Content-Type: multipart/mixed; boundary="-------------0407161252791" This is a multi-part message in MIME format. ---------------0407161252791 Content-Type: text/plain; name="04-218.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-218.keywords" Pauli-Fierz Hamiltonians, Fock spaces, quadratic form, spectral theory ---------------0407161252791 Content-Type: application/postscript; name="ground.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ground.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: ground.dvi %%Pages: 26 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips ground.dvi -o ground.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2004.07.16:1334 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 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Fn(3)24 b Fp(\(\010)p Fo(;)14 b Fp(\011\))23 b Fn(7!)g Fp(\()p Fn(j)p Fo(h)p Fn(j)2258 3976 y Fk(1)p Fi(=)p Fk(2)2363 4011 y Fp(\010)p Fn(j)k Fp(sgn)p Fo(h)g Fn(j)p Fo(h)p Fn(j)2763 3976 y Fk(1)p Fi(=)p Fk(2)2867 4011 y Fp(\011\))p Fo(:)125 4182 y Fp(If)h(\011)22 b Fn(2)i Fp(Dom)548 4194 y Fk(r)578 4182 y Fo(v)s Fp(,)k(then)g(w)n(e)f(set)1004 4354 y Fn(k)p Fo(v)s Fp(\011)p Fn(k)22 b Fp(:=)h(sup)p Fn(fj)p Fp(\(\010)p Fn(j)p Fo(v)s Fp(\011\))p Fn(j)51 b Fp(:)f(\010)23 b Fn(2)h Fp(Dom)2257 4366 y Fk(l)2280 4354 y Fo(v)s(;)69 b Fn(k)p Fp(\010)p Fn(k)22 b Fp(=)h(1)p Fn(g)p Fo(:)0 4526 y Fp(Note)i(that)h(the)g(ab)r(o)n(v)n(e)e(notation)g(agrees)g (with)i(the)f(usual)g(notation)g(if)h Fo(v)i Fp(is)d(giv)n(en)g(b)n(y)g (an)g(op)r(erator)f(and)h(\011)d Fn(2)i Fp(Dom)18 b Fo(v)s Fp(.)125 4625 y(W)-7 b(e)28 b(will)f(also)g(write)602 4797 y Fn(k)p Fo(v)s Fn(k)22 b Fp(:=)h(sup)p Fn(fj)p Fp(\(\010)p Fn(j)p Fo(v)s Fp(\011\))p Fn(j)50 b Fp(:)h(\010)23 b Fn(2)h Fp(Dom)1790 4809 y Fk(l)1813 4797 y 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y Fk(r)2354 5168 y Fo(h)2402 5138 y Fl(\003)2402 5189 y Fk(2)2440 5168 y Fo(v)s(h)2531 5180 y Fk(1)2596 5168 y Fp(:=)c Fn(f)p Fp(\011)h Fn(2)g Fp(Dom)18 b Fo(h)3169 5180 y Fk(1)3207 5168 y Fn(j)p Fo(h)3278 5180 y Fk(1)3315 5168 y Fp(\011)27 b Fn(2)i Fp(Dom)3665 5180 y Fk(r)3695 5168 y Fo(v)s Fn(g)0 5268 y Fp(and)e(Dom)336 5280 y Fk(l)358 5268 y Fo(h)406 5238 y Fl(\003)406 5288 y Fk(2)444 5268 y Fo(v)s(h)535 5280 y Fk(1)596 5268 y Fp(:=)c Fn(f)p Fp(\010)f Fn(2)h Fp(Dom)c Fo(h)1150 5280 y Fk(2)1187 5268 y Fn(j)p Fo(h)1258 5280 y Fk(2)1295 5268 y Fp(\010)k Fn(2)h Fp(Dom)1631 5280 y Fk(l)1654 5268 y Fo(v)s Fn(g)p Fo(:)1869 5753 y Fp(4)p eop %%Page: 5 5 5 4 bop 0 390 a Ff(2.3)112 b(T)-9 b(ensor)38 b(pro)s(ducts)f(and)i(F)-9 b(o)s(c)m(k)37 b(spaces)125 573 y Fp(If)30 b Fn(K)273 585 y Fk(0)341 573 y Fp(and)f Fn(Z)564 585 y Fk(0)632 573 y Fp(are)g(v)n(ector)g(spaces,)g(then)i Fn(K)1560 585 y Fk(0)1613 508 y Fl(\016)1597 573 y Fn(\012Z)1722 585 y Fk(0)1789 573 y Fp(will)g(denote)f(their)g(algebraic)e(tensor)h (pro)r(duct.)3385 507 y Fl(\016)3376 581 y Fp(\000)3428 585 y Fk(s)3458 573 y Fp(\()p Fn(Z)3550 585 y Fk(0)3588 573 y Fp(\))h(will)0 672 y(denote)e(the)f(algebraic)f(symmetric)i(F)-7 b(o)r(c)n(k)27 b(space)g(o)n(v)n(er)e Fn(Z)1822 684 y Fk(0)1860 672 y Fp(.)125 772 y(If)41 b Fn(K)h Fp(and)f Fn(Z)47 b Fp(are)40 b(Hilb)r(ert)i(spaces,)h(then)e Fn(K)29 b(\012)e(Z)47 b Fp(will)41 b(denote)g(their)g(Hilb)r(ert)g(space)f (tensor)g(pro)r(duct)h(\(the)0 902 y(completion)34 b(of)f Fn(K)611 837 y Fl(\016)595 902 y Fn(\012Z)7 b Fp(\).)56 b(\000)890 914 y Fk(s)920 902 y Fp(\()p Fn(Z)7 b Fp(\))34 b(will)g(denote)g(the)g(symmetric)g(F)-7 b(o)r(c)n(k)33 b(space)g(o)n(v)n(er)f Fn(Z)41 b Fp(\(the)34 b(completion)g(of)3519 836 y Fl(\016)3511 910 y Fp(\000)3562 914 y Fk(s)3593 902 y Fp(\()p Fn(Z)7 b Fp(\)\).)0 1002 y(\000)52 971 y Fi(n)52 1022 y Fk(s)97 1002 y Fp(\()p Fn(Z)g Fp(\))29 b(will)h(stand)e(for)h(its)g Fo(n)g Fp(particle)f(subspace)g(\(the)i (completion)f(of)g(the)g(algebraic)e(symmetric)i(tensor)f(p)r(o)n(w)n (er)g(of)0 1136 y Fn(Z)60 1148 y Fk(0)97 1136 y Fp(,)g(denoted)g(b)n(y) 586 1070 y Fl(\016)577 1144 y Fp(\000)629 1056 y Fi(n)629 1157 y Fk(s)674 1136 y Fp(\()p Fn(Z)766 1148 y Fk(0)804 1136 y Fp(\)\).)125 1236 y(If)j Fo(v)i Fp(is)e(a)f(form)g(from)g Fn(H)911 1248 y Fk(1)979 1236 y Fp(to)h Fn(H)1154 1248 y Fk(2)1222 1236 y Fp(and)f Fo(w)j Fp(is)e(another)f(form)g(from)g Fn(K)2334 1248 y Fk(1)2402 1236 y Fp(to)h Fn(K)2570 1248 y Fk(2)2607 1236 y Fp(,)g(then)h(the)e(tensor)g(pro)r(duct)h(of)f(the)0 1335 y(forms)35 b Fo(v)40 b Fp(and)35 b Fo(w)k Fp(is)d(de\014ned)g(as)f (the)i(form)e Fo(v)28 b Fn(\012)1569 1305 y Fk(f)1622 1335 y Fo(w)38 b Fp(from)e Fn(H)1994 1347 y Fk(1)2055 1335 y Fn(\012)24 b(K)2207 1347 y Fk(1)2280 1335 y Fp(to)36 b Fn(H)2460 1347 y Fk(2)2522 1335 y Fn(\012)23 b(K)2673 1347 y Fk(2)2711 1335 y Fp(,)38 b(suc)n(h)d(that)i(Dom)3330 1347 y Fi(r)3367 1335 y Fp(\()p Fo(v)27 b Fn(\012)3531 1305 y Fk(f)3584 1335 y Fo(w)r Fp(\))38 b(=)0 1465 y(Dom)174 1477 y Fk(r)204 1465 y Fp(\()p Fo(v)s Fp(\))327 1400 y Fl(\016)311 1465 y Fn(\012)q Fp(Dom)551 1477 y Fk(r)581 1465 y Fp(\()p Fo(w)r Fp(\),)29 b(Dom)932 1477 y Fi(l)957 1465 y Fp(\()p Fo(v)22 b Fn(\012)1116 1435 y Fk(f)1164 1465 y Fo(w)r Fp(\))i(=)e(Dom)1542 1477 y Fk(l)1565 1465 y Fp(\()p Fo(v)s Fp(\))1689 1400 y Fl(\016)1672 1465 y Fn(\012)q Fp(Dom)1912 1477 y Fk(l)1935 1465 y Fp(\()p Fo(w)r Fp(\))29 b(and)981 1571 y Fm(\000)1019 1638 y Fp(\011)1084 1650 y Fk(2)1140 1638 y Fn(\012)18 b Fp(\004)1278 1650 y Fk(2)1315 1638 y Fn(j)p Fp(\()p Fo(v)k Fn(\012)1497 1604 y Fk(f)1545 1638 y Fo(w)r Fp(\)\011)1703 1650 y Fk(1)1759 1638 y Fn(\012)c Fp(\004)1897 1650 y Fk(1)1934 1571 y Fm(\001)1996 1638 y Fp(:=)k(\(\011)2203 1650 y Fk(2)2240 1638 y Fn(j)p Fo(v)s Fp(\011)2371 1650 y Fk(1)2408 1638 y Fp(\)\(\004)2527 1650 y Fk(2)2566 1638 y Fn(j)p Fo(w)r Fp(\004)2705 1650 y Fk(1)2743 1638 y Fp(\))p Fo(:)125 1811 y Fp(If)j Fo(v)j Fp(is)c(an)h(op)r(erator)e(from)h Fn(H)1061 1823 y Fk(1)1123 1811 y Fp(to)h Fn(H)1292 1823 y Fk(2)1354 1811 y Fp(and)g Fo(w)i Fp(is)e(an)f(op)r(erator)f(from)i Fn(K)2381 1823 y Fk(1)2443 1811 y Fp(to)g Fn(K)2605 1823 y Fk(2)2642 1811 y Fp(,)h(then)f(the)g(tensor)f(pro)r(duct)h(of)f(the)0 1911 y(op)r(erators)h Fo(v)31 b Fp(and)c Fo(w)k Fp(is)c(de\014ned)h(as) e(the)i(op)r(erator)e Fo(v)21 b Fn(\012)d Fo(w)30 b Fp(from)d Fn(H)2136 1923 y Fk(1)2191 1911 y Fn(\012)18 b(K)2337 1923 y Fk(1)2402 1911 y Fp(to)27 b Fn(H)2573 1923 y Fk(2)2629 1911 y Fn(\012)17 b(K)2774 1923 y Fk(2)2812 1911 y Fp(,)27 b(suc)n(h)g(that)h(Dom)19 b(\()p Fo(v)i Fn(\012)d Fo(w)r Fp(\))24 b(=)0 2041 y(Dom)19 b(\()p Fo(v)s Fp(\))316 1976 y Fl(\016)300 2041 y Fn(\012)p Fp(Dom)f(\()p Fo(w)r Fp(\))29 b(and)1337 2140 y(\()p Fo(v)22 b Fn(\012)c Fo(w)r Fp(\)\011)h Fn(\012)f Fp(\004)24 b(:=)f(\()p Fo(v)s Fp(\011\))18 b Fn(\012)g Fp(\()p Fo(w)r Fp(\004\))p Fo(:)125 2284 y Fp(Th)n(us)26 b(if)h Fo(v)i Fp(and)e Fo(w)i Fp(are)c(op)r(erators,)g (then)i(w)n(e)f(ha)n(v)n(e)f(t)n(w)n(o)h(sligh)n(tly)g(di\013eren)n(t)g (tensor)g(pro)r(ducts)g(of)g Fo(v)k Fp(and)c Fo(w)r Fp(.)38 b(In)26 b(the)0 2409 y(case)i(of)g(the)h(form)f(tensor)g(pro)r(duct)g (Dom)1350 2421 y Fk(l)1373 2409 y Fp(\()p Fo(v)23 b Fn(\012)1533 2379 y Fk(f)1581 2409 y Fo(w)r Fp(\))i(=)f Fn(H)1858 2421 y Fk(2)1911 2344 y Fl(\016)1896 2409 y Fn(\012)o(K)2023 2421 y Fk(2)2089 2409 y Fp(and)29 b(in)f(the)h(case)f(of)g(the)h(op)r (erator)e(tensor)h(pro)r(duct)0 2508 y(Dom)174 2520 y Fk(l)197 2508 y Fp(\()p Fo(v)22 b Fn(\012)c Fo(w)r Fp(\))24 b(=)f Fn(H)649 2520 y Fk(2)705 2508 y Fn(\012)18 b(K)851 2520 y Fk(2)916 2508 y Fp(.)125 2608 y(If)28 b Fo(v)k Fp(and)d Fo(w)i Fp(are)c(closed)h(op)r(erators,)f(then)i(the)g(ab)r(o)n (v)n(e)e(de\014ned)i Fo(v)22 b Fn(\012)d Fo(w)31 b Fp(is)e(a)f (closable)f(op)r(erator.)38 b(W)-7 b(e)29 b(will)g(denote)0 2708 y(b)n(y)h(the)h(same)f(sym)n(b)r(ol)g(its)h(closure.)44 b(If)31 b(they)g(are)e(densely)i(de\014ned,)g(then)g(so)f(is)g Fo(v)24 b Fn(\012)c Fo(w)r Fp(.)46 b(If)31 b Fo(v)j Fp(and)c Fo(w)k Fp(are)29 b(b)r(ounded,)0 2807 y(then)f Fo(v)22 b Fn(\012)c Fo(w)30 b Fp(is)e(b)r(ounded)g(as)f(w)n(ell.)125 2907 y(If)h Fo(h)g Fp(is)g(a)g(closed)f(op)r(erator)f(on)i Fn(Z)7 b Fp(,)28 b(then)h(d\000\()p Fo(h)p Fp(\))f(denotes)g(the)g (closed)g(op)r(erator)e(on)i(\000)2873 2919 y Fk(s)2903 2907 y Fp(\()p Fn(Z)7 b Fp(\))29 b(de\014ned)f(in)h(the)f(usual)0 3007 y(w)n(a)n(y)-7 b(.)34 b(If)24 b Fo(q)j Fp(is)c(a)g(con)n(traction) e(from)i Fn(Z)1162 3019 y Fk(1)1223 3007 y Fp(to)h Fn(Z)1381 3019 y Fk(2)1418 3007 y Fp(,)g(then)g(\000\()p Fo(q)s Fp(\))g(denotes)f(the)h(con)n(traction)e(from)h(\000)2941 3019 y Fk(s)2971 3007 y Fp(\()p Fn(Z)3063 3019 y Fk(1)3101 3007 y Fp(\))g(to)g(\000)3305 3019 y Fk(s)3336 3007 y Fp(\()p Fn(Z)3428 3019 y Fk(2)3466 3007 y Fp(\))g(de\014ned)0 3106 y(in)28 b(the)g(usual)f(w)n(a)n(y)f(\(see)i(e.g.)36 b([5,)28 b(18)o(]\).)125 3206 y(Finally)-7 b(,)27 b Fo(N)37 b Fp(will)27 b(denote)h(the)g(n)n(um)n(b)r(er)f(op)r(erator,)f Fg(i.e.)39 b Fo(N)31 b Fp(=)23 b(d\000\(1\).)0 3437 y Ff(2.4)112 b(Creation)37 b(and)h(annihilation)e(forms)125 3590 y Fp(Let)23 b Fn(K)332 3602 y Fk(0)393 3590 y Fp(and)g Fn(Z)610 3602 y Fk(0)670 3590 y Fp(b)r(e)h(subspaces)e(of)h(Hilb)r(ert) h(spaces)e Fn(K)j Fp(and)e Fn(Z)7 b Fp(,)24 b(resp)r(ectiv)n(ely)-7 b(.)35 b(Let)23 b Fo(v)k Fp(b)r(e)c(a)g(form)g(from)g Fn(K)h Fp(to)f Fn(K)11 b(\012)f(Z)0 3715 y Fp(with)28 b(the)g(righ)n(t)f(domain)g Fn(K)891 3727 y Fk(0)956 3715 y Fp(and)h(the)g(left)g(domain)f Fn(K)1764 3727 y Fk(0)1817 3650 y Fl(\016)1802 3715 y Fn(\012)o(Z)1926 3727 y Fk(0)1964 3715 y Fp(.)125 3814 y(W)-7 b(e)39 b(de\014ne)h(the)f (annihilation)g(form)g Fo(a)1410 3784 y Fk(f)1439 3814 y Fp(\()p Fo(v)s Fp(\))h(as)f(a)g(form)f(on)h Fn(K)28 b(\012)e Fp(\000)2349 3826 y Fk(s)2379 3814 y Fp(\()p Fn(Z)7 b Fp(\))40 