(\000\()p Fb(h)1458 5266 y Fa(?)1458 5301 y Fh(\033)1511 5284 y Fm(\))1563 5262 y Fp(\012)20 b Fr(d\000\()7 b(~)-52 b Fo(!)1854 5276 y Fk(\033)n(;)p Fm(1)1951 5262 y Fr(\))21 b(+)e Fo(H)2180 5225 y Fm(2)2173 5285 y Fk(\033)2240 5262 y Fp(\012)h Fr(1)-20 b(l)2381 5281 y Fm(\000\()p Fb(h)2488 5289 y Fh(\033)2531 5281 y Fm(\))2562 5262 y Fo(;)1867 5517 y Fr(5)p eop %%Page: 6 6 6 5 bop 0 162 a Fr(where)45 b(~)-52 b Fo(!)328 176 y Fk(\033)n(;)p Fm(1)463 162 y Fr(=)45 b(~)-52 b Fo(!)629 181 y Fk(\033)r Fn(j)p Fb(h)727 189 y Fh(\033)773 162 y Fr(.)63 b(No)m(w)39 b Fo(H)1159 129 y Fm(2)1152 185 y Fk(\033)1236 162 y Fr(has)f(a)h(ground)e(state)i Fo( )j Fr(if)37 b(and)g(only)g(if)g Fo(U)2806 139 y Fr(~)2782 162 y Fo(H)2858 176 y Fk(\033)2905 162 y Fo(U)2977 129 y Fn(\003)3054 162 y Fr(or)h Fo(U)10 b(H)3321 176 y Fk(\033)3367 162 y Fo(U)3439 129 y Fn(\003)3517 162 y Fr(ha)m(v)m(e)39 b(a)0 275 y(ground)30 b(state)j(\(equal)e(to)h Fo( )24 b Fp(\012)d Fr(\012,)31 b(where)g(\012)26 b Fp(2)h Fr(\000\()p Fl(h)1797 289 y Fk(\033)1844 275 y Fr(\))32 b(is)e(the)i(v)-5 b(acuum)30 b(v)m(ector\).)46 b(This)29 b(pro)m(v)m(es)j(the)g(lemma.)0 388 y Fc(2)141 501 y Fr(The)e(follo)m(wing)f(result)g(is)h(essen)m (tially)f(w)m(ell)g(kno)m(wn)h(\(see)h([AH)q(],)g([BFS]\))g(and)f (rather)g(easy)h(to)g(sho)m(w.)0 685 y Fd(Prop)s(osition)36 b(3.3)46 b Fq(Assume)25 b(hyp)-5 b(otheses)28 b(\(H0\),)f(\(H1\),)f (\(I1\).)41 b(Then)25 b(for)h(any)g Fo(\033)i(>)d Fr(0)h Fo(H)3081 699 y Fk(\033)3152 685 y Fq(admits)h(a)e(gr)-5 b(ound)0 798 y(state.)0 982 y Fd(Pro)s(of.)46 b Fr(By)26 b(Lemma)h(3.2)g(it)f(su\016ces)f(to)i(sho)m(w)f(that)1865 959 y(~)1841 982 y Fo(H)1917 996 y Fk(\033)1990 982 y Fr(admits)f(a)h(ground)g(state.)40 b(Let)27 b(for)f Fo(\034)35 b Fp(2)25 b Fr(I)-15 b(N)3507 959 y(~)3483 982 y Fo(H)3559 996 y Fk(\033)n(;\034)3686 982 y Fr(b)s(e)0 1112 y(the)30 b(Hamiltonian)f(asso)s(ciated)i(with)d(the)j(quadratic)f(form)f Fp(k)p Fo(K)2235 1049 y Fi(1)p 2235 1061 31 4 v 2235 1102 a(2)2280 1112 y Fo(u)p Fp(k)2377 1079 y Fm(2)2437 1112 y Fr(+)2527 1043 y Fj(R)2605 1112 y Fr(~)-52 b Fo(!)2655 1126 y Fk(\033)2701 1112 y Fr(\()p Fo(k)s Fr(\))p Fp(k)p Fo(a)p Fr(\()p