b(with)f(the)h(\(left)g(and)f(righ)n(t\))g(domain)0 3949 y Fn(K)63 3961 y Fk(0)116 3884 y Fl(\016)101 3949 y Fn(\012)174 3883 y Fl(\016)165 3957 y Fp(\000)217 3961 y Fk(s)247 3949 y Fp(\()p Fn(Z)339 3961 y Fk(0)377 3949 y Fp(\).)e(It)28 b(is)g(de\014ned)g(for)f(\010)c Fn(2)g(K)1280 3961 y Fk(0)1333 3884 y Fl(\016)1318 3949 y Fn(\012)1391 3883 y Fl(\016)1382 3957 y Fp(\000)1434 3868 y Fi(m)1434 3969 y Fk(s)1497 3949 y Fp(\()p Fn(Z)1589 3961 y Fk(0)1627 3949 y Fp(\),)28 b(\011)23 b Fn(2)g(K)1939 3961 y Fk(0)1992 3884 y Fl(\016)1977 3949 y Fn(\012)2050 3883 y Fl(\016)2041 3957 y Fp(\000)2093 3868 y Fi(n)2093 3969 y Fk(s)2138 3949 y Fp(\()p Fn(Z)2230 3961 y Fk(0)2268 3949 y Fp(\))28 b(as)886 4198 y(\(\010)p Fn(j)p Fo(a)1045 4163 y Fk(f)1075 4198 y Fp(\()p Fo(v)s Fp(\)\011\))23 b(:=)1413 4056 y Fm(\()1522 4121 y Fp(0)p Fo(;)855 b(m)23 b Fn(6)p Fp(=)g Fo(n)18 b Fn(\000)g Fp(1)p Fo(;)1522 4212 y Fn(p)p 1591 4212 50 4 v 60 x Fo(n)p Fp(\(\010)p Fn(j)p Fo(v)1799 4242 y Fl(\003)p Fk(f)1881 4272 y Fn(\012)g Fp(1)2006 4242 y Fk(\()p Fi(n)p Fl(\000)p Fk(1\))p Fl(\012)2239 4272 y Fp(\011\))p Fo(;)83 b(m)23 b Fp(=)g Fo(n)18 b Fn(\000)g Fp(1)p Fo(:)0 4455 y Fp(The)28 b(creation)e(form)h Fo(a)729 4424 y Fl(\003)p Fk(f)793 4455 y Fp(\()p Fo(v)s Fp(\))h(is)g(de\014ned)g(as)1537 4652 y Fo(a)1581 4618 y Fl(\003)p Fk(f)1644 4652 y Fp(\()p Fo(v)s Fp(\))c(:=)1885 4585 y Fm(\000)1924 4652 y Fo(a)1968 4618 y Fk(f)1997 4652 y Fp(\()p Fo(v)s Fp(\))2104 4585 y Fm(\001)2143 4598 y Fl(\003)p Fk(f)2220 4652 y Fo(:)125 4825 y Fp(Note)j(that)h(if)f Fo(v)k Fp(is)c(b)r(ounded)h(with)g(the)f(righ)n(t)g(domain)g(equal)g (to)g Fn(K)q Fp(,)h(then)g(the)g(form)f Fo(a)2907 4795 y Fk(f)2936 4825 y Fp(\()p Fo(v)s Fp(\))h(is)f(asso)r(ciated)f(with)i (a)0 4925 y(densely)f(de\014ned)h(closable)e(op)r(erator.)35 b(Then)28 b(w)n(e)f(use)g(the)h(sym)n(b)r(ol)e Fo(a)p Fp(\()p Fo(v)s Fp(\))j(for)d(the)i(closure)e(of)i(this)f(op)r(erator.) 35 b(W)-7 b(e)28 b(can)0 5024 y(then)g(in)n(tro)r(duce)f(the)h (creation)f(op)r(erator)1582 5124 y Fo(a)1626 5090 y Fl(\003)1664 5124 y Fp(\()p Fo(v)s Fp(\))d(:=)f(\()p Fo(a)p Fp(\()p Fo(v)s Fp(\)\))2122 5082 y Fl(\003)2174 5124 y Fo(:)125 5268 y Fp(The)k(follo)n(wing)g(lemma)g(is)h(essen)n (tially)e(pro)n(v)n(en)g(in)i([8].)1869 5753 y(5)p eop %%Page: 6 6 6 5 bop 0 408 a Fj(Lemma)38 b(2.1.)44 b Fg(L)l(et)36 b Fo(h)h Fg(b)l(e)f(a)h(p)l(ositive)h(op)l(er)l(ator)f(on)g Fn(Z)7 b Fg(.)58 b(Supp)l(ose)37 b(that)f Fn(Z)2437 420 y Fk(0)2510 408 y Fn(\032)f Fp(Dom)18 b Fo(h)2850 378 y Fk(1)p Fi(=)p Fk(2)2991 408 y Fg(and)37 b Fp(\010)e Fn(2)h(K)3408 420 y Fk(0)3460 343 y Fl(\016)3445 408 y Fn(\012)3519 343 y Fl(\016)3510 416 y Fp(\000)3562 420 y Fk(s)3592 408 y Fp(\()p Fn(Z)3684 420 y Fk(0)3722 408 y Fp(\))p Fg(.)0 508 y(Then)31 632 y Fp(\(1\))42 b(\010)23 b Fn(2)g Fp(Dom)c(\(1)f Fn(\012)g Fp(d\000\()p Fo(h)p Fp(\))918 602 y Fk(1)p Fi(=)p Fk(2)1023 632 y Fp(\))30 b Fg(and)1267 784 y Fn(k)p Fo(a)1353 749 y Fk(f)1382 784 y Fp(\()p Fo(v)s Fp(\)\010)p Fn(k)1591 749 y Fk(2)1652 784 y Fn(\024)22 b(k)p Fo(h)1829 749 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1985 784 y Fo(v)s Fn(k)2070 749 y Fk(2)2107 784 y Fp(\(\010)p Fn(j)p Fp(1)c Fn(\012)g Fp(d\000\()p Fo(h)p Fp(\)\010\))p Fo(:)31 962 y Fp(\(2\))42 b Fg(If,)30 b(mor)l(e)l(over,)h Fo(v)c Fn(2)c(B)s Fp(\()p Fn(K)q Fo(;)14 b Fn(K)20 b(\012)e(Z)7 b Fp(\))p Fg(,)30 b(then)975 1113 y Fn(k)p Fo(a)1061 1078 y Fl(\003)1098 1113 y Fp(\()p Fo(v)s Fp(\)\010)p Fn(k)1307 1078 y Fk(2)1368 1113 y Fn(\024)22 b Fp(\(\010)p Fn(j)p Fo(v)1613 1078 y Fl(\003)1652 1113 y Fo(v)g Fn(\012)c Fp(1\010\))g(+)g Fn(k)p Fo(h)2122 1078 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2278 1113 y Fo(v)s Fn(k)2363 1078 y Fk(2)2400 1113 y Fp(\(\010)p Fn(j)p Fp(1)g Fn(\012)g Fp(d\000\()p Fo(h)p Fp(\)\010\))p Fo(:)0 1291 y Fj(Remark)31 b(2.1.)40 b Fg(If)30 b Fo(B)d Fn(2)d(B)s Fp(\()p Fn(K)q Fp(\))29 b Fg(and)i Fo(z)26 b Fn(2)d(Z)7 b Fo(;)30 b Fg(then)925 1442 y Fo(a)969 1408 y Fl(\003)1020 1442 y Fp(\()q Fo(B)22 b Fn(\012)c(j)p Fo(z)t Fp(\)\))23 b(=)g Fo(B)g Fn(\012)18 b Fo(a)1675 1408 y Fl(\003)1713 1442 y Fp(\()p Fo(z)t Fp(\))p Fo(;)132 b(a)14 b Fp(\()p Fo(B)23 b Fn(\012)18 b(j)p Fo(z)t Fp(\)\))23 b(=)f Fo(B)2541 1408 y Fl(\003)2598 1442 y Fn(\012)c Fo(a)p Fp(\()p Fo(z)t Fp(\))p Fo(;)0 1593 y Fg(wher)l(e)30 b Fo(a)278 1563 y Fl(\003)316 1593 y Fp(\()p Fo(z)t Fp(\))g Fg(and)g Fo(a)p Fp(\()p Fo(z)t Fp(\))g Fg(ar)l(e)g(the)g(usual)f(cr)l(e)l(ation) h(and)h(annihilation)g(op)l(er)l(ators)g(and)f Fn(j)p Fo(z)t Fp(\))f Fg(is)i(de\014ne)l(d)f(in)f(\(2.5\).)0 1820 y Ff(2.5)112 b(P)m(auli-Fierz)36 b(Hamiltonians)f(de\014ned)k(as)f (forms)125 1973 y Fp(Let)27 b Fn(K)i Fp(and)f Fn(Z)34 b Fp(b)r(e)28 b(Hilb)r(ert)g(spaces.)36 b(The)28 b(main)g(space)e(used) i(in)g(our)f(pap)r(er)g(will)h(b)r(e)g Fn(H)c Fp(:=)e Fn(K)e(\012)e Fp(\000)3267 1985 y Fk(s)3298 1973 y Fp(\()p Fn(Z)7 b Fp(\).)125 2073 y(Let)28 b Fo(K)35 b Fp(b)r(e)29 b(a)f(p)r(ositiv)n(e)g(op)r(erator)f(on)h Fn(K)j Fp(and)d Fo(h)h Fp(b)r(e)g(a)f(p)r(ositiv)n(e)g(op)r(erator)f(on)i Fn(Z)7 b Fp(.)39 b Fo(m)25 b Fp(:=)g(inf)20 b Fo(h)29 b Fp(will)g(b)r(e)g(sometimes)0 2172 y(called)h(the)g(mass.)43 b(\(Recall)30 b(that)g(inf)21 b Fo(h)29 b Fp(denotes)h(the)g(in\014m)n (um)h(of)e(the)i(sp)r(ectrum)f(of)g(the)g(self-adjoin)n(t)f(op)r (erator)f Fo(h)p Fp(\).)0 2272 y(The)g(free)f(P)n(auli-Fierz)f(op)r (erator)g(is)h(de\014ned)h(as)f(the)h(self-adjoin)n(t)f(op)r(erator)f (on)h Fn(H)i Fp(giv)n(en)e(b)n(y)1414 2423 y Fo(H)1483 2435 y Fk(fr)1557 2423 y Fp(:=)c Fo(K)h Fn(\012)18 b Fp(1)g(+)g(1)g Fn(\012)g Fp(d\000\()p Fo(h)p Fp(\))p Fo(:)125 2574 y Fp(Let)31 b Fn(K)340 2586 y Fk(0)410 2574 y Fp(b)r(e)h(a)f(dense)h(subspace)f(of)h Fn(K)h Fp(con)n(tained)e(in)h(Dom)40 b Fo(K)2149 2544 y Fk(1)p Fi(=)p Fk(2)2284 2574 y Fp(and)32 b Fn(Z)2510 2586 y Fk(0)2579 2574 y Fp(a)f(dense)h(subspace)f(of)h Fn(Z)38 b Fp(con)n(tained)0 2674 y(in)31 b(Dom)40 b Fo(h)362 2644 y Fk(1)p Fi(=)p Fk(2)466 2674 y Fp(.)47 b(Let)32 b Fo(v)i Fp(b)r(e)e(suc)n(h)f(as)f(in)h(the)h(previous)e(subsection.)47 b(W)-7 b(e)31 b(will)h(refer)e(to)h Fo(v)k Fp(as)30 b(a)h(coupling)f (form.)48 b(The)0 2799 y(P)n(auli-Fierz)26 b(in)n(teraction)h(is)g (de\014ned)h(as)f(a)g(form)g(on)h Fn(K)20 b(\012)e Fp(\000)1915 2811 y Fk(s)1945 2799 y Fp(\()p Fn(Z)7 b Fp(\))p Fo(;)28 b Fp(with)g(the)g(domain)f Fn(K)2817 2811 y Fk(0)2870 2734 y Fl(\016)2855 2799 y Fn(\012)2928 2734 y Fl(\016)2919 2807 y Fp(\000)2971 2811 y Fk(s)3002 2799 y Fp(\()p Fn(Z)3094 2811 y Fk(0)3131 2799 y Fp(\))p Fo(;)h Fp(equal)f(to)1529 2951 y Fo(V)42 b Fp(:=)23 b Fo(a)1774 2916 y Fk(f)5 b Fl(\003)1837 2951 y Fp(\()p Fo(v)s Fp(\))19 b(+)f Fo(a)2090 2916 y Fk(f)2120 2951 y Fp(\()p Fo(v)s Fp(\))p Fo(:)0 3102 y Fp(In)31 b(order)f(to)i(abbreviate)e(the)h(notation,)h(in)g (what)f(follo)n(ws)f(w)n(e)h(will)h(omit)f(the)h(sup)r(erscripts)e(f)39 b(in)31 b(the)h(annihilation)0 3201 y(and)27 b(creation)g(forms.)36 b(W)-7 b(e)28 b(will)g(also)f(often)g(omit)h(the)g(factors)f(of)g(1.)0 3342 y Fj(Theorem)62 b(2.1.)40 b Fg(Supp)l(ose)30 b(that)1174 3493 y Fo(\013)24 b Fp(:=)e(lim)14 b(sup)1410 3559 y Fi(t)p Fl(!1)1629 3493 y Fn(k)p Fo(h)1719 3459 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1875 3493 y Fo(v)s Fp(\()p Fo(t)19 b Fp(+)f Fo(K)6 b Fp(\))2191 3459 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2347 3493 y Fn(k)23 b Fo(<)f Fn(1)p Fo(:)1004 b Fp(\(2.7\))0 3701 y Fg(Then,)27 b(for)f(any)g Fo(t)d(>)g Fp(0)p Fg(,)j Fo(h)803 3671 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)959 3701 y Fo(v)s Fp(\()p Fo(t)9 b Fp(+)g Fo(K)d Fp(\))1256 3671 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1437 3701 y Fg(is)25 b(b)l(ounde)l(d)h(and)g Fo(t)d Fn(7!)g(k)p Fo(h)2239 3671 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2394 3701 y Fo(v)s Fp(\()p Fo(t)9 b Fp(+)g Fo(K)d Fp(\))2691 3671 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2847 3701 y Fn(k)25 b Fg(is)g(a)h(de)l(cr)l(e)l(asing)g(function.)0 3801 y(Ther)l(efor)l(e,) 31 b Fp(lim)14 b(sup)28 b Fg(in)h(\(2.7\))g(c)l(an)g(b)l(e)f(r)l(eplac) l(e)l(d)i(with)f Fp(lim)p Fg(.)39 b(Mor)l(e)l(over,)30 b(the)f(form)g Fo(V)47 b Fg(is)29 b(form)g(b)l(ounde)l(d)g(wrt)f Fo(H)3550 3813 y Fk(fr)3629 3801 y Fg(with)0 3900 y(the)i Fo(H)207 3912 y Fk(fr)258 3900 y Fg(-form)g(b)l(ound)g Fn(\024)22 b Fo(\013)p Fg(.)0 4041 y Fj(Pro)s(of.)55 b Fp(Let)28 b Fo(t)23 b(>)f Fp(0)262 4178 y Fn(j)p Fp(\(\010)p Fn(j)p Fo(V)e Fp(\010\))p Fn(j)83 b Fp(=)23 b(2)p Fn(j)p Fp(\(\010)p Fn(j)p Fo(a)p Fp(\()p Fo(v)s Fp(\)\010\))p Fn(j)83 b(\024)23 b Fp(2)p Fn(k)p Fp(\()p Fo(t)18 b Fp(+)g Fo(K)6 b Fp(\))1727 4148 y Fk(1)p Fi(=)p Fk(2)1831 4178 y Fp(\010)p Fn(kk)p Fp(\()p Fo(t)18 b Fp(+)g Fo(K)6 b Fp(\))2247 4148 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2403 4178 y Fo(a)p Fp(\()p Fo(v)s Fp(\)\010)p Fn(k)1283 4352 y(\024)23 b Fo(\017)p Fp(\(\010)p Fn(j)p Fp(\()p Fo(t)c Fp(+)f Fo(K)6 b Fp(\)\010\))18 b(+)g Fo(\017)2020 4322 y Fl(\000)p Fk(1)2109 4352 y Fn(k)p Fp(\()p Fo(t)g Fp(+)g Fo(K)6 b Fp(\))2423 4322 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2579 4352 y Fo(a)p Fp(\()p Fo(v)s Fp(\)\010)p Fn(k)2832 4322 y Fk(2)1283 4527 y Fn(\024)23 b Fo(\017)p Fp(\(\010)p Fn(j)p Fp(\()p Fo(t)c Fp(+)f Fo(K)6 b Fp(\)\010\))18 b(+)g Fo(\017)2020 4497 y Fl(\000)p Fk(1)2109 4527 y Fn(k)p Fo(h)2199 4497 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2355 4527 y Fo(v)s Fp(\()p Fo(t)h Fp(+)f Fo(K)6 b Fp(\))2671 4497 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2827 4527 y Fn(k)2869 4497 y Fk(2)2905 4527 y Fp(\(\010)p Fn(j)p Fp(d\000\()p Fo(h)p Fp(\)\010\))p Fo(:)3609 4351 y Fp(\(2.8\))0 4690 y(In)28 b(the)g(last)f(step)h(w)n(e)f(used)h(the)g(iden)n(tit)n(y)f(\() p Fo(t)19 b Fp(+)f Fo(K)6 b