Fo(k)s Fr(\))p Fo(u)p Fp(k)3131 1079 y Fm(2)3173 1112 y Fr(d)p Fo(k)23 b Fr(+)d Fo(V)3438 1126 y Fk(\033)n(;\034)3539 1112 y Fr(\()p Fo(u;)15 b(u)p Fr(\).)0 1225 y(Let)1089 1315 y(~)1068 1338 y Fo(E)1135 1352 y Fk(\033)n(;\034)1262 1338 y Fr(=)25 b(inf)c(sp)s(ec\()1713 1315 y(~)1689 1338 y Fo(H)1765 1352 y Fk(\033)n(;\034)1866 1338 y Fr(\))p Fo(;)1983 1315 y Fr(~)1962 1338 y Fo(E)2029 1352 y Fk(\033)2101 1338 y Fr(=)k(inf)c(sp)s(ec\()2552 1315 y(~)2528 1338 y Fo(H)2604 1352 y Fk(\033)2651 1338 y Fr(\))p Fo(:)0 1502 y Fr(Applying)28 b(Lemma)i(A.2,)i(w)m(e)f(ha)m(v) m(e)g(for)f Fo(z)g Fp(2)p 1579 1502 4 56 v 25 w Fr(C)o Fp(n)p Fr(I)-15 b(R)1223 1702 y(\()p Fo(z)25 b Fp(\000)1440 1679 y Fr(~)1416 1702 y Fo(H)1492 1716 y Fk(\033)1538 1702 y Fr(\))1573 1664 y Fn(\000)p Fm(1)1693 1702 y Fr(=)81 b(lim)1789 1756 y Fk(n)p Fn(!)p Fm(+)p Fn(1)2028 1702 y Fr(\()p Fo(z)25 b Fp(\000)2244 1679 y Fr(~)2221 1702 y Fo(H)2297 1716 y Fk(\033)n(;n)2402 1702 y Fr(\))2437 1664 y Fn(\000)p Fm(1)2531 1702 y Fo(:)-2556 b Fr(\(3.4\))0 1950 y(On)37 b(the)h(other)g(hand)e(applying)g(the)i(b)s(ounds)d(in)i (Lemma)h(A.1)g(w)m(e)g(ha)m(v)m(e)h Fo(D)s Fr(\()2785 1927 y(~)2761 1950 y Fo(H)2837 1964 y Fk(\033)n(;\034)2939 1950 y Fr(\))e(=)h Fo(D)s Fr(\()p Fo(K)32 b Fr(+)24 b(d\000\()7 b(~)-52 b Fo(!)3637 1964 y Fk(\033)3684 1950 y Fr(\)\).)0 2063 y(The)33 b(Hamiltonian)739 2040 y(~)716 2063 y Fo(H)792 2077 y Fk(\033)n(;\034)926 2063 y Fr(is)g(v)m(ery)g(similar)e(to)j(the) g(class)f(of)h(massiv)m(e)f(P)m(auli-Fierz)g(Hamiltonians)e(studied)0 2176 y(in)i([DG)q(].)53 b(It)35 b(is)e(easy)i(to)g(see)g(that)g(the)g (argumen)m(ts)g(of)f([DG)q(])h(extend)f(to)2627 2153 y(~)2603 2176 y Fo(H)2679 2190 y Fk(\033)n(;\034)2781 2176 y Fr(.)52 b(In)34 b(particular,)g(follo)m(wing)0 2289 y(the)40 b(pro)s(ofs)f(of)h([DG)q(,)j(Lemma)d(3.4],)k([DG)q(,)e (Thm.)68 b(4.1],)44 b(w)m(e)d(obtain)e(that)h Fo(\037)p Fr(\()2825 2266 y(~)2801 2289 y Fo(H)2877 2303 y Fk(\033)n(;\034)2979 2289 y Fr(\))g(is)f(compact)i(if)e Fo(\037)i Fp(2)0 2402 y Fo(C)72 2369 y Fn(1)65 2425 y Fm(0)146 2402 y Fr(\(])27 b Fp(\000)f(1)p Fo(;)483 2379 y Fr(~)461 2402 y Fo(E)528 2416 y Fk(\033)n(;\034)657 2402 y Fr(+)g Fo(\033)s(=)p Fr(2[\).)70 b(Using)39 b(\(3.4\))i(and)e(the)h(fact)g(that)2313 2379 y(~)2292 2402 y Fo(E)2359 2416 y Fk(\033)2446 2402 y Fr(=)h(lim)2684 2416 y Fk(n)p Fn(!)p Fm(+)p Fn(1)2963 2379 y Fr(~)2942 2402 y Fo(E)3009 2416 y Fk(\033)n(;\034)3111 2402 y Fr(,)h(w)m(e)e(obtain)f(that)0 2515 y Fo(\037)p Fr(\()131 2492 y(~)92 2515 y Fo(H)168 2529 y Fk(\033)215 2515 y Fr(\))c(is)e(compact)j(if)e Fo(\037)e Fp(2)g Fo(C)1090 2482 y Fn(1)1083 2538 y Fm(0)1164 2515 y Fr(\(])24 b Fp(\000)f(1)p Fo(;)1494 2492 y Fr(~)1473 2515 y Fo(E)1540 2529 y Fk(\033)1610 2515 y Fr(+)g Fo(\033)s(=)p Fr(2[\).)55 b(This)33 b(implies)f(that)2739 2492 y(~)2715 2515 y Fo(H)2791 2529 y Fk(\033)2872 2515 y Fr(and)i(hence)h Fo(H)3386 2529 y Fk(\033)3467 2515 y Fr(admit)f(a)0 2628 y(ground)29 b(state.)42 b Fc(2)0 2870 y Fe(3.4)112 b(The)38 b(pullthrough)e(form)m(ula)0 3042 y Fr(As)g(in)f([BFS)q(],)j(w)m(e)e (shall)f(use)h(the)g(pullthrough)d(form)m(ula)j(to)g(get)i(con)m(trol)e (on)h(the)f(ground)f(states)i(of)g Fo(H)3708 3056 y Fk(\033)3754 3042 y Fr(.)0 3155 y(Since)e(the)i(domain)e Fo(H)811 3169 y Fk(\033)893 3155 y Fr(is)h(not)g(explicitely)e(kno)m(wn)i(under) f(assumption)f Fq(\(I1\))p Fr(,)39 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y Fr(=)25 b Fo(a)p Fr(\()p Fo(k)s Fr(\)\()p Fo(H)973 5281 y Fk(\033)n(;\034)1096 5267 y Fp(\000)20 b Fo(z)t Fr(\))1268 5234 y Fn(\000)p Fm(1)1363 5267 y Fo( )j Fr(+)d(\()p Fo(H)1647 5281 y Fk(\033)n(;\034)1769 5267 y Fr(+)g Fo(!)s Fr(\()p Fo(k)s Fr(\))h Fp(\000)f Fo(z)t Fr(\))2233 5234 y Fn(\000)p Fm(1)2328 5267 y Fo(v)2372 5281 y Fk(\033)n(;\034)2473 5267 y Fr(\()p Fo(k)s Fr(\)\()p Fo(H)2704 5281 y Fk(\033)n(;\034)2827 5267 y Fp(\000)g Fo(z)t Fr(\))2999 5234 y Fn(\000)p Fm(1)3094 5267 y Fo( )s(;)0 5175 y Fr(\(3.5\))1867 5517 y(6)p eop %%Page: 7 7 7 6 bop 0 162 a Fr(holds)29 b(as)h(an)h(iden)m(tit)m(y)f(in)f Fo(L)980 129 y Fm(2)980 189 y(lo)r(c)1072 162 y Fr(\(I)-15 b(R)1192 126 y Fk(d)1232 162 y Fp(nf)p Fr(0)p Fp(g)p Fo(;)15 b Fr(d)p Fo(k)s Fr(;)g Fp(H)q Fr(\).)141 292 y(By)36 b Fq(\(I1\))p Fr(,)j(\()p Fo(v)587 306 y Fk(\033)n(;\034)689 292 y Fr(\()p Fo(k)s Fr(\))25 b Fp(\000)e Fo(v)972 306 y Fk(\033)1019 292 