Fp(\))1614 4660 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1770 4690 y Fo(a)p Fp(\()p Fo(v)s Fp(\))24 b(=)e Fo(a)2090 4623 y Fm(\000)2128 4690 y Fo(v)s Fp(\()p Fo(t)d Fp(+)f Fo(K)6 b Fp(\))2444 4660 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2600 4623 y Fm(\001)2666 4690 y Fp(and)27 b(Lemma)h(2.1)e(\(1\).)125 4801 y(Let)i Fo(\017)c(>)f(\013)p Fp(.)40 b(W)-7 b(e)28 b(c)n(ho)r(ose)f Fo(t)h Fp(suc)n(h)g(that)h Fn(k)p Fo(h)1460 4771 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1615 4801 y Fo(v)s Fp(\()p Fo(t)20 b Fp(+)e Fo(K)6 b Fp(\))1932 4771 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2088 4801 y Fn(k)24 b(\024)g Fo(\017)p Fp(.)38 b(Then)28 b(the)h(righ)n(t)f(hand)g(side)g (of)g(\(2.8\))g(is)g(less)0 4901 y(than)g(or)e(equal)i(to)1481 5000 y Fn(\024)23 b Fo(t\017)p Fn(k)p Fp(\010)p Fn(k)1777 4966 y Fk(2)1831 5000 y Fp(+)18 b Fo(\017)p Fp(\(\010)p Fn(j)p Fo(H)2132 5012 y Fk(fr)2183 5000 y Fp(\010\))p Fo(:)3717 5131 y Fe(2)125 5268 y Fp(Throughout)26 b(the)i(pap)r(er)f(w) n(e)g(will)h(mak)n(e)f(the)h(assumption)1869 5753 y(6)p eop %%Page: 7 7 7 6 bop 0 390 a Fj(Assumption)29 b(A.)63 b Fp(lim)670 440 y Fi(t)p Fl(!1)841 390 y Fn(k)p Fo(h)931 360 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1086 390 y Fo(v)s Fp(\()p Fo(t)19 b Fp(+)f Fo(K)6 b Fp(\))1402 360 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1558 390 y Fn(k)23 b Fo(<)g Fp(1)p Fg(.)125 581 y Fp(The)k(KLMN)h(theorem)f([18)o(])h(implies)f(the)h(follo)n(wing)f (result.)0 744 y Fj(Theorem)62 b(2.2.)40 b Fg(Supp)l(ose)30 b(that)g(Assumption)f(A)h(holds.)40 b(Then)30 b(the)g(op)l(er)l(ator) 1641 922 y Fo(H)g Fp(:=)23 b Fo(H)1920 934 y Fk(fr)1989 922 y Fp(+)18 b Fo(V)0 1099 y Fg(is)25 b(wel)t(l)g(de\014ne)l(d)g(as)g (a)g(form)h(sum.)36 b(The)25 b(form)h(domains)g(of)f Fo(H)1942 1111 y Fk(fr)2017 1099 y Fg(and)g Fo(H)32 b Fg(c)l(oincide,)27 b(that)e(is)g Fp(Dom)18 b Fn(j)p Fo(H)7 b Fn(j)3190 1047 y Fh(1)p 3190 1056 29 3 v 3190 1089 a(2)3256 1099 y Fp(=)22 b(Dom)d Fn(j)p Fo(H)3628 1111 y Fk(fr)3679 1099 y Fn(j)3712 1047 y Fh(1)p 3712 1056 V 3712 1089 a(2)3754 1099 y Fg(.)0 1262 y Fp(The)28 b(op)r(erator)e Fo(H)34 b Fp(de\014ned)28 b(in)g(Theorem)f(2.2)f(will)i(b)r(e)g(called) f(the)h(P)n(auli-Fierz)e(Hamiltonian.)125 1361 y(W)-7 b(e)28 b(will)f(need)h(also)f(the)h(follo)n(wing)e(assumption:)0 1523 y Fj(Assumption)j(B.)41 b Fo(h)713 1493 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)869 1523 y Fo(v)s Fp(\(1)19 b(+)f Fo(K)6 b Fp(\))1197 1493 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1383 1523 y Fg(is)30 b(c)l(omp)l(act.)125 1686 y Fp(Note)j(that)g (Assumption)h(B)f(implies)g(A.)54 b(In)33 b(fact,)i(Assumption)f(B)f (implies)54 b(lim)2648 1735 y Fi(t)p Fl(!1)2819 1686 y Fn(k)p Fo(h)2909 1655 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)3065 1686 y Fo(v)s Fp(\()p Fo(t)22 b Fp(+)g Fo(K)6 b Fp(\))3388 1655 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)3544 1686 y Fn(k)32 b Fp(=)g(0.)0 1810 y(Hence,)c(Assumption)g(B)f(can)g(b)r(e)h(used)g(in) g(Theorem)f(2.2.)125 1909 y(In)h(some)f(of)h(our)g(argumen)n(ts)e(w)n (e)i(will)g(need)h(to)e(use)h(a)g(whole)g(family)g(of)g(P)n(auli-Fierz) e(op)r(erators.)37 b(Eac)n(h)26 b(time)j(w)n(e)0 2009 y(will)22 b(k)n(eep)f Fo(K)28 b Fp(and)21 b Fo(h)h Fp(\014xed,)h(and)e (w)n(e)h(will)g(v)-5 b(ary)21 b(the)h(coupling)f(form)g Fo(v)s Fp(.)36 b(In)22 b(suc)n(h)f(situations)g(the)h(follo)n(wing)f (assumption)0 2109 y(will)28 b(b)r(e)g(often)g(helpful:)0 2271 y Fj(Assumption)h(C.)1427 2370 y Fn(k)p Fo(h)1517 2336 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1672 2370 y Fo(v)s Fp(\()p Fo(t)19 b Fp(+)f Fo(K)6 b Fp(\))1988 2336 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2144 2370 y Fn(k)23 b Fo(<)f(\017:)0 2533 y Fp(W)-7 b(e)28 b(will)g(refer)f(to)g(the)h(ab)r(o)n(v)n(e)e 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b(that)g(w)n(e)f(ha)n(v)n(e)f(the)i (w)n(ell)f(kno)n(wn)g(unitary)h(iden)n(ti\014cation)f(\(see)g(e.g.)37 b([5)o(]\))1291 4926 y Fo(U)32 b Fp(:)23 b(\000)1478 4938 y Fk(s)1508 4926 y Fp(\()p Fn(Z)j(\010)18 b(Z)7 b Fp(\))23 b Fn(!)g Fp(\000)1989 4938 y Fk(s)2019 4926 y Fp(\()p Fn(Z)7 b Fp(\))19 b Fn(\012)f Fp(\000)2304 4938 y Fk(s)2334 4926 y Fp(\()p Fn(Z)7 b Fp(\))p Fo(:)0 5097 y Fp(Th)n(us)27 b(w)n(e)h(ha)n(v)n(e)e(a)h(natural)g(unitary)g (iden)n(ti\014cation)1381 5268 y(1)17 b Fn(\012)i Fo(U)50 b Fp(:)41 b Fn(H)1766 5234 y Fk(ext)1884 5268 y Fn(!)23 b(H)c(\012)f Fp(\000)2214 5280 y Fk(s)2245 5268 y Fp(\()p Fn(Z)7 b Fp(\))p Fo(:)1869 5753 y Fp(9)p eop %%Page: 10 10 10 9 bop 125 390 a Fp(W)-7 b(e)28 b(in)n(tro)r(duce)f(the)h(extended)g (free)f(P)n(auli-Fierz)f(Hamiltonian)1314 573 y Fo(H)1390 539 y Fk(ext)1383 593 y(fr)1508 573 y Fp(:=)c Fo(K)i Fn(\012)18 b Fp(1)g(+)g(1)g Fn(\012)g Fp(d\000\()p Fo(h)h Fn(\010)f Fo(h)p Fp(\))p Fo(:)0 756 y Fp(W)-7 b(e)24 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y(\000\()p Fo(j)403 4910 y Fk(0)441 4898 y Fo(;)14 b(j)512 4910 y Fl(1)583 4898 y Fp(\))p Fo(H)25 b Fn(\000)18 b Fo(H)868 4868 y Fk(ext)962 4898 y Fp(\000\()p Fo(j)1080 4910 y Fk(0)1118 4898 y Fo(;)c(j)1189 4910 y Fl(1)1259 4898 y Fp(\))24 b(=)e Fo(R)1465 4910 y Fk(1)1521 4898 y Fp(+)c Fo(R)1667 4910 y Fk(2)1723 4898 y Fp(+)g Fo(R)1869 4910 y Fk(3)1906 4898 y Fo(;)285 5069 y Fp(\()p Fo(z)k Fn(\000)c Fo(H)537 5039 y Fk(ext)632 5069 y Fp(\))664 5039 y Fl(\000)p Fk(1)753 5069 y Fp(\000\()p Fo(j)871 5081 y Fk(0)909 5069 y Fo(;)c(j)980 5081 y Fl(1)1050 5069 y Fp(\))19 b Fn(\000)f Fp(\000\()p Fo(j)1302 5081 y Fk(0)1340 5069 y Fo(;)c(j)1411 5081 y Fl(1)1481 5069 y Fp(\)\()p Fo(z)22 b Fn(\000)c Fo(H)7 b Fp(\))1797 5039 y Fl(\000)p Fk(1)1910 5069 y Fp(=)22 b Fn(\000)p Fp(\()p Fo(z)g Fn(\000)c Fo(H)2314 5039 y Fk(ext)2408 5069 y Fp(\))2440 5039 y Fl(\000)p Fk(1)2529 5069 y Fp(\()p Fo(R)2624 5081 y Fk(1)2680 5069 y Fp(+)g Fo(R)2826 5081 y Fk(2)2882 5069 y Fp(+)g Fo(R)3028 5081 y Fk(3)3066 5069 y Fp(\)\()p Fo(z)k Fn(\000)c Fo(H)7 b Fp(\))3382 5039 y Fl(\000)p Fk(1)3471 5069 y Fo(;)1848 5753 y Fp(10)p eop %%Page: 11 11 11 10 bop 0 390 a Fp(where)1239 468 y Fo(R)1302 480 y Fk(1)1363 468 y Fp(:=)22 b(\000\()p Fo(j)1591 480 y Fk(0)1629 468 y Fo(;)14 b(j)1700 480 y Fl(1)1770 468 y Fp(\))p Fo(a)p Fp(\(\(1)19 b Fn(\000)f Fo(j)2093 438 y Fl(\003)2088 489 y Fk(0)2131 468 y Fp(\))p Fo(v)s Fp(\))p Fo(;)1239 639 y(R)1302 651 y Fk(2)1363 639 y Fp(:=)k Fo(a)1517 609 y Fl(\003)1569 639 y Fp(\()q(\()p Fo(j)1668 651 y Fk(0)1724 639 y Fn(\000)c Fp(1\))p Fo(v)s(;)c(j)1995 651 y Fl(1)2065 639 y Fo(v)s Fp(\))h(\000\()p Fo(j)2273 651 y Fk(0)2310 639 y Fo(;)f(j)2381 651 y Fl(1)2452 639 y Fp(\))p Fo(;)1239 809 y(R)1302 821 y Fk(3)1363 809 y Fp(:=)22 b(d\000)14 b(\()q(\()p Fo(j)1684 821 y Fk(0)1721 809 y Fo(;)g(j)1792 821 y Fl(1)1863 809 y Fp(\);)g(\([)p Fo(j)2021 821 y Fk(0)2059 809 y Fo(;)g(h)p Fp(])p Fo(;)g Fp([)p Fo(j)2261 821 y Fl(1)2331 809 y Fo(;)g(h)p Fp(]\)\))g Fo(:)0 976 y Fp(F)-7 b(or)27 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x Fi(z)1949 2944 y Fp(~)1931 2966 y Fo(f)9 b Fp(\()p Fo(z)t Fp(\)\()p Fo(z)22 b Fn(\000)c Fo(H)2340 2932 y Fk(ext)2434 2966 y Fp(\))2466 2932 y Fl(\000)p Fk(1)2555 2966 y Fp(\()p Fo(R)2650 2978 y Fk(1)2706 2966 y Fp(+)g Fo(R)2852 2978 y Fk(2)2908 2966 y Fp(+)g Fo(R)3054 2978 y Fk(3)3091 2966 y Fp(\)\()p Fo(z)23 b Fn(\000)18 b Fo(H)7 b Fp(\))3408 2932 y Fl(\000)p Fk(1)3497 2966 y Fp(d)p Fo(z)t Fp(d)p 3632 2920 43 4 v Fo(z)s(:)0 3206 y Fp(Then)28 b(w)n(e)f(use)g Fn(j)p Fo(@)p 549 3185 39 3 v 13 x Fi(z)606 3184 y Fp(~)588 3206 y Fo(f)8 b Fp(\()p Fo(z)t Fp(\))p Fn(j)23 b(\024)g Fo(C)937 3218 y Fi(N)1000 3206 y Fn(j)p Fp(Im)q Fo(z)t Fn(j)1189 3176 y Fi(N)1251 3206 y Fp(,)974 3389 y Fn(k)p Fp(\()p Fo(z)e Fn(\000)d Fo(H)7 b Fp(\))1299 3354 y Fl(\000)p Fk(1)1389 3389 y Fp(\()p Fo(c)18 b Fp(+)g Fo(H)7 b Fp(\))1666 3354 y Fk(1)p Fi(=)p Fk(2)1771 3389 y Fn(k)22 b(\024)h Fo(c)1959 3401 y Fk(1)1996 3389 y Fp(\()p Fn(j)p Fp(Im)q Fo(z)t Fn(j)2217 3354 y Fl(\000)p Fk(1)2323 3389 y Fp(+)18 b Fn(j)p 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Fo(h)p Fp(\()p Fo(\030)t Fp(\))p Fn(k)1879 4572 y Fl(B)q Fk(\()p Fc(R)2006 4555 y Fb(d)2040 4572 y Fl(\012)p Fc(C)2146 4555 y Fb(n)2187 4572 y Fi(;)p Fc(C)2261 4555 y Fb(n)2301 4572 y Fk(\))2384 4555 y Fp(:)53 b Fo(\030)27 b Fn(2)d Fj(R)2674 4521 y Fi(d)2712 4555 y Fn(g)p Fo(:)0 4737 y Fj(Pro)s(of.)55 b Fp(If)28 b Fo(A)g Fp(is)f(a)g(p)r(ositiv)n(e)h (self-adjoin)n(t)f(op)r(erator,)f(then)i(w)n(e)f(ha)n(v)n(e)1422 4973 y Fo(A)1484 4939 y Fk(1)p Fi(=)p Fk(2)1612 4973 y Fp(=)1714 4917 y(1)p 1709 4954 51 4 v 1709 5030 a Fo(\031)1783 4860 y Fm(Z)1866 4881 y Fk(+)p Fl(1)1830 5049 y Fk(0)2077 4917 y Fo(A)p 2012 4954 194 4 v 2012 5030 a(t)18 b Fp(+)g Fo(A)2237 4917 y Fp(d)p Fo(t)p 2225 4954 100 4 v 2225 4971 a Fn(p)p 2294 4971 30 4 v 67 x Fo(t)2334 4973 y(:)1848 5753 y Fp(11)p eop %%Page: 12 12 12 11 bop 0 390 a Fp(Th)n(us,)299 626 y Fn(k)p Fo(h)389 592 y Fk(1)p Fi(=)p Fk(2)493 626 y Fp(\()p Fo(\030)23 b Fn(\000)18 b Fo(\021)s Fp(\))h Fn(\000)f Fo(h)893 592 y Fk(1)p Fi(=)p Fk(2)997 626 y Fp(\()p Fo(\030)t Fp(\))p Fn(k)83 b(\024)1388 570 y Fp(1)p 1384 607 51 4 v 1384 683 a Fo(\031)1458 513 y Fm(Z)1541 534 y Fk(+)p Fl(1)1504 702 y Fk(0)1698 570 y Fp(d)p Fo(t)p 1686 607 100 4 v 1686 624 a Fn(p)p 1755 624 30 4 v 67 x Fo(t)1809 506 y Fm(\015)1809 556 y(\015)1809 605 y(\015)1809 655 y(\015)1931 570 y Fo(h)p Fp(\()p Fo(\030)23 b Fn(\000)18 b Fo(\021)s Fp(\))p 1865 607 430 4 v 1865 683 a Fo(t)h Fp(+)f Fo(h)p Fp(\()p Fo(\030)k Fn(\000)c Fo(\021)s Fp(\))2323 626 y Fn(\000)2482 570 y Fo(h)p Fp(\()p Fo(\030)t Fp(\))p 2416 607 284 4 v 2416 683 a Fo(t)h Fp(+)f Fo(h)p Fp(\()p Fo(\030)t Fp(\))2710 506 y Fm(\015)2710 556 y(\015)2710 605 y(\015)2710 655 y(\015)1226 867 y Fn(\024)1388 810 y Fp(1)p 1384 848 51 4 v 1384 924 a Fo(\031)1458 754 y Fm(Z)1541 774 y Fk(+)p Fl(1)1504 942 y Fk(0)1676 867 y Fp(d)p Fo(t)1752 795 y Fn(p)p 1822 795 30 4 v 1822 867 a Fo(t)1865 746 y Fm(\015)1865 796 y(\015)1865 846 y(\015)1865 896 y(\015)2115 810 y Fp(1)p 1921 848 430 4 v 1921 924 a Fo(t)h Fp(+)f Fo(h)p Fp(\()p Fo(\030)23 b Fn(\000)18 b Fo(\021)s Fp(\))2361 746 y Fm(\015)2361 796 y(\015)2361 846 y(\015)2361 896 y(\015)2421 867 y Fn(k)p Fo(h)p Fp(\()p Fo(\030)k Fn(\000)c Fo(\021)s Fp(\))h Fn(\000)f Fo(h)p Fp(\()p Fo(\030)t Fp(\))p Fn(k)3070 746 y Fm(\015)3070 796 y(\015)3070 846 y(\015)3070 896 y(\015)3247 810 y Fp(1)p 3126 848 284 4 v 3126 924 a Fo(t)h Fp(+)f Fo(h)p Fp(\()p Fo(\030)t Fp(\))3420 746 y Fm(\015)3420 796 y(\015)3420 846 y(\015)3420 896 y(\015)1226 1109 y Fn(\024)1388 1053 y Fp(1)p 1384 1090 51 4 v 1384 1166 a Fo(\031)1458 996 y Fm(Z)1541 1017 y Fk(+)p Fl(1)1504 1185 y Fk(0)1676 1109 y Fp(d)p Fo(t)1866 985 y Fn(p)p 1935 985 30 4 v 68 x Fo(t)p 1762 1090 307 4 v 1762 1166 a Fp(\()p Fo(t)h Fp(+)f Fo(m)p Fp(\))2031 1142 y Fk(2)2078 1109 y Fn(kr)p Fo(h)p Fn(k)2279 1121 y Fl(1)2349 1109 y Fn(j)p Fo(\021)s Fn(j)p Fo(:)3717 1315 y Fe(2)125 1464 y Fp(Recall)27 b(that)h Fo(x)g Fp(is)f(an)h(auxiliary)e(op)r(erator)g (that)h(app)r(ears)g(in)h(Assumption)g(E.)0 1630 y Fj(Prop)s(osition)36 b(3.2.)43 b Fg(Supp)l(ose)35 b(Assumption)f(E)g(holds)i(and)f Fp(inf)21 b Fo(h)31 b(>)g Fp(0)p Fg(.)52 b(L)l(et)34 b Fo(g)i Fg(b)l(e)f(a)g(me)l(asur)l(able)f(function)h(on)f Fj(R)3741 1600 y Fi(d)0 1730 y Fg(with)180 1663 y Fm(R)250 1730 y Fn(j)p Fo(\021)6 b Fp(^)-45 b Fo(g)r Fp(\()p Fo(\021)s Fp(\))p Fn(j)p Fp(d)p Fo(\021)28 b(<)22 b Fn(1)p Fg(.)39 b(Then)31 1855 y Fp(\(1\))j Fn(k)p Fo(h)269 1824 y Fk(1)p Fi(=)p Fk(2)372 1855 y Fo(g)s Fp(\()p Fo(x)p Fp(\))p Fo(h)574 1824 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)750 1855 y Fn(\000)18 b Fo(g)s Fp(\()p Fo(x)p Fp(\))p Fn(k)23 b(\024)f Fo(c)1189 1788 y Fm(R)1258 1855 y Fn(j)p Fo(\021)6 b Fp(^)-45 b Fo(g)s Fp(\()p Fo(\021)s Fp(\))p Fn(j)p Fp(d)p Fo(\021)s Fg(.)31 1987 y Fp(\(2\))42 b Fn(k)p Fo(h)269 1957 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)424 1987 y Fp([)p Fo(g)s Fp(\()p Fo(x)p Fp(\))p Fo(;)14 b(h)p Fp(])p Fo(h)757 1957 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)914 1987 y Fn(k)22 b(\024)h Fp(2)p Fo(c)1158 1921 y Fm(R)1227 1987 y Fn(j)p Fo(\021)6 b Fp(^)-45 b Fo(g)r Fp(\()p Fo(\021)s Fp(\))p Fn(j)p Fp(d)p Fo(\021)s Fg(.)0 2169 y Fj(Pro)s(of.)55 b Fp(If)28 b(w)n(e)f(write)g Fo(g)s Fp(\()p Fo(x)p Fp(\))d(=)f(\(2)p Fo(\031)s Fp(\))1153 2139 y Fl(\000)1215 2116 y Fb(d)p 1215 2125 31 3 v 1216 2159 a Fh(2)1274 2102 y Fm(R)1343 2169 y Fp(e)1380 2139 y Fk(i)p Fi(\021)r Fl(\001)p Fi(x)1499 2169 y Fp(^)-45 b Fo(g)s Fp(\()p Fo(\021)s Fp(\)d)p Fo(\021)s Fp(,)29 b(then)434 2398 y Fo(h)482 2364 y Fk(1)p Fi(=)p Fk(2)586 2398 y Fo(g)s Fp(\()p Fo(x)p Fp(\))p Fo(h)788 2364 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)963 2398 y Fn(\000)18 b Fo(g)s Fp(\()p Fo(x)p Fp(\))84 b(=)e(\(2)p Fo(\031)s Fp(\))1587 2364 y Fl(\000)1649 2342 y Fb(d)p 1649 2351 V 1650 2384 a Fh(2)1708 2285 y Fm(Z)1805 2398 y Fp(e)1842 2364 y Fk(i)p Fi(\021)r Fl(\001)p Fi(x)1958 2331 y Fm(\000)1996 2398 y Fp(e)2033 2364 y Fl(\000)p Fk(i)p Fi(\021)r Fl(\001)p Fi(x)2202 2398 y Fo(h)2250 2364 y Fk(1)p Fi(=)p Fk(2)2354 2398 y Fp(e)2391 2364 y Fk(i)p Fi(\021)r Fl(\001)p Fi(x)2508 2398 y Fo(h)2556 2364 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2730 2398 y Fn(\000)18 b Fp(1)2855 2331 y Fm(\001)2895 2398 y Fp(^)-44 b Fo(g)r Fp(\()p Fo(\021)s Fp(\)d)p Fo(\021)1284 2616 y Fp(=)82 b(\(2)p Fo(\031)s Fp(\))1587 2582 y Fl(\000)1649 2560 y Fb(d)p 1649 2569 V 1650 2602 a Fh(2)1708 2503 y Fm(Z)1805 2616 y Fp(e)1842 2582 y Fk(i)p Fi(\021)r Fl(\001)p Fi(x)1958 2549 y Fm(\000)1996 2616 y Fo(h)2044 2582 y Fk(1)p Fi(=)p Fk(2)2149 2616 y Fp(\()p Fo(\030)23 b Fn(\000)18 b Fo(\021)s Fp(\))p Fo(h)2447 2582 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2603 2616 y Fp(\()p Fo(\030)t Fp(\))h Fn(\000)f Fp(1)2851 2549 y Fm(\001)2892 2616 y Fp(^)-45 b Fo(g)r Fp(\()p Fo(\021)s Fp(\)d)p Fo(\021)1284 2834 y Fp(=)82 b(\(2)p Fo(\031)s Fp(\))1587 2800 y Fl(\000)1649 2777 y Fb(d)p 1649 2786 V 1650 2820 a Fh(2)1708 2721 y Fm(Z)1805 2834 y Fp(e)1842 2800 y Fk(i)p Fi(\021)r Fl(\001)p Fi(x)1958 2767 y Fm(\000)1996 2834 y Fo(h)2044 2800 y Fk(1)p Fi(=)p Fk(2)2149 2834 y Fp(\()p Fo(\030)23 b Fn(\000)18 b Fo(\021)s Fp(\))g Fn(\000)h Fo(h)2549 2800 y Fk(1)p Fi(=)p Fk(2)2653 2834 y Fp(\()p Fo(\030)t Fp(\))2757 2767 y Fm(\001)2795 2834 y Fo(h)2843 2800 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)3000 2834 y Fp(\()p Fo(\030)t Fp(\))s(^)-45 b Fo(g)s Fp(\()p Fo(\021)s Fp(\)d)p Fo(\021)0 3054 y Fp(Using)27 b(Lemma)h(3.2,)f(the)h(norm)f(of)g(this)h(can)f(b)r (e)h(b)r(ounded)g(b)n(y)1211 3275 y(\(2)p Fo(\031)s Fp(\))1367 3241 y Fl(\000)1430 3218 y Fb(d)p 1430 3227 V 1431 3261 a Fh(2)1488 3162 y Fm(Z)1585 3275 y Fp(d)p Fo(\021)s Fn(kr)p Fo(h)p Fn(k)1876 3287 y Fl(1)1946 3275 y Fn(k)p Fo(h)2036 3241 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2191 3275 y Fn(k)2233 3287 y Fl(1)2303 3275 y Fn(j)p Fo(\021)6 b Fp(^)-45 b Fo(g)s Fp(\()p Fo(\021)s Fp(\))p Fn(j)p Fo(:)0 3511 y Fp(Using)27 b(Assumption)h(E)f(and)h Fn(k)p Fo(h)1029 3481 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1184 3511 y Fn(k)1226 3523 y Fl(1)1319 3511 y Fo(<)23 b Fn(1)p Fp(,)28 b(this)g(pro)n(v)n(es)d(\(1\).)0 3611 y(No)n(w,)i(\(2\))h (follo)n(ws)f(from)g(\(1\))h(and)f(the)h(follo)n(wing)e(iden)n(tit)n(y) 658 3794 y Fo(h)706 3759 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)862 3794 y Fp([)p Fo(g)s Fp(\()p Fo(x)p Fp(\))p Fo(;)14 b(h)p Fp(])p Fo(h)1195 3759 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1375 3794 y Fp(=)23 b Fo(h)1511 3759 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1667 3794 y Fo(g)s Fp(\()p Fo(x)p Fp(\))p Fo(h)1869 3759 y Fk(1)p Fi(=)p Fk(2)1992 3794 y Fn(\000)18 b Fo(g)s Fp(\()p Fo(x)p Fp(\))h Fn(\000)f Fo(h)2379 3759 y Fk(1)p Fi(=)p Fk(2)2483 3794 y Fo(g)s Fp(\()p Fo(x)p Fp(\))p Fo(h)2685 3759 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2860 3794 y Fp(+)g Fo(g)s Fp(\()p Fo(x)p Fp(\))p Fo(:)3717 3976 y Fe(2)125 4126 y Fp(Assume)25 b(no)n(w)g(that)i Fo(j)814 4138 y Fk(0)874 4126 y Fn(2)c Fo(C)1017 4096 y Fl(1)1011 4146 y Fk(0)1088 4126 y Fp(\()p Fj(R)p Fp(\),)j Fo(j)1307 4138 y Fl(1)1401 4126 y Fn(2)d Fo(C)1544 4096 y Fl(1)1615 4126 y Fp(\()p Fj(R)p Fp(\))j(are)f(p)r(ositiv)n(e)g(functions)h (satisfying)f Fo(j)2982 4096 y Fk(2)2977 4146 y(0)3034 4126 y Fp(+)15 b Fo(j)3153 4096 y Fk(2)3148 4146 y Fl(1)3241 4126 y Fp(=)23 b(1)i(and)h Fo(j)3590 4138 y Fk(0)3650 4126 y Fp(=)d(1)0 4225 y(on)k(a)g(neigh)n(b)r(orho)r(o)r(d)g(of)g(0.)37 b(F)-7 b(or)27 b Fo(r)f(>)c Fp(0)28 b(w)n(e)f(de\014ne)h(the)g(op)r (erators)d(on)j Fn(Z)1267 4408 y Fo(j)1306 4374 y Fi(r)1301 4429 y Fk(0)1366 4408 y Fp(:=)22 b Fo(j)1510 4420 y Fk(0)1548 4408 y Fp(\()p Fn(j)p Fo(x)p Fn(j)p Fo(=r)r Fp(\))p Fo(;)98 b(j)1946 4374 y Fi(r)1941 4429 y Fl(1)2035 4408 y Fp(:=)23 b Fo(j)2180 4420 y Fl(1)2250 4408 y Fp(\()p Fn(j)p Fo(x)p Fn(j)p Fo(=r)r Fp(\))p Fo(:)0 4591 y Fj(Lemma)30 b(3.3.)40 b Fg(Supp)l(ose)30 b(Assumptions)f(B,)i(E,)f(ar)l(e)g(satis\014e)l(d,)h (and)f Fp(inf)21 b Fo(h)i(>)f Fp(0)p Fo(:)30 b Fg(L)l(et)f Fo(f)j Fn(2)23 b Fo(C)2931 4560 y Fl(1)2925 4611 y Fk(0)3002 4591 y Fp(\()p Fj(R)p Fp(\))p Fg(.)38 b(Then)31 b(as)f Fo(r)c Fn(!)d(1)1110 4773 y Fo(f)9 b Fp(\()p Fo(H)1268 4739 y Fk(ext)1363 4773 y Fp(\)\000\()p Fo(j)1518 4739 y Fi(r)1513 4794 y Fk(0)1555 4773 y Fo(;)14 b(j)1631 4739 y Fi(r)1626 4794 y Fl(1)1696 4773 y Fp(\))19 b Fn(\000)f Fp(\000\()p Fo(j)1953 4739 y Fi(r)1948 4794 y Fk(0)1990 4773 y Fo(;)c(j)2066 4739 y Fi(r)2061 4794 y Fl(1)2131 4773 y Fp(\))p Fo(f)9 b Fp(\()p Fo(H)e Fp(\))24 b(=)e Fo(o)p Fp(\()p Fo(r)2575 4739 y Fk(0)2614 4773 y Fp(\))p Fo(:)0 4967 y Fj(Pro)s(of.)55 b Fp(Note)27 b(that)709 4948 y Fm(b)694 4967 y Fo(j)733 4938 y Fi(r)728 4989 y Fk(0)770 4967 y Fp(\()p Fo(t)p Fp(\))d(=)e Fo(r)1013 4948 y Fp(^)1014 4967 y Fo(j)1048 4979 y Fk(0)1086 4967 y Fp(\()p Fo(r)r(t)p Fp(\),)1282 4948 y Fm(c)1271 4967 y Fo(j)1310 4943 y Fi(r)1305 4987 y Fl(1)1376 4967 y Fp(\()p Fo(t)p Fp(\))i(=)e Fo(r)1619 4948 y Fp(^)1620 4967 y Fo(j)1654 4979 y Fl(1)1726 4967 y Fp(\()p Fo(r)r(t)p Fp(\).)38 b(Therefore,)1107 5080 y Fm(Z)1204 5193 y Fn(j)1242 5174 y Fm(b)1227 5193 y Fo(j)1266 5165 y Fi(r)1261 5215 y Fk(0)1303 5193 y Fp(\()p Fo(t)p Fp(\))p Fo(t)p Fn(j)p Fp(d)p Fo(t)24 b Fp(=)e Fo(c)1673 5205 y Fk(0)1711 5193 y Fo(=r)n(;)1907 5080 y Fm(Z)2004 5193 y Fn(j)2038 5174 y Fm(c)2027 5193 y Fo(j)2066 5169 y Fi(r)2061 5214 y Fl(1)2132 5193 y Fp(\()p Fo(t)p Fp(\))p Fo(t)p Fn(j)p Fp(d)p Fo(t)h Fp(=)g Fo(c)2502 5205 y Fl(1)2572 5193 y Fo(=r)n(:)1848 5753 y Fp(12)p eop %%Page: 13 13 13 12 bop 0 390 a Fp(Th)n(us,)27 b(using)h(Prop)r(osition)e(3.2)g (\(2\))819 573 y Fo(h)867 539 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1023 573 y Fp([)p Fo(j)1085 539 y Fi(r)1080 593 y Fl(1)1151 573 y Fo(;)14 b(h)p Fp(])p Fo(h)1307 539 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1486 573 y Fp(=)23 b Fo(O)r Fp(\()p Fo(r)1710 539 y Fl(\000)p Fk(1)1800 573 y Fp(\))p Fo(;)98 b(h)2001 539 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2157 573 y Fp([)p Fo(j)2219 539 y Fi(r)2214 593 y Fk(0)2255 573 y Fo(;)14 b(h)p Fp(])p Fo(h)2411 539 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2590 573 y Fp(=)23 b Fo(O)r Fp(\()p Fo(r)2814 539 y Fl(\000)p Fk(1)2905 573 y Fp(\))p Fo(:)0 756 y Fp(Next,)357 937 y Fo(h)405 907 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)561 937 y Fp(\()p Fo(j)632 907 y Fi(r)627 957 y Fk(0)688 937 y Fn(\000)18 b Fp(1\))p Fo(v)s Fp(\(1)g(+)g