y Fr(\()p Fo(k)s Fr(\)\)\()p Fo(H)1285 306 y Fm(0)1350 292 y Fr(+)g(1\))1524 256 y Fn(\000)1589 229 y Fi(1)p 1590 241 31 4 v 1590 282 a(2)1670 292 y Fr(tends)36 b(to)h(0)f(in)f Fo(L)2291 259 y Fm(2)2330 292 y Fr(\(I)-15 b(R)2450 256 y Fk(d)2490 292 y Fp(nf)p Fr(0)p Fp(g)p Fo(;)15 b Fr(d)p Fo(k)s Fr(;)g Fo(B)5 b Fr(\()p Fp(K)q Fr(\)\))40 b(when)35 b Fo(\034)44 b Fp(!)35 b Fr(+)p Fp(1)p Fr(.)0 405 y(Using)30 b(also)g(\(3.2\))i(and)e(letting) g Fo(\034)35 b Fp(!)25 b Fr(+)p Fp(1)30 b Fr(w)m(e)h(obtain)211 596 y(\()p Fo(H)322 610 y Fk(\033)389 596 y Fr(+)20 b Fo(!)s Fr(\()p Fo(k)s Fr(\))h Fp(\000)f Fo(z)t Fr(\))853 559 y Fn(\000)p Fm(1)948 596 y Fo(a)p Fr(\()p Fo(k)s Fr(\))p Fo( )29 b Fr(=)c Fo(a)p Fr(\()p Fo(k)s Fr(\)\()p Fo(H)1579 610 y Fk(\033)1647 596 y Fp(\000)20 b Fo(z)t Fr(\))1819 559 y Fn(\000)p Fm(1)1914 596 y Fo( )k Fr(+)c(\()p Fo(H)2199 610 y Fk(\033)2266 596 y Fr(+)f Fo(!)s Fr(\()p Fo(k)s Fr(\))i Fp(\000)f Fo(z)t Fr(\))2729 559 y Fn(\000)p Fm(1)2824 596 y Fo(v)2868 610 y Fk(\033)2915 596 y Fr(\()p Fo(k)s Fr(\)\()p Fo(H)3146 610 y Fk(\033)3214 596 y Fp(\000)g Fo(z)t Fr(\))3386 559 y Fn(\000)p Fm(1)3481 596 y Fo( )s(;)0 787 y Fr(as)31 b(claimed.)39 b Fc(2)0 1072 y Fs(4)135 b(Pro)t(of)45 b(of)g(Thm.)59 b(1)0 1275 y Fr(Let)1138 1388 y Fo(E)1205 1402 y Fk(\033)1278 1388 y Fr(:=)25 b(inf)c(sp)s(ec)o(\()p Fo(H)1805 1402 y Fk(\033)1852 1388 y Fr(\))p Fo(;)36 b(E)31 b Fr(:=)25 b(inf)c(sp)s(ec)o(\()p Fo(H)7 b Fr(\))p Fo(:)0 1548 y Fr(W)-8 b(e)29 b(denote)g(b)m(y)f Fo( )628 1562 y Fk(\033)675 1548 y Fr(,)g Fo(\033)h(>)c Fr(0)j(a)h(normalized)d(ground)h(state)i(of)g Fo(H)2218 1562 y Fk(\033)2264 1548 y Fr(.)40 b(Applying)25 b(the)k(pullthrough)c (form)m(ula)i(to)0 1661 y Fo( )59 1675 y Fk(\033)106 1661 y Fr(,)k(w)m(e)f(obtain)g(easily)g(the)g(follo)m(wing)f(iden)m (tit)m(y)h(on)g Fo(L)1893 1628 y Fm(2)1933 1661 y Fr(\(I)-15 b(R)2053 1625 y Fk(d)2093 1661 y Fo(;)15 b Fr(d)p Fo(k)s Fr(;)g Fp(H)q Fr(\):)1118 1852 y Fo(a)p Fr(\()p Fo(k)s Fr(\))p Fo( )1345 1866 y Fk(\033)1418 1852 y Fr(=)25 b(\()p Fo(E)1616 1866 y Fk(\033)1684 1852 y Fp(\000)20 b Fo(H)1851 1866 y Fk(\033)1917 1852 y Fp(\000)g Fo(!)s Fr(\()p Fo(k)s Fr(\)\))2223 1815 y