Fo(K)6 b Fp(\))1172 907 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1351 937 y Fp(=)83 b(\()p Fo(h)1579 907 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1735 937 y Fp(\()p Fo(j)1806 907 y Fi(r)1801 957 y Fk(0)1861 937 y Fn(\000)18 b Fp(1\))p Fo(h)2066 907 y Fk(1)p Fi(=)p Fk(2)2189 937 y Fn(\000)g Fo(j)2311 907 y Fi(r)2306 957 y Fk(0)2366 937 y Fp(+)g(1\))p Fo(h)2571 907 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2727 937 y Fo(v)s Fp(\(1)g(+)g Fo(K)6 b Fp(\))3054 907 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1499 1111 y Fp(+\()p Fo(j)1635 1081 y Fi(r)1630 1132 y Fk(0)1690 1111 y Fn(\000)18 b Fp(1\))p Fo(h)1895 1081 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2050 1111 y Fo(v)s Fp(\(1)h(+)f Fo(K)6 b Fp(\))2378 1081 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2534 1111 y Fo(:)3567 1023 y Fp(\(3.12\))0 1300 y(Using)28 b(Prop)r(osition)e(3.2)h(\(1\),)h(one)g(sees)f(that)h (the)g(\014rst)g(term)g(on)g(the)g(righ)n(t)f(is)h Fo(O)r Fp(\()p Fo(r)2666 1270 y Fl(\000)p Fk(1)2757 1300 y Fp(\))p Fo(;)g Fp(and)g(b)n(y)f(Assumption)h(B)g(the)0 1400 y(second)g(is)g Fo(o)p Fp(\()p Fo(r)464 1369 y Fk(0)503 1400 y Fp(\).)40 b(Therefore)28 b(\(3.12\))f(is)i Fo(o)p Fp(\()p Fo(r)1412 1369 y Fk(0)1450 1400 y Fp(\).)41 b(A)29 b(similar)e(argumen)n(t)h(sho) n(ws)f(that)i Fo(h)2751 1369 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2907 1400 y Fp(\(\()p Fo(j)3010 1369 y Fi(r)3005 1420 y Fk(0)3048 1400 y Fp(\))3080 1369 y Fl(\003)3137 1400 y Fn(\000)19 b Fp(1\))p Fo(v)s Fp(\(1)g(+)g Fo(K)6 b Fp(\))3624 1369 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)0 1499 y Fp(and)27 b Fo(h)209 1469 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)366 1499 y Fo(j)405 1469 y Fi(r)400 1520 y Fl(1)470 1499 y Fo(v)s Fp(\(1)19 b(+)f Fo(K)6 b Fp(\))798 1469 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)981 1499 y Fp(are)27 b Fo(o)p Fp(\()p Fo(r)1231 1469 y Fk(0)1269 1499 y Fp(\))p Fo(:)2393 b Fe(2)0 1715 y Fj(Lemma)25 b(3.4.)37 b Fg(Suppp)l(ose)27 b(Assumptions)f(D)g(and)g(E)h(hold.)39 b(Supp)l(ose)27 b Fp(inf)20 b Fo(h)j(>)g Fp(0)p Fo(:)j Fg(L)l(et)g Fo(g)f Fn(2)f Fo(C)2929 1685 y Fl(1)2923 1736 y Fk(0)2999 1715 y Fp(\()p Fj(R)p Fp(\))j Fg(such)f(that)g Fn(j)p Fo(g)s Fn(j)d(\024)g Fp(1)p Fg(.)0 1815 y(Then)30 b Fp(\000\()p Fo(g)s Fp(\()p Fn(j)p Fo(x)p Fn(j)p Fp(\)\)\(1)20 b(+)e Fo(H)778 1827 y Fk(fr)829 1815 y Fp(\))861 1785 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1047 1815 y Fg(is)30 b(c)l(omp)l(act.)0 1981 y Fj(Pro)s(of.)62 b Fp(W)-7 b(e)32 b(kno)n(w)f(that)h(\(1)20 b(+)h Fo(K)6 b Fp(\))1162 1951 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1349 1981 y Fp(is)32 b(compact.)48 b(Hence)31 b(for)g(an)n(y)g Fo(\017)e(>)g Fp(0,)j(w)n(e)f(can)g(\014nd)h(a)f(\014nite)h (dimensional)0 2080 y(pro)5 b(jection)27 b Fo(P)39 b Fp(comm)n(uting)27 b(with)h Fo(K)34 b Fp(suc)n(h)27 b(that)h Fn(k)p Fp(\(1)17 b(+)h Fo(K)6 b Fp(\))1904 2050 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2060 2080 y Fp(\(1)19 b Fn(\000)f Fo(P)12 b Fp(\))p Fn(k)22 b(\024)h Fo(\017)p Fp(.)37 b(No)n(w)127 2266 y(\000\()p Fo(g)s Fp(\()p Fn(j)p Fo(x)p Fn(j)p Fp(\)\)\(1)19 b(+)f Fo(H)688 2278 y Fk(fr)739 2266 y Fp(\))771 2236 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)951 2266 y Fp(=)1098 2199 y Fm(\000)1136 2266 y Fo(P)31 b Fn(\012)18 b Fp(\000\()p Fo(g)s Fp(\()p Fn(j)p Fo(x)p Fn(j)p Fp(\)\)\(1)h(+)f(d\000\()p Fo(h)p Fp(\)\))2037 2236 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2194 2199 y Fm(\001)42 b(\000)2312 2266 y Fp(1)18 b Fn(\012)g Fp(\(1)g(+)g(d\000\()p Fo(h)p Fp(\)\))2872 2236 y Fk(1)p Fi(=)p Fk(2)2977 2199 y Fm(\001)3057 2266 y Fp(\(1)g(+)g Fo(H)3301 2278 y Fk(fr)3351 2266 y Fp(\))3383 2236 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1098 2441 y Fp(+)c(\(1)k Fn(\012)g Fp(\000\()p Fo(g)s Fp(\()p Fn(j)p Fo(x)p Fn(j)p Fp(\)\)\))43 b(\(1)18 b(+)g Fo(H)1987 2453 y Fk(fr)2038 2441 y Fp(\))2070 2410 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2268 2373 y Fm(\000)2306 2441 y Fp(\(1)g(+)g Fo(K)6 b Fp(\))2590 2410 y Fk(1)p Fi(=)p Fk(2)2722 2441 y Fp(\(1)18 b(+)g Fo(K)6 b Fp(\))3006 2410 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)3162 2441 y Fp(\(1)19 b Fn(\000)f Fo(P)12 b Fp(\))18 b Fn(\012)g Fp(1)3578 2373 y Fm(\001)3630 2441 y Fo(:)0 2636 y Fp(By)38 b(Assumption)g(E)g(and)g(b)r(ecause)g (inf)21 b Fo(h)41 b(>)f Fp(0,)h(\000\()p Fo(g)s Fp(\()p Fn(j)p Fo(x)p Fn(j)p Fp(\)\)\(1)26 b(+)f(d\000\()p Fo(h)p Fp(\)\))2359 2605 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2555 2636 y Fp(is)38 b(a)g(compact)f(op)r(erator)g(on)h(\000)3595 2648 y Fk(s)3625 2636 y Fp(\()p Fn(Z)7 b Fp(\).)0 2735 y(Moreo)n(v)n(er)24 b Fo(P)39 b Fp(is)27 b(a)f(compact)g(op)r(erator)f (on)i Fn(K)q Fp(.)37 b(Hence)27 b(the)g(\014rst)g(term)g(on)f(the)i (righ)n(t)e(is)g(compact.)37 b(The)26 b(second)h(term)0 2835 y(is)g(less)h(than)f Fo(c\017)p Fp(.)3194 b Fe(2)0 3117 y Ff(3.5)112 b(Pro)s(of)37 b(of)h(theorem)f(2.3)0 3270 y Fj(Pro)s(of)32 b(of)h(Theorem)e(2.3.)36 b Fp(W)-7 b(e)29 b(\014rst)f(pro)n(v)n(e)e(that)j(sp)1765 3291 y Fk(ess)1852 3270 y Fo(H)h Fn(\032)24 b Fp([inf)d Fo(H)k Fp(+)19 b(inf)h Fo(h;)14 b Fp(+)p Fn(1)p Fp([)p Fo(:)28 b Fp(It)h(is)f(enough)f(to)h(assume)f(that)0 3370 y(inf)21 b Fo(h)i Fp(=)f(inf)f Fo(h)p Fp(\()p Fo(\030)t Fp(\))j Fo(>)e Fp(0.)35 b(Let)24 b Fo(f)31 b Fn(2)24 b Fo(C)1112 3340 y Fl(1)1106 3390 y Fk(0)1182 3370 y Fp(\()p Fj(R)p Fp(\),)h(supp)p Fo(f)31 b Fn(\032)p Fp(])9 b Fn(\000)g(1)p Fo(;)14 b Fp(inf)21 b Fo(H)16 b Fp(+)9 b(inf)21 b Fo(h)p Fp([.)35 b(W)-7 b(e)24 b(pro)n(v)n(e)d(that)j Fo(f)9 b Fp(\()p Fo(H)e Fp(\))23 b(is)g(compact.)35 b(Note)0 3470 y(that)28 b(b)r(ecause)g(of)g(the)g(supp)r(ort)g(of)g Fo(f)9 b Fp(,)28 b Fo(f)9 b Fp(\()p Fo(H)1388 3439 y Fk(ext)1482 3470 y Fp(\))24 b(=)g(\000\(1)18 b Fn(\010)g Fp(0\))p Fo(f)9 b Fp(\()p Fo(H)2086 3439 y Fk(ext)2180 3470 y Fp(\).)39 b(Moreo)n(v)n(er,)25 b(\000\()p Fo(j)2785 3439 y Fi(r)2780 3490 y Fk(0)2822 3470 y Fo(;)14 b(j)2898 3439 y Fi(r)2893 3490 y Fl(1)2964 3470 y Fp(\))2996 3439 y Fl(\003)3034 3470 y Fp(\000\(1)k Fn(\010)h Fp(0\))k(=)h(\000\()p Fo(j)3571 3439 y Fi(r)3566 3490 y Fk(0)3608 3470 y Fo(;)14 b Fp(0\))3719 3439 y Fl(\003)3756 3470 y Fp(.)0 3569 y(No)n(w,)27 b(since)h(inf)20 b Fo(h)j(>)g Fp(0,)k(w)n(e)g(can)h(apply) f(Lemma)g(3.3.)36 b(Therefore,)1142 3746 y Fo(f)9 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y Fk(0)2401 4132 y Fo(;)14 b(j)2477 4102 y Fi(r)2472 4152 y Fl(1)2542 4132 y Fp(\))1415 4293 y Fi(o)p Fk(\()p Fi(r)1507 4268 y Fh(0)1539 4293 y Fk(\))1458 4347 y Fp(=)125 b(\000)1714 4280 y Fm(\000)1751 4347 y Fp(\()p Fo(j)1822 4317 y Fi(r)1817 4368 y Fk(0)1859 4347 y Fp(\))1891 4317 y Fk(2)1929 4280 y Fm(\001)1981 4347 y Fo(f)9 b Fp(\()p Fo(H)e Fp(\))p Fo(;)0 4583 y Fp(where)238 4528 y Fi(o)p Fk(\()p Fi(r)330 4503 y Fh(0)362 4528 y Fk(\))281 4583 y Fp(=)67 b(means)25 b(that)h(the)g(equalit)n(y)f (holds)g(up)h(to)g(a)f Fo(o)p Fp(\()p Fo(r)1909 4552 y Fk(0)1947 4583 y Fp(\))h(term.)37 b(Finally)-7 b(,)26 b(b)n(y)f(Lemma)g(3.4,)g(the)h(righ)n(t)f(hand)h(side)0 4682 y(is)h(compact.)125 4782 y(W)-7 b(e)29 b(no)n(w)g(pro)n(v)n(e)f (that)h([inf)21 b Fo(H)27 b Fp(+)19 b(inf)i Fo(h;)14 b Fp(+)p Fn(1)p Fp([)p Fn(\032)25 b Fp(sp)1709 4802 y Fk(ess)1795 4782 y Fo(H)r(:)30 b Fp(Let)f Fo(E)i Fp(=)26 b(inf)21 b Fo(H)36 b Fp(and)29 b Fo(\025)e(>)e Fp(inf)c Fo(h:)29 b 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Fn(k)22 b Fp(=)g(1)1848 5753 y(13)p eop %%Page: 14 14 14 13 bop 125 390 a Fp(Using)27 b(Lemma)g(2.1)g(with)h Fo(v)e Fp(=)d(1)18 b Fn(\012)g(j)p Fo(z)t Fp(\))27 b(w)n(e)h(see)f (that)h(there)f(exists)g Fo(C)j(>)22 b Fp(0)27 b(suc)n(h)h(that,)g(for) f(an)n(y)g Fo(z)f Fn(2)d(Z)7 b Fp(,)1273 565 y Fn(k)p Fp(\()p Fo(H)25 b Fp(+)18 b Fo(c)p Fp(\))1592 531 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1748 565 y Fo(a)1792 531 y Fl(\003)1830 565 y Fp(\()p Fo(z)t Fp(\))p Fn(k)23 b(\024)f Fo(C)6 b Fn(k)p Fo(h)2244 531 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2400 565 y Fo(z)t Fn(k)p Fo(:)125 741 y Fp(W)-7 b(e)28 b(consider)e(v)n (ectors)g(of)i(the)g(form)f Fo(a)1351 710 y Fl(\003)1389 741 y Fp(\()p Fo(z)1460 753 y Fi(n)1505 741 y Fp(\)\010)p Fo(:)h Fp(One)f(has)292 849 y Fm(\000)331 916 y Fp(\()p Fo(H)e Fp(+)18 b Fo(c)p Fp(\))608 882 y Fl(\000)p Fk(1)716 916 y Fn(\000)g Fp(\()p Fo(E)24 b Fp(+)18 b Fo(\025)h Fp(+)f Fo(c)p Fp(\))1217 882 y Fl(\000)p Fk(1)1306 849 y Fm(\001)1344 916 y Fo(a)1388 882 y Fl(\003)1426 916 y 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y Fi(n)3153 1174 y Fp(\)\010)1865 1299 y(+\()p Fo(E)23 b Fp(+)18 b Fo(\025)h Fp(+)f Fo(c)p Fp(\))2347 1265 y Fl(\000)p Fk(1)2437 1299 y Fp(\()p Fo(H)25 b Fp(+)18 b Fo(c)p Fp(\))2714 1265 y Fl(\000)p Fk(1)2803 1299 y Fo(a)2847 1265 y Fl(\003)2886 1299 y Fp(\()p Fo(z)2957 1311 y Fi(n)3002 1299 y Fp(\)\()p Fo(H)25 b Fn(\000)18 b Fo(E)5 b Fp(\)\010)1717 1423 y(=)83 b Fo(I)25 b Fp(+)18 b Fo(I)7 b(I)26 b Fp(+)18 b Fo(I)7 b(I)g(I)g(:)0 1599 y Fp(F)-7 b(rom)27 b(no)n(w)g(on,)g(w)n(e)h(will)g(denote)f(b)n(y)g (the)h(letter)g Fo(C)34 b Fp(an)n(y)27 b(constan)n(t)g(whic)n(h)g(do)r (es)g(not)h(dep)r(end)g(on)g Fo(n)f Fp(and)h Fo(\017:)f Fp(No)n(w)957 1774 y Fn(k)p Fo(I)7 b Fn(k)82 b(\024)g Fp(\()p Fo(E)24 b Fp(+)18 b Fo(\025)h Fp(+)f Fo(c)p Fp(\))1731 1739 y Fl(\000)p Fk(1)1820 1774 y Fn(k)p Fp(\()p Fo(H)25 b Fp(+)18 b Fo(c)p Fp(\))2139 1739 y Fl(\000)p Fk(1)2229 1774 y Fo(a)2273 1739 y Fl(\003)2311 1774 y Fp(\(\()p Fo(h)h Fn(\000)f Fo(\025)p Fp(\))p Fo(z)2644 1786 y Fi(n)2689 1774 y Fp(\)\010)p Fn(k)1166 1898 y(\024)82 b Fo(C)6 b Fn(k)p Fo(h)1468 1864 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1624 1898 y Fp(\()p Fo(h)19 b Fn(\000)f Fo(\025)p Fp(\))p Fo(z)1925 1910 y Fi(n)1970 1898 y Fn(k)23 b(\024)f Fo(C)6 b(\017:)891 2149 y Fn(k)p Fo(I)h(I)g(I)g Fn(k)82 b(\024)g Fp(\()p Fo(E)24 b Fp(+)18 b Fo(\025)h Fp(+)f Fo(c)p Fp(\))1751 2115 y Fl(\000)p Fk(1)1841 2149 y Fn(k)p Fp(\()p Fo(H)25 b Fp(+)18 b Fo(c)p Fp(\))2160 2115 y Fl(\000)p Fk(1)2249 2149 y Fo(a)2293 2115 y Fl(\003)2331 2149 y Fp(\()p Fo(z)2402 2161 y Fi(n)2447 2149 y Fp(\)\()p Fo(H)26 b Fn(\000)18 b Fo(E)5 b Fp(\)\010)p Fn(k)1186 2274 y(\024)82 b Fo(C)6 b Fn(k)p Fo(h)1488 2239 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1644 2274 y Fo(z)1683 2286 y Fi(n)1728 2274 y Fn(k)p Fo(\017)22 b Fn(\024)h Fo(C)6 b(\017:)380 2524 y Fn(k)p Fo(I)h(I)g Fn(k)82 b(\024)g Fp(\()p Fo(E)24 b Fp(+)18 b Fo(\025)h Fp(+)f Fo(c)p Fp(\))1197 2490 y Fl(\000)p Fk(1)1286 2524 y Fn(k)p Fp(\()p Fo(H)25 b Fp(+)18 b Fo(c)p Fp(\))1605 2490 y Fl(\000)p Fk(1)1695 2524 y Fo(v)1738 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y Fp(+)18 b Fn(k)p Fo(a)p Fp(\()p Fo(z)2339 3440 y Fi(n)2383 3428 y Fp(\)\010)p Fn(k)2517 3394 y Fk(2)2577 3428 y Fn(!)23 b Fp(1)0 3603 y(Th)n(us)41 b(c)n(ho)r(osing)e Fo(n)j Fp(large)d(enough)i(and)g(setting)g(\010)1703 3615 y Fi(\017)1780 3603 y Fp(:=)k Fo(a)1957 3573 y Fl(\003)1995 3603 y Fp(\()p Fo(z)2066 3615 y Fi(n)2111 3603 y Fp(\)\010)p Fo(=)p Fn(k)p Fo(a)2331 3573 y Fl(\003)2369 3603 y Fp(\()p Fo(z)2440 3615 y Fi(n)2485 3603 y Fp(\)\010)p Fn(k)c Fp(w)n(e)f(obtain)h(a)g(family)g(of)g(v)n(ectors)0 3703 y(satisfying)925 3732 y Fm(\015)925 3782 y(\015)971 3735 y(\000)1009 3803 y Fp(\()p Fo(H)26 b Fp(+)18 b Fo(c)p Fp(\))1287 3768 y Fl(\000)p Fk(1)1395 3803 y Fn(\000)g Fp(\()p Fo(E)24 b Fp(+)18 b Fo(\025)g Fp(+)g Fo(c)p Fp(\))1895 3768 y Fl(\000)p Fk(1)1985 3735 y Fm(\001)2037 3803 y Fp(\010)2097 3815 y Fi(\017)2128 3732 y Fm(\015)2128 3782 y(\015)2198 3803 y Fn(\024)k Fo(C)6 b(\017;)97 b Fn(k)p Fp(\010)2606 3815 y Fi(\017)2638 3803 y Fn(k)22 b Fp(=)h(1)p Fo(:)0 3948 y Fp(for)k 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Fp(\000)1663 5280 y Fk(s)1694 5268 y Fp(\()p Fn(Z)7 b Fp(\))23 b Fn(!)1963 5202 y Fl(\016)1954 5276 y Fp(\000)2006 5280 y Fk(s)2037 5268 y Fp(\()p Fn(Z)7 b Fp(\))2183 5203 y Fl(\016)2168 5268 y Fn(\012Z)1848 5753 y Fp(14)p eop %%Page: 15 15 15 14 bop 0 390 a Fp(satisfying)27 b(for)g Fo(z)f Fn(2)e(Z)34 b Fp(the)28 b(follo)n(wing)f(condition)1453 570 y Fo(Az)1558 536 y Fl(\012)p Fi(n)1677 570 y Fp(=)1765 507 y Fn(p)p 1834 507 50 4 v 63 x Fo(nz)1927 536 y Fl(\012)p Fk(\()p Fi(n)p Fl(\000)p Fk(1\))2179 570 y Fn(\012)18 b Fo(z)t(:)0 751 y Fp(The)30 b(op)r(erator)f Fo(A)h Fp(extends)g(to)g(a)g(unique)g (closed)g(op)r(erator)e(from)i(\000)2192 763 y Fk(s)2222 751 y Fp(\()p Fn(Z)7 b Fp(\))31 b(to)f(\000)2540 763 y Fk(s)2570 751 y Fp(\()p Fn(Z)7 b Fp(\))21 b Fn(\012)f(Z)37 b Fp(also)29 b(denoted)h(b)n(y)g Fo(A)p Fp(.)45 b(The)0 850 y(op)r(erator)20 b Fo(A)j Fp(will)f(b)r(e)g(called)g(the)g (Annihilation)g(Pullthrough)f(Op)r(erator.)34 b(It)22 b(is)g(easy)f(to)h(see)f(that,)j(for)d Fo(z)3283 862 y Fk(1)3320 850 y Fo(;)14 b(:)g(:)g(:)g(;)g(z)3544 862 y Fi(n)3611 850 y Fn(2)24 b(Z)7 b Fp(,)633 1093 y Fo(A)28 b(z)762 1105 y Fk(1)798 1093 y Fn(\012)863 1105 y Fk(s)907 1093 y Fn(\001)14 b(\001)g(\001)19 b(\012)1088 1105 y Fk(s)1136 1093 y Fo(z)1175 1105 y Fi(n)1243 1093 y Fp(=)k Fo(n)1381 1058 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1590 989 y Fi(n)1550 1014 y Fm(X)1553 1191 y Fi(j)s Fk(=1)1684 1093 y Fp(\()p Fo(z)1755 1105 y Fk(1)1792 1093 y Fn(\012)1857 1105 y Fk(s)1901 1093 y Fn(\001)14 b(\001)g(\001)g(\012)2077 1105 y Fk(s)2107 1093 y Fo(z)2146 1105 y Fi(j)s Fl(\000)p Fk(1)2265 1093 y Fn(\012)2330 1105 y Fk(s)2361 1093 y Fo(z)2400 1105 y Fi(j)s Fk(+1)2518 1093 y Fn(\012)2583 1105 y Fk(s)2627 1093 y Fn(\001)g(\001)g(\001)g(\012)2803 1105 y Fk(s)2833 1093 y Fo(z)2872 1105 y Fi(n)2917 1093 y Fp(\))k Fn(\012)g Fo(z)3089 1105 y Fi(j)3124 1093 y Fo(:)420 b Fp(\(4.13\))0 1357 y Fo(A)62 1327 y Fl(\003)100 1357 y Fp(,)28 b(called)f(the)h(Creation)f(Pullthrough)g(Op)r(erator,)f (satis\014es)573 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Fg(.)47 b(We)32 b(have)i(the)f(fol)t(lowing) i(identities)e(on)3465 1842 y Fl(\016)3456 1915 y Fp(\000)3508 1919 y Fk(s)3538 1907 y Fp(\()p Fn(Z)7 b Fp(\))33 b Fg(or)0 2038 y(on)128 1972 y Fl(\016)119 2046 y Fp(\000)171 2050 y Fk(s)201 2038 y Fp(\(Dom)k Fo(b)p Fp(\))p Fg(:)31 2162 y Fp(\(1\))42 b Fo(A)30 b(a)315 2132 y Fl(\003)353 2162 y Fp(\()p Fo(z)t Fp(\))18 b Fn(\000)g Fo(a)605 2132 y Fl(\003)643 2162 y Fp(\()p Fo(z)t Fp(\))p Fn(\012)p Fp(1)29 b Fo(A)23 b Fp(=)g(1)p Fn(\012)o(j)p Fo(z)t Fp(\))p Fo(;)31 2294 y Fp(\(2\))42 b Fo(A)30 b(a)p Fp(\()p Fo(z)t Fp(\))18 b Fn(\000)g Fo(a)p Fp(\()p Fo(z)t Fp(\))p Fn(\012)o Fp(1)29 b Fo(A)24 b Fp(=)e(0)p Fo(;)31 2427 y Fp(\(3\))42 b Fo(A)30 b Fp(d\000\()p Fo(b)p Fp(\))18 b Fn(\000)g Fp(d\000\()p Fo(b)p Fp(\))p Fn(\012)p Fp(1)29 b Fo(A)24 b Fp(=)e(1)p Fn(\012)o Fo(b)30 b(A;)31 2559 y Fp(\(4\))42 b Fo(A)241 2529 y Fl(\003)309 2559 y Fp(1)p Fn(\012)o Fo(b)29 b(A)24 b Fp(=)e(d\000\()p Fo(b)p Fp(\))p Fg(.)31 2692 y Fp(\(5\))42 b Fo(A)241 2661 y 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1390 3528 271 4 v 1461 3604 a Fo(p)p Fp(!)p Fo(q)s Fp(!)1671 3547 y(\010)g Fn(\012)1814 3559 y Fk(s)1863 3547 y Fp(\011)p Fo(;)96 b Fp(\010)23 b Fn(2)h Fp(\000)2261 3513 y Fi(p)2261 3568 y Fk(s)2299 3547 y Fp(\()p Fn(Z)7 b Fp(\))p Fo(;)28 b Fp(\011)23 b Fn(2)g Fp(\000)2699 3513 y Fi(q)2699 3568 y Fk(s)2736 3547 y Fp(\()p Fn(Z)7 b Fp(\))p Fo(:)0 3788 y Fp(Another)27 b(form)n(ula)g(de\014ning)h Fo(I)34 b Fp(is)1640 3887 y Fo(I)c Fp(:=)23 b(\000\()p Fo(i)p Fp(\))p Fo(U)2028 3853 y Fl(\000)p Fk(1)2117 3887 y Fo(;)0 4035 y Fp(where)k Fo(U)32 b Fp(:)23 b(\000)427 4047 y Fk(s)457 4035 y Fp(\()p Fn(Z)j(\010)18 b(Z)7 b Fp(\))23 b Fn(7!)g Fp(\000)938 4047 y Fk(s)968 4035 y Fp(\()p Fn(Z)7 b Fp(\))19 b Fn(\012)f Fp(\000)1253 4047 y Fk(s)1284 4035 y Fp(\()p Fn(Z)7 b Fp(\))28 b(is)f(the)h(unitary)f(iden)n(ti\014cation)h(in)n(tro)r(duced) f(in)h(Section)f(3.2)g(and)1482 4215 y Fo(i)22 b Fp(:)83 b Fn(Z)26 b(\010)18 b(Z)29 b(!)23 b(Z)1639 4315 y Fp(\()p Fo(z)1710 4327 y Fk(1)1747 4315 y Fo(;)14 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Fk(s)1531 4969 y Fp(\()p Fn(Z)7 b Fp(\))21 b(on)n(to)f(\000)1915 4939 y Fi(n)1915 4989 y Fk(s)1960 4969 y Fp(\()p Fn(Z)7 b Fp(\))p Fo(:)21 b Fp(W)-7 b(e)21 b(call)f Fo(A)2478 4939 y Fk(\()p Fi(n)p Fk(\))2596 4969 y Fp(the)h Fo(n)2782 4939 y Fk(th)2870 4969 y Fp(Annihilation)g(Pullthrough)0 5069 y(Op)r(erator.)35 b(Note)28 b(that)g(Dom)18 b Fo(A)1025 5038 y Fk(\()p Fi(n)p Fk(\))1146 5069 y Fp(=)k(Dom)d Fo(N)1502 5038 y Fi(n=)p Fk(2)1642 5069 y Fp(and)27 b Fo(A)c Fp(=)g Fo(A)2038 5038 y Fk(\(1\))2127 5069 y Fo(:)125 5168 y Fp(The)h(follo)n(wing)f(iden)n(tities)i(ab)r(out)f(the)h(scattering)e (iden)n(ti\014cation)h(op)r(erator)e Fo(I)32 b Fp(follo)n(w)23 b(from)h(standard)g(prop)r(erties)0 5268 y(of)k(d\000)f(and)h Fo(U)36 b Fp([5].)h(They)27 b(easily)g(imply)h(Lemma)f(4.1.)1848 5753 y(15)p eop %%Page: 16 16 16 15 bop 0 408 a Fj(Lemma)30 b(4.2.)71 b Fp(\(1\))41 b Fg(L)l(et)30 b Fo(b)f Fg(b)l(e)h(an)g(op)l(er)l(ator)g(on)g Fn(Z)7 b Fg(,)30 b(then)g(on)2016 343 y Fl(\016)2007 416 y Fp(\000)2059 420 y Fk(s)2090 408 y Fp(\(Dom)19 b(\()p Fo(b)p Fp(\)\))2462 343 y Fl(\016)2447 408 y Fn(\012)2521 343 y Fl(\016)2512 416 y Fp(\000)2564 420 y Fk(s)2594 408 y Fp(\(Dom)g(\()p Fo(b)p Fp(\)\))1340 591 y(d\000\()p Fo(b)p Fp(\))p Fo(I)30 b Fp(=)23 b Fo(I)1735 524 y Fm(\000)1773 591 y Fp(d\000\()p Fo(b)p Fp(\))c Fn(\012)f Fp(1)g(+)g(1)g Fn(\012)g Fp(d\000\()p Fo(b)p Fp(\))2557 524 y Fm(\001)2595 591 y Fo(:)31 846 y Fp(\(2\))42 b Fg(F)-6 b(or)30 b Fo(z)c Fn(2)d(Z)7 b Fo(;)30 b Fg(on)725 780 y Fl(\016)716 854 y Fp(\000)768 858 y Fk(s)798 846 y Fp(\()p Fn(Z)7 b Fp(\))945 781 y Fl(\016)929 846 y Fn(\012)1003 780 y Fl(\016)994 854 y Fp(\000)1046 858 y Fk(s)1076 846 y Fp(\()p Fn(Z)g Fp(\))30 b Fg(we)g(have)1371 1028 y Fo(a)p Fp(\()p Fo(z)t Fp(\))p Fo(I)89 b Fp(=)83 b Fo(I)1838 961 y Fm(\000)1876 1028 y Fo(a)p Fp(\()p Fo(z)t Fp(\))18 b Fn(\012)g Fp(1)g(+)g(1)g Fn(\012)g Fo(a)p Fp(\()p Fo(z)t Fp(\))2565 961 y Fm(\001)2603 1028 y Fo(;)1332 1129 y(a)1376 1099 y Fl(\003)1414 1129 y Fp(\()p Fo(z)t Fp(\))p Fo(I)90 b Fp(=)83 b Fo(I)1838 1061 y Fm(\000)1876 1129 y Fo(a)1920 1099 y Fl(\003)1958 1129 y Fp(\()p Fo(z)t Fp(\))18 b Fn(\012)g Fp(1)2208 1061 y Fm(\001)2246 1129 y Fo(:)125 1308 y Fp(The)27 b(follo)n(wing)g(prop)r(osition)f(describ)r(es)h(the)h(relation)f(b)r (et)n(w)n(een)g(the)h(di\013eren)n(t)g(Pullthrough)f(Op)r(erators.)0 1474 y Fj(Prop)s(osition)j(4.1.)1137 1608 y Fo(A)1199 1573 y Fk(\()p Fi(n)p Fk(\))1319 1608 y Fp(=)1467 1552 y(1)p 1417 1589 143 4 v 1417 1605 a Fn(p)p 1486 1605 73 4 v 71 x Fo(n)p Fp(!)1569 1540 y Fm(\000)1607 1608 y Fo(A)19 b Fn(\012)f Fp(1)1813 1634 y Fk(\000)1854 1607 y Fb(n)p Fa(\000)p Fh(1)1854 1643 y(s)1967 1634 y Fk(\()p Fl(Z)5 b Fk(\))2076 1540 y Fm(\001)2127 1608 y Fn(\001)14 b(\001)g(\001)g Fp(\()p Fo(A)p Fn(\012)p Fp(1)2439 1620 y Fl(Z)2496 1608 y Fp(\))29 b Fo(A:)0 1822 y Fj(Pro)s(of.)55 b Fp(On)27 b(\000)504 1792 y Fi(m)504 1842 y(s)567 1822 y Fp(\()p Fn(Z)7 b Fp(\))p Fo(;)28 b Fp(w)n(e)f(ha)n(v)n(e)116 2005 y Fo(A)178 1970 y Fk(\()p Fi(n)p Fk(\))275 2005 y Fo(z)314 2017 y Fk(1)351 2005 y Fn(\012)416 2017 y Fk(s)460 2005 y Fn(\001)14 b(\001)g(\001)f(\012)635 2017 y Fk(s)665 2005 y Fo(z)704 2017 y Fi(m)850 2005 y Fp(=)83 b(\(1)18 b Fn(\012)g Fo(P)1226 2017 y Fi(n)1271 2005 y Fp(\))d Fo(I)1361 1970 y Fl(\003)1399 2005 y Fo(z)1438 2017 y Fk(1)1475 2005 y Fn(\012)1540 2017 y Fk(s)1583 2005 y Fn(\001)f(\001)g(\001)g(\012)1759 2017 y Fk(s)1789 2005 y Fo(z)1828 2017 y Fi(m)850 2271 y Fp(=)83 b(\(1)18 b Fn(\012)g Fo(P)1226 2283 y Fi(n)1271 2271 