Fn(\000)p Fm(1)2319 1852 y Fo(v)2363 1866 y Fk(\033)2409 1852 y Fr(\()p Fo(k)s Fr(\))p Fo( )2588 1866 y Fk(\033)2636 1852 y Fo(:)-2661 b Fr(\(4.1\))0 2043 y(The)30 b(\014rst)f(rather)i(ob)m(vious)f(b)s (ound)e(on)i(the)h(family)d(of)j(ground)e(states)j Fo( )2547 2057 y Fk(\033)2624 2043 y Fr(is)e(the)g(follo)m(wing.)0 2220 y Fd(Lemma)j(4.1)46 b Fq(Assume)32 b(hyp)-5 b(otheses)35 b(\(H0\),)e(\(H1\),)g(\(I1\).)42 b(Then)1154 2411 y Fr(\()p Fo( )1248 2425 y Fk(\033)1295 2411 y Fo(;)15 b(H)1411 2425 y Fm(0)1451 2411 y Fo( )1510 2425 y Fk(\033)1557 2411 y Fr(\))25 b Fp(\024)g Fo(C)q(;)49 b Fq(uniformly)33 b(in)g Fo(\033)28 b(>)d Fr(0)p Fo(:)-2625 b Fr(\(4.2\))0 2779 y(The)31 b(b)s(ound)e(\(4.2\))34 b(follo)m(ws)c(immediately)g (from)h(the)g(fact)i(that)f(the)f(quadratic)g(forms)g Fo(Q)3147 2793 y Fk(\033)3225 2779 y Fr(are)h(equiv)-5 b(alen)m(t)0 2892 y(to)36 b(the)f Fo(Q)349 2906 y Fm(0)388 2892 y Fr(,)h(uniformly)c(in)i Fo(\033)s Fr(.)55 b(The)34 b(follo)m(wing)f(lemma)i(is)f(also)h(w)m(ell-kno)m(wn)e(\(see)j(eg)g ([BFS)q(,)g(Thm.)53 b(I)s(I.5],)0 3005 y([AH)q(,)30 b(Lemma)h(4.3]\).) 42 b(W)-8 b(e)32 b(denote)f(b)m(y)f Fo(N)40 b Fr(the)31 b(n)m(um)m(b)s(er)e(op)s(erator)h(on)h(\000\()p Fl(h)p Fr(\).)0 3182 y Fd(Lemma)i(4.2)46 b Fq(Assume)32 b(hyp)-5 b(otheses)35 b(\(H0\),)e(\(H1\),)g(\(I1\),)g(\(I2\).)43 b(Then)1170 3373 y Fr(\()p Fo( )1264 3387 y Fk(\033)1311 3373 y Fo(;)15 b(N)10 b( )1493 3387 y Fk(\033)1541 3373 y Fr(\))25 b Fp(\024)g Fo(C)q(;)48 b Fq(uniformly)34 b(in)f Fo(\033)28 b(>)d Fr(0)p Fo(:)-2609 b Fr(\(4.3\))0 3741 y Fd(Pro)s(of.)47 b Fr(W)-8 b(e)32 b(ha)m(v)m(e)g(using)c(\(4.1\)) 709 3930 y(\()p Fo( )803 3944 y Fk(\033)850 3930 y Fo(;)15 b(N)10 b( )1032 3944 y Fk(\033)1080 3930 y Fr(\))83 b(=)1294 3861 y Fj(R)1365 3930 y Fp(k)p Fo(a)p Fr(\()p Fo(k)s Fr(\))p Fo( )1637 3944 y Fk(\033)1685 3930 y Fp(k)1730 3897 y Fm(2)1770 3930 y Fr(d)p Fo(k)1198 4114 y Fr(=)1294 4045 y Fj(R)1365 4114 y Fp(k)p Fr(\()p Fo(E)1512 4128 y Fk(\033)1580 4114 y Fp(\000)20 b Fo(H)1747 4128 y Fk(\033)1793 4114 y Fr(\()p Fo(k)s Fr(\))h Fp(\000)f Fo(!)s Fr(\()p Fo(k)s Fr(\)\))2240 4081 y Fn(\000)p Fm(1)2335 4114 y Fo(v)2379 4128 y Fk(\033)2426 4114 y Fr(\()p Fo(k)s Fr(\))p Fo( )2605 4128 y Fk(\033)2653 4114 y Fp(k)2698 4081 y Fm(2)2738 4114 y Fr(d)p Fo(k)1198 4313 y Fp(\024)25 b(k)p Fr(\()p Fo(H)1450 4327 y Fm(0)1510 4313 y Fr(+)20 b(1\))1691 4250 y Fi(1)p 1692 4262 V 1692 4304 a(2)1736 4313 y Fo( )1795 4327 y Fk(\033)1843 4313 y Fp(k)1888 4280 y Fm(2)1943 4245 y Fj(R)2093 4278 y Fm(1)p 2023 4293 175 4 v 2023 4347 a Fk(!)r Fm(\()p Fk(k)r Fm(\))2162 4328 y Fi(2)2207 4313 y Fp(k)p Fo(v)2296 4327 y Fk(\033)2344 4313 y Fr(\()p Fo(k)s Fr(\)\()p Fo(K)28 b Fr(+)20 b(1\))2775 4278 y Fn(\000)2840 4250 y Fi(1)p 2840 4262 31 4 v 2840 4304 a(2)2885 4313 y Fp(k)2930 4280 y Fm(2)2970 4313 y Fr(d)p Fo(k)1198 4497 y Fp(\024)25 b Fo(C)q(;)0 4686 y Fr(uniformly)i(in)i Fo(\033)g(>)c Fr(0)31 b(using)e Fq(\(I2\))i Fr(and)e(\(4.2\).)43 b Fc(2)0 4885 y Fd(Lemma)33 b(4.3)46 b Fq(Assume)32 b(hyp)-5 b(otheses)35 b(\(H0\),)e(\(H1\),)g(\(I1\),)g(\(I2\).)43 b(Then)1588 5077 y Fo(E)25 b Fp(\000)20 b Fo(E)1838 5091 y Fk(\033)1911 5077 y Fp(2)k Fo(o)p Fr(\()p Fo(\033)s Fr(\))p Fo(:)-2190 b Fr(\(4.4\))1867 5517 y(7)p eop %%Page: 8 8 8 7 bop 0 162 a Fd(Pro)s(of.)47 b Fr(Let)31 b(0)26 b Fo(<)f(\033)714 129 y Fn(0)763 162 y Fo(<)g(\033)s Fr(.)41 b(W)-8 b(e)31 b(ha)m(v)m(e)784 343 y Fo(E)851 358 y Fk(\033)893 339 y Fa(0)941 343 y Fp(\000)20 b Fo(E)1099 357 y Fk(\033)1171 343 y Fp(\024)25 b Fr(\()p Fo(Q)1374 358 y Fk(\033)1416 339 y Fa(0)1464 343 y Fp(\000)20 b Fo(Q)1627 357 y Fk(\033)1673 343 y Fr(\)\()p Fo( )1802 357 y Fk(\033)1850 343 y Fo(;)15 b( )1949 357 y Fk(\033)1997 343 y Fr(\))25 b(=)g(\()p Fo(V)2241 358 y Fk(\033)2283 339 y Fa(0)2331 343 y Fp(\000)20 b Fo(V)2475 357 y Fk(\033)2522 343 y Fr(\)\()p Fo( )2651 357 y Fk(\033)2699 343 y Fo(;)15 b( )2798 357 y Fk(\033)2845 343 y Fr(\))p 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Fi(1)p 2272 891 31 4 v 2272 933 a(2)2317 944 y Fr(\()p Fo(u;)15 b Fr(\()p Fo(K)28 b Fr(+)20 b(1\))p Fo(u)p Fr(\))2852 879 y Fi(1)p 2853 891 V 2853 933 a(2)2897 944 y Fo(;)-2922 b Fr(\(4.6\))0 1147 y(for)932 1270 y Fo(C)7 b Fr(\()p Fo(\033)1094 1232 y Fn(0)1118 1270 y Fo(;)15 b(\033)s Fr(\))26 b(=)f(\()1420 1155 y Fj(Z)1466 1344 y Fn(f)p Fk(\033)1543 1325 y Fa(0)1567 1344 y Fk(