y Fp(\))1348 2167 y Fi(m)1318 2192 y Fm(X)1318 2371 y Fi(k)q Fk(=0)1452 2136 y Fm(r)p 1535 2136 397 4 v 1545 2215 a Fp(\()p Fo(m)h Fn(\000)f Fo(k)s Fp(\)!)p Fo(k)s Fp(!)p 1545 2252 377 4 v 1685 2328 a Fo(m)p Fp(!)1946 2192 y Fm(X)2079 2204 y(\000)2117 2271 y Fo(z)2156 2283 y Fi(i)2179 2291 y Fh(1)2216 2271 y Fn(\012)2281 2283 y Fk(s)2325 2271 y Fn(\001)c(\001)g(\001)g(\012)2501 2283 y Fk(s)2531 2271 y Fo(z)2570 2283 y Fi(i)2593 2292 y Fb(m)p Fa(\000)p Fb(k)2729 2204 y Fm(\001)2786 2271 y Fn(\012)2869 2204 y Fm(\000)2907 2271 y Fo(z)2946 2283 y Fi(i)2969 2292 y Fb(m)p Fa(\000)p Fb(k)q Fh(+1)3176 2271 y Fn(\012)3241 2283 y Fk(s)3285 2271 y Fn(\001)g(\001)g(\001)g(\012)3461 2283 y Fk(s)3491 2271 y Fo(z)3530 2283 y Fi(i)3553 2291 y Fb(m)3612 2204 y Fm(\001)850 2648 y Fp(=)998 2513 y Fm(r)p 1081 2513 405 4 v 1091 2592 a Fp(\()p Fo(m)k Fn(\000)g Fo(n)p Fp(\)!)p Fo(n)p Fp(!)p 1091 2629 385 4 v 1235 2705 a Fo(m)p Fp(!)1499 2569 y Fm(X)1633 2581 y(\000)1671 2648 y Fo(z)1710 2660 y Fi(i)1733 2668 y Fh(1)1770 2648 y Fn(\012)1835 2660 y Fk(s)1878 2648 y Fn(\001)c(\001)g(\001)g(\012) 2054 2660 y Fk(s)2084 2648 y Fo(z)2123 2660 y Fi(i)2146 2668 y Fb(m)p Fa(\000)p Fb(n)2287 2581 y Fm(\001)2344 2648 y Fn(\012)2427 2581 y Fm(\000)2465 2648 y Fo(z)2504 2660 y Fi(i)2527 2668 y Fb(m)p Fa(\000)p Fb(n)p Fh(+1)2738 2648 y Fn(\012)2803 2660 y Fk(s)2847 2648 y Fn(\001)g(\001)g(\001)g (\012)3023 2660 y Fk(s)3053 2648 y Fo(z)3092 2660 y Fi(i)3115 2668 y Fb(m)3174 2581 y Fm(\001)3226 2648 y Fo(;)0 2863 y Fp(where)29 b(the)h(sum)f(is)g(o)n(v)n(er)f(the)i(set)f(of)h(indices) f Fn(f)p Fo(i)1547 2875 y Fk(1)1583 2863 y Fo(;)14 b(:)g(:)g(:)g(;)g(i) 1797 2875 y Fi(m)1860 2863 y Fn(g)28 b Fp(suc)n(h)i(that)f Fo(i)2330 2875 y Fk(1)2393 2863 y Fo(<)d Fn(\001)14 b(\001)g(\001)26 b Fo(<)f(i)2726 2875 y Fi(m)p Fl(\000)p Fi(k)2878 2863 y Fo(;)k(i)2959 2875 y Fi(m)p Fl(\000)p Fi(k)q Fk(+1)3220 2863 y Fo(<)d Fn(\001)14 b(\001)g(\001)26 b Fo(<)g(i)3554 2875 y Fi(m)3646 2863 y Fp(and)0 2962 y Fn(f)p Fo(i)71 2974 y Fk(1)107 2962 y Fo(;)14 b(:)g(:)g(:)g(;)g(i)321 2974 y Fi(m)p Fl(\000)p Fi(k)472 2962 y Fn(g)20 b(\\)h(f)p Fo(i)681 2974 y Fi(m)p Fl(\000)p Fi(k)q Fk(+1)916 2962 y Fo(;)14 b(:)g(:)g(:)f(;)h(i)1129 2974 y Fi(m)1192 2962 y Fn(g)28 b Fp(=)g Fn(;)i Fp(in)h(the)g(second)f(line,)i(and)f(o)n(v)n (er)e(the)i(same)f(set)h(with)g Fo(k)g Fp(=)d Fo(n)j Fp(in)g(the)g(last)0 3062 y(line.)125 3162 y(On)c(the)h(other)f(hand,)h (using)f(\(4.13\),)g(one)g(has)445 3277 y Fm(\000)483 3344 y Fo(A)p Fn(\012)p Fp(1)652 3370 y Fk(\000)693 3343 y Fb(n)p Fa(\000)p Fh(1)693 3380 y(s)806 3370 y Fk(\()p Fl(Z)5 b Fk(\))915 3277 y Fm(\001)967 3344 y Fn(\001)14 b(\001)g(\001)f Fp(\()p Fo(A)19 b Fn(\012)f Fp(1)1315 3356 y Fl(Z)1372 3344 y Fp(\))p Fo(Az)1505 3356 y Fk(1)1542 3344 y Fn(\012)1607 3356 y Fk(s)1651 3344 y Fn(\001)c(\001)g(\001)g (\012)1827 3356 y Fk(s)1857 3344 y Fo(z)1896 3356 y Fi(m)297 3630 y Fp(=)445 3496 y Fm(r)p 528 3496 332 4 v 538 3574 a Fp(\()p Fo(m)k Fn(\000)g Fo(n)p Fp(\)!)p 538 3611 312 4 v 646 3687 a Fo(m)p Fp(!)1308 3552 y Fm(X)941 3727 y Fb(i)963 3739 y Fh(1)995 3727 y Fb(<)p Fa(\001\001\001)p Fb()d Fp(0)p Fg(,)30 b Fo(H)2454 635 y Fi(\033)2528 623 y Fg(has)h(a)f(gr)l(ound)f (state.)0 789 y Fj(Pro)s(of.)55 b Fp(\(1\))28 b(is)f(straigh)n(tforw)n (ard.)125 888 y(Let)35 b(us)h(pro)n(v)n(e)e(\(2\).)62 b(By)36 b(\(1\))g(and)f(Prop)r(osition)f(2.1,)j(w)n(e)f(can)f(\014nd)i Fo(c)e Fp(suc)n(h)h(that)g(\()p Fo(H)2913 900 y Fi(\033)2982 888 y Fp(+)23 b Fo(c)p Fp(\))3138 858 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)3295 888 y Fp(\()p Fo(H)31 b Fp(+)23 b Fo(c)p Fp(\))3583 858 y Fk(1)p Fi(=)p Fk(2)3724 888 y Fp(is)0 988 y(b)r(ounded)28 b(uniformly)f(in)h Fo(\033)n(:)h Fp(No)n(w,)e(w)n(e)g(can)g(write)104 1171 y(\()p Fo(H)205 1183 y Fi(\033)269 1171 y Fp(+)18 b Fo(c)p Fp(\))420 1136 y Fl(\000)p Fk(1)527 1171 y Fn(\000)g Fp(\()p Fo(H)26 b Fp(+)18 b Fo(c)p Fp(\))888 1136 y Fl(\000)p Fk(1)1060 1171 y Fp(=)83 b(\()p Fo(H)1309 1183 y Fi(\033)1372 1171 y Fp(+)18 b Fo(c)p Fp(\))1523 1136 y Fl(\000)p Fk(1)1613 1171 y Fp(\()p Fo(H)25 b Fn(\000)18 b Fo(H)1891 1183 y Fi(\033)1936 1171 y Fp(\)\()p Fo(H)26 b Fp(+)18 b Fo(c)p Fp(\))2246 1136 y Fl(\000)p Fk(1)1060 1295 y Fp(=)83 b(\()p Fo(H)1309 1307 y Fi(\033)1372 1295 y Fp(+)18 b Fo(c)p Fp(\))1523 1261 y Fl(\000)p Fk(1)1613 1295 y Fp(\()p Fo(H)25 b Fp(+)18 b Fo(c)p Fp(\))1890 1261 y Fk(1)p Fi(=)p Fk(2)1995 1295 y Fp(\()p Fo(H)25 b Fp(+)18 b Fo(c)p Fp(\))2272 1261 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2429 1295 y Fp(\()p Fo(H)25 b Fn(\000)18 b Fo(H)2707 1307 y Fi(\033)2752 1295 y Fp(\)\()p Fo(H)26 b Fp(+)18 b Fo(c)p Fp(\))3062 1261 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)3218 1295 y Fp(\()p Fo(H)26 b Fp(+)18 b Fo(c)p Fp(\))3496 1261 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)3652 1295 y Fo(:)0 1478 y Fp(Th)n(us,)27 b(it)h(su\016ces)g(to)f(sho)n(w)g(that)h(\()p Fo(H)d Fp(+)18 b Fo(c)p Fp(\))1363 1448 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1520 1478 y Fp(\()p Fo(H)25 b Fn(\000)18 b Fo(H)1798 1490 y Fi(\033)1843 1478 y Fp(\)\()p Fo(H)26 b Fp(+)18 b Fo(c)p Fp(\))2153 1448 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2337 1478 y Fp(go)r(es)27 b(to)g(zero)g(in)g(norm)g(as)g Fo(\033)k Fp(go)r(es)c(to)h(zero.)125 1577 y(Let)f(\010)p Fo(;)14 b Fp(\011)23 b Fn(2)g(H)q Fo(;)28 b Fp(then)413 1685 y Fm(\014)413 1735 y(\014)413 1785 y(\014)440 1780 y Fp(\(\010)p Fn(j)p Fp(\()p Fo(H)e Fp(+)18 b Fo(c)p Fp(\))833 1746 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)990 1780 y Fp(\()p Fo(H)1091 1792 y Fi(\033)1154 1780 y Fn(\000)g Fo(H)7 b Fp(\)\()p Fo(H)26 b Fp(+)18 b Fo(c)p Fp(\))1623 1746 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1779 1780 y Fp(\011\))1876 1685 y Fm(\014)1876 1735 y(\014)1876 1785 y(\014)265 2034 y Fp(=)413 1938 y Fm(\014)413 1988 y(\014)413 2038 y(\014)440 2034 y Fp(\(\()p Fo(H)26 b Fp(+)18 b Fo(c)p Fp(\))750 2000 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)907 2034 y Fp(\010)p Fn(j)p Fo(a)1034 2000 y Fk(f)1063 2034 y Fp(\()p Fo(v)k Fn(\000)c Fo(v)1280 2046 y Fi(\033)1325 2034 y Fp(\)\()p Fo(H)26 b Fp(+)18 b Fo(c)p Fp(\))1635 2000 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1791 2034 y Fp(\011\))g(+)g(\()p Fo(a)2065 2000 y Fk(f)2095 2034 y Fp(\()p Fo(v)k Fn(\000)c Fo(v)2312 2046 y Fi(\033)2357 2034 y Fp(\)\()p Fo(H)26 b Fp(+)18 b Fo(c)p Fp(\))2667 2000 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2823 2034 y Fp(\010)p Fn(j)p Fp(\()p Fo(H)26 b Fp(+)18 b Fo(c)p Fp(\))3184 2000 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)3340 2034 y Fp(\011\))3437 1938 y Fm(\014)3437 1988 y(\014)3437 2038 y(\014)265 2270 y Fn(\024)83 b(k)p Fp(\(1)17 b(+)h Fo(K)6 b Fp(\))738 2236 y Fk(1)p Fi(=)p Fk(2)843 2270 y Fp(\()p Fo(H)25 b Fp(+)18 b Fo(c)p Fp(\))1120 2236 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1277 2270 y Fp(\010)p Fn(kk)p Fp(\(1)f(+)h Fo(K)6 b Fp(\))1704 2236 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1860 2270 y Fo(a)p Fp(\()p Fo(v)22 b Fn(\000)c Fo(v)2121 2282 y Fi(\033)2166 2270 y Fp(\)\()p Fo(H)25 b Fp(+)18 b Fo(c)p Fp(\))2475 2236 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2632 2270 y Fp(\011)p Fn(k)579 2465 y Fp(+)p Fn(k)p Fp(\(1)f(+)h Fo(K)6 b Fp(\))969 2431 y Fk(1)p Fi(=)p Fk(2)1073 2465 y Fp(\()p Fo(H)26 b Fp(+)18 b Fo(c)p Fp(\))1351 2431 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1507 2465 y Fp(\011)p Fn(kk)p Fp(\(1)f(+)h Fo(K)6 b Fp(\))1939 2431 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2095 2465 y Fo(a)p Fp(\()p Fo(v)22 b Fn(\000)c Fo(v)2356 2477 y Fi(\033)2401 2465 y Fp(\)\()p Fo(H)26 b Fp(+)18 b Fo(c)p Fp(\))2711 2431 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2867 2465 y Fp(\010)p Fn(k)265 2661 y(\024)83 b(k)p Fp(\(1)17 b(+)h Fo(H)698 2673 y Fk(fr)749 2661 y Fp(\))781 2626 y Fk(1)p Fi(=)p Fk(2)886 2661 y Fp(\()p Fo(H)25 b Fp(+)18 b Fo(c)p Fp(\))1163 2626 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1320 2661 y Fp(\010)p Fn(k)c(k)p Fo(h)1526 2626 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1681 2661 y Fp(\()p Fo(v)22 b Fn(\000)c Fo(v)1898 2673 y Fi(\033)1943 2661 y Fp(\)\(1)g(+)g Fo(K)6 b Fp(\))2259 2626 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2415 2661 y Fn(k)14 b(k)p Fp(d\000\()p Fo(h)p Fp(\))2723 2626 y Fk(1)p Fi(=)p Fk(2)2827 2661 y Fp(\()p Fo(H)25 b Fp(+)18 b Fo(c)p Fp(\))3104 2626 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)3261 2661 y Fp(\011)p Fn(k)579 2856 y Fp(+)p Fn(k)p Fp(\(1)f(+)h Fo(H)929 2868 y Fk(fr)980 2856 y Fp(\))1012 2822 y Fk(1)p Fi(=)p Fk(2)1116 2856 y Fp(\()p Fo(H)26 b Fp(+)18 b Fo(c)p Fp(\))1394 2822 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1550 2856 y Fp(\011)p Fn(k)c(k)p Fo(h)1761 2822 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1916 2856 y Fp(\()p Fo(v)22 b Fn(\000)c Fo(v)2133 2868 y Fi(\033)2178 2856 y Fp(\)\(1)g(+)g Fo(K)6 b Fp(\))2494 2822 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2650 2856 y Fn(k)14 b(k)p Fp(d\000\()p Fo(h)p Fp(\))2958 2822 y Fk(1)p 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Fo(\030)t Fp(\))p Fo(=\033)s Fp(\)\))15 b Fo(:)0 3627 y Fp(Then)28 b Fo(v)257 3639 y Fi(\033)329 3627 y Fp(and)g Fo(h)539 3639 y Fi(\033)611 3627 y Fp(satisfy)f(Assumptions)h(C\()p Fo(\017;)14 b(t)p Fp(\),)28 b(D)g(and)f(E.)37 b(Moreo)n(v)n(er,)25 b Fo(h)2444 3639 y Fi(\033)2511 3627 y Fn(\025)e Fo(\033)s(=)p Fp(2.)36 b(Set)1319 3789 y(~)1298 3810 y Fo(H)1367 3822 y Fi(\033)1434 3810 y Fp(:=)23 b Fo(K)h Fn(\012)18 b Fp(1)g(+)g(1)g Fn(\012)g Fp(d\000\()p Fo(h)2187 3822 y Fi(\033)2232 3810 y Fp(\))h(+)f Fo(V)2414 3822 y Fi(\033)2459 3810 y Fo(:)0 3993 y Fp(By)27 b(Theorem)g(2.3,)660 3972 y(~)638 3993 y Fo(H)707 4005 y Fi(\033)779 3993 y Fp(has)g(a)h(ground)e (state.)125 4092 y(W)-7 b(e)38 b(de\014ne)g Fn(Z)588 4104 y Fi(\033)673 4092 y Fp(:=)i(1)843 4107 y Fk([0)p Fi(;\033)r Fk([)978 4092 y Fp(\()p Fo(h)p Fp(\))p Fn(Z)45 b Fp(and)38 b Fn(Z)1434 4062 y Fi(\033)1519 4092 y Fp(:=)i(1)1689 4107 y Fk([)p Fi(\033)o(;)p Fk(+)p Fl(1)p Fk([)1904 4092 y Fp(\()p Fo(h)p Fp(\))p Fn(Z)7 b Fo(:)39 b 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4706 y Fi(\033)377 4736 y Fp(=)i Fo(K)10 b Fn(\012)t Fp(1)t(+)t(1)t Fn(\012)t Fp(d\000\()p Fo(h)p Fp(1)1064 4751 y Fk([)p Fi(\033)o(;)p Fk(+)p Fl(1)p Fk([)1278 4736 y Fp(\()p Fo(h)p Fp(\)\))t(+)t Fo(V)1543 4748 y Fi(\033)1589 4736 y Fo(:)21 b Fp(Clearly)-7 b(,)21 b(the)g(ground)e(state)i(of)2648 4715 y(~)2626 4736 y Fo(H)2695 4748 y Fi(\033)2760 4736 y Fp(is)g(of)f(the)h(form)3261 4715 y(~)3249 4736 y(\011)3314 4748 y Fi(\033)3382 4736 y Fp(=)3481 4715 y(~)3469 4736 y(\011)3534 4706 y Fi(\033)3583 4736 y Fn(\012)t Fp(\012)3712 4748 y Fi(\033)3756 4736 y Fp(,)0 4836 y(where)27 b(\012)300 4848 y Fi(\033)372 4836 y Fp(is)h(the)g(v)-5 b(acuum)28 b(of)f(\000)1052 4848 y Fk(s)1082 4836 y Fp(\()p Fn(Z)1174 4848 y Fi(\033)1220 4836 y Fp(\).)37 b(Hence,)28 b(it)g(is)f(also)g(a)g(ground)g(state)g (of)g Fo(H)2632 4848 y Fi(\033)2677 4836 y Fp(.)1017 b Fe(2)125 4985 y Fp(Let)35 b(us)f(no)n(w)g(extend)i(Prop)r(osition)d (4.3)h(to)g(the)i(case)d(where)i(w)n(e)f(do)h(not)g(assume)f(the)h (existence)f(of)h(a)g(ground)0 5085 y(state.)1848 5753 y(20)p eop %%Page: 21 21 21 20 bop 0 390 a Fj(Prop)s(osition)36 b(4.6.)43 b Fg(Fix)35 b Fo(\017)f Fg(and)h Fo(t)c(<)g Fp(1)p Fg(.)53 b(Supp)l(ose)34 b(Assumptions)g(B,)j(D)d(and)h(E)g(hold.)54 b(Then)35 b(ther)l(e)g(exists)f Fo(c)g Fg(with)0 490 y(the)f(fol)t(lowing)i(pr)l (op)l(erties.)49 b(L)l(et)32 b Fo(v)1107 502 y Fk(1)1145 490 y Fg(,)i Fo(v)1244 502 y Fk(2)1314 490 y Fg(b)l(e)f(c)l(oupling)g (forms)h(satisfying)g(Assumption)e(C\()p Fo(\017;)14 b(t)p Fg(\).)48 b(L)l(et)32 b Fo(H)3331 502 y Fk(1)3369 490 y Fg(,)h Fo(H)3496 502 y Fk(2)3566 490 y Fg(b)l(e)g(the)0 589 y(c)l(orr)l(esp)l(onding)e(Pauli-Fierz)g(op)l(er)l(ators)g(and)f Fp(inf)21 b Fo(H)1666 601 y Fk(1)1726 589 y Fp(=)i Fo(E)1875 601 y Fk(1)1912 589 y Fg(,)31 b Fp(inf)21 b Fo(H)2152 601 y Fk(2)2212 589 y Fp(=)h Fo(E)2360 601 y Fk(2)2398 589 y Fg(.)39 b(Then)338 772 y Fn(j)p Fo(E)422 784 y Fk(1)478 772 y Fn(\000)18 b Fo(E)622 784 y Fk(2)659 772 y Fn(j)24 b(\024)e Fo(c)p Fn(k)p Fp(\()p Fo(v)943 784 y Fk(1)999 772 y Fn(\000)c Fo(v)1122 784 y Fk(2)1159 772 y Fp(\)\(1)h(+)f Fo(K)6 b Fp(\))1476 737 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1632 772 y Fn(k)14 b Fp(max)o(\()p Fn(k)p Fo(h)1964 737 y Fl(\000)p Fk(1)2052 772 y Fo(v)2092 784 y Fk(1)2130 772 y Fp(\(1)k(+)g Fo(K)6 b Fp(\))2414 737 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2570 772 y Fn(k)p Fo(;)14 b Fn(k)p Fo(h)2739 737 y Fl(\000)p Fk(1)2827 772 y Fo(v)2867 784 y Fk(2)2905 772 y Fp(\(1)k(+)g Fo(K)6 b Fp(\))3189 737 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)3345 772 y Fn(k)p Fp(\))p Fo(:)0 954 y Fj(Pro)s(of.)63 b Fp(By)32 b(applying)f(the)h(cuto\013)h(pro)r(cedure)e(to)g Fo(H)1749 966 y Fk(1)1819 954 y Fp(and)g Fo(H)2053 966 y Fk(2)2122 954 y Fp(w)n(e)h(can)f(appro)n(ximate)g(them)h(b)n(y)g Fo(H)3295 966 y Fk(1)p Fi(;\033)3424 954 y Fp(and)g Fo(H)3659 966 y Fk(2)p Fi(;\033)3756 954 y Fp(,)0 1053 y(whic)n(h)c(p)r(ossess)e (ground)h(states)g(and)g(ground)g(state)g(energies)g Fo(E)2063 1065 y Fk(1)p Fi(;\033)2160 1053 y Fp(,)h Fo(E)2272 1065 y Fk(2)p Fi(;\033)2370 1053 y Fp(.)37 b(Then)28 b(w)n(e)f(use)g(Prop)r(osition)f(4.3,)1353 1235 y(lim)1341 1287 y Fi(\033)r Fl(!)p Fk(0)1495 1235 y Fn(j)p Fo(E)1579 1247 y Fk(1)p Fi(;\033)1695 1235 y Fn(\000)18 b Fo(E)1839 1247 y Fk(2)p Fi(;\033)1937 1235 y Fn(j)23 b Fp(=)g Fn(j)p Fo(E)2155 1247 y Fk(1)2211 1235 y Fn(\000)18 b Fo(E)2355 1247 y Fk(2)2392 1235 y Fn(j)p Fo(;)0 1443 y Fp(whic)n(h)28 b(follo)n(ws)e(from)h(Prop)r(osition)f(4.5,)h(and)734 1625 y Fn(k)p Fp(\()p Fo(v)848 1637 y Fk(1)p Fi(;\033)964 1625 y Fn(\000)18 b Fo(v)1087 1637 y Fk(2)p Fi(;\033)1185 1625 y Fp(\)\(1)g(+)g Fo(K)6 b Fp(\))1501 1591 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1657 1625 y Fn(k)92 b Fp(=)g Fn(k)p Fo(f)9 b Fp(\()p Fo(h=\033)s Fp(\)\()p Fo(v)2316 1637 y Fk(1)2371 1625 y Fn(\000)18 b Fo(v)2494 1637 y Fk(2)2532 1625 y Fp(\)\(1)g(+)g Fo(K)6 b Fp(\))2848 1591 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)3004 1625 y Fn(k)1782 1831 y(\045)83 b(k)p Fp(\()p Fo(v)2062 1843 y Fk(1)2117 1831 y Fn(\000)18 b Fo(v)2240 1843 y Fk(2)2278 1831 y Fp(\)\(1)g(+)g Fo(K)6 b Fp(\))2594 1797 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2750 1831 y Fn(k)p Fo(;)806 2095 y Fn(k)p Fo(h)896 2061 y Fl(\000)p Fk(1)984 2095 y Fo(v)1024 2107 y Fi(i;\033)1113 2095 y Fp(\(1)18 b(+)g Fo(K)6 b Fp(\))1397 2061 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1553 2095 y Fn(k)92 b Fp(=)f Fn(k)p Fo(f)9 b Fp(\()p Fo(h=\033)s Fp(\))p Fo(h)2187 2061 y Fl(\000)p Fk(1)2276 2095 y Fo(v)2316 2107 y Fi(i)2344 2095 y Fp(\(1)18 b(+)g Fo(K)6 b Fp(\))2628 2061 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2784 2095 y Fn(k)1677 2301 y(\045)83 b(k)p Fo(h)1933 2267 y Fl(\000)p Fk(1)2022 2301 y Fo(v)2062 2313 y Fi(i)2090 2301 y Fp(\(1)18 b(+)g Fo(K)6 b Fp(\))2374 2267 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2530 2301 y Fn(k)p Fo(;)96 b(i)23 b Fp(=)g(1)p Fo(;)14 b Fp(2)p Fo(:)3717 2483 y Fe(2)125 2632 y Fp(T)-7 b(ak)n(e)26 b(no)n(w,)h(for)g(an)n(y)g Fo(\033)g(>)22 b Fp(0)p Fo(;)27 b Fp(a)h(normalized)e(ground)h(state)g (\011)2102 2644 y Fi(\033)2174 2632 y Fp(of)h Fo(H)2338 2644 y Fi(\033)2410 2632 y Fp(,)g(that)g(means)1372 2814 y Fo(H)1441 2826 y Fi(\033)1485 2814 y Fp(\011)1550 2826 y Fi(\033)1618 2814 y Fp(=)22 b Fo(E)1766 2826 y Fi(\033)1811 2814 y Fp(\011)1876 2826 y Fi(\033)1921 2814 y Fo(;)96 b Fn(k)p Fp(\011)2147 2826 y Fi(\033)2191 2814 y Fn(k)23 b Fp(=)f(1)p Fo(:)0 2997 y Fj(Prop)s(osition)30 b(4.7.)40 b Fg(Supp)l(ose)30 b(Assumptions)g(B,)g(D,)g(E)g(and)g(F)g(ar)l(e)g (true.)38 b(Then)31 3121 y Fp(\(1\))k Fo(E)23 b Fn(\000)18 b Fo(E)407 3133 y Fi(\033)475 3121 y Fp(=)23 b Fo(o)p Fp(\()p Fo(\033)s Fp(\))p Fo(:)31 3254 y Fp(\(2\))54 b(lim)179 3305 y Fi(\033)r Fl(!)p Fk(0)332 3187 y Fm(\000)370 3254 y Fo(A)p Fp(\011)497 3266 y Fi(\033)560 3254 y Fp(+)18 b(\()p Fo(H)26 b Fn(\012)18 b Fp(1)g(+)g(1)g Fn(\012)g Fo(h)g Fn(\000)g Fo(E)5 b Fp(\))1386 3224 y Fl(\000)p Fk(1)1476 3254 y Fo(v)s Fp(\011)1584 3266 y Fi(\033)1628 3187 y Fm(\001)1689 3254 y Fp(=)23 b(0)p Fo(:)31 3417 y Fp(\(3\))42 b Fg(L)l(et)29 b Fo(b)g Fg(b)l(e)h(a)g(b)l(ounde)l(d)g(p) l(ositive)i(op)l(er)l(ator)e(on)g Fn(Z)7 b Fg(.)39 b(Then)30 b(ther)l(e)g(exists)f(a)h(c)l(onstant)f Fo(c)h Fg(such)g(that)1075 3599 y Fp(lim)14 b(sup)1132 3667 y Fi(\033)r Fl(!)p Fk(0)1329 3599 y Fp(\(\011)1426 3611 y Fi(\033)1470 3599 y Fn(j)p Fp(d\000\()p Fo(b)p Fp(\)\011)1756 3611 y Fi(\033)1801 3599 y Fp(\))23 b Fn(\024)g Fo(c)p Fn(k)p Fo(b)2058 3565 y Fk(1)p Fi(=)p Fk(2)2161 3599 y Fo(h)2209 3565 y Fl(\000)p Fk(1)2298 3599 y Fo(v)s Fp(\(1)c(+)f Fo(K)6 b Fp(\))2626 3565 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2782 3599 y Fn(k)2824 3565 y Fk(2)2861 3599 y Fo(:)31 3861 y Fp(\(4\))67 b(lim)179 3911 y Fi(r)r Fl(!1)358 3861 y Fp(lim)14 b(sup)415 3928 y Fi(\033)r Fl(!)p Fk(0)612 3861 y Fp(\(\011)709 3873 y Fi(\033)753 3861 y Fn(j)p Fp(d\000\(1)948 3876 y Fk([)p Fi(r)n(;)p Fl(1)p Fk([)1105 3861 y Fp(\()p Fn(j)p Fo(x)p Fn(j)p Fp(\)\)\011)1359 3873 y Fi(\033)1405 3861 y Fp(\))23 b(=)g(0)p Fo(:)0 4098 y Fj(Pro)s(of.)55 b Fp(\(1\))28 b(By)f(Prop)r(osition)f(4.6)424 4280 y Fn(j)p Fo(E)e Fn(\000)18 b Fo(E)676 4292 y Fi(\033)721 4280 y Fn(j)83 b(\024)g Fo(c)p Fn(k)p Fp(\(1)17 b Fn(\000)h Fo(f)9 b Fp(\()p Fo(h=\033)s Fp(\)\))p Fo(v)s Fp(\(1)19 b(+)f Fo(K)6 b Fp(\))1841 4246 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1997 4280 y Fn(kk)p Fo(h)2129 4246 y Fl(\000)p Fk(1)2217 4280 y Fo(v)s Fp(\(1)19 b(+)f Fo(K)6 b Fp(\))2545 4246 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2701 4280 y Fn(k)827 4476 y(\024)83 b Fo(c)p Fn(k)p Fp(\(1)17 b Fn(\000)h Fo(f)9 b Fp(\()p Fo(h=\033)s Fp(\)\))p Fo(h)p Fn(kk)p Fp(1)1687 4491 y Fk([0)p Fi(;)p Fk(2)p Fi(\033)r Fk(])1854 4476 y Fp(\()p Fo(h)p Fp(\))p Fo(h)2014 4441 y Fl(\000)p Fk(1)2104 4476 y Fo(v)s Fp(\(1)18 b(+)g Fo(K)6 b Fp(\))2431 4441 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2587 4476 y Fn(kk)p Fo(h)2719 4441 y Fl(\000)p Fk(1)2807 4476 y Fo(v)s Fp(\(1)19 b(+)f Fo(K)6 b Fp(\))3135 4441 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)3291 4476 y Fn(k)p Fo(:)211 b Fp(\(4.18\))0 4658 y(The)28 b(\014rst)f(factor)g(of)g(\(4.18\))g(is)h Fo(O)r Fp(\()p Fo(\033)s Fp(\).)38 b(By)28 b(Assumption)f(F,)h(the)g(second)f(factor)g (is)h Fo(o)p Fp(\()p Fo(\033)2788 4627 y Fk(0)2826 4658 y Fp(\).)125 4757 y(\(2\))f(Using)h(\(4.16\),)f(one)g(has)718 4938 y Fo(A)p Fp(\011)845 4950 y Fi(\033)908 4938 y Fp(+)18 b(\()p Fo(H)26 b Fn(\012)18 b Fp(1)g(+)g(1)g Fn(\012)g Fo(h)g Fn(\000)g Fo(E)5 b Fp(\))1734 4908 y Fl(\000)p Fk(1)1824 4938 y Fo(v)s Fp(\011)1932 4950 y Fi(\033)718 5109 y Fp(=)23 b Fn(\000)p Fp(\()p Fo(H)972 5121 y Fi(\033)1035 5109 y Fn(\012)18 b Fp(1)g(+)g(1)g Fn(\012)g Fo(h)g Fn(\000)g Fo(E)1614 5121 y Fi(\033)1659 5109 y Fp(\))1691 5079 y Fl(\000)p Fk(1)1781 5109 y Fo(v)1821 5121 y Fi(\033)1866 5109 y Fp(\011)1931 5121 y Fi(\033)1993 5109 y Fp(+)g(\()p Fo(H)26 b Fn(\012)18 b Fp(1)g(+)g(1)g Fn(\012)g Fo(h)g Fn(\000)g Fo(E)5 b Fp(\))2819 5079 y Fl(\000)p Fk(1)2909 5109 y Fo(v)s Fp(\011)3017 5121 y Fi(\033)718 5279 y Fp(=)23 b Fo(R)869 5291 y Fk(1)925 5279 y Fp(+)18 b Fo(R)1071 5291 y Fk(2)1126 5279 y Fp(+)g Fo(R)1272 5291 y Fk(3)1328 5279 y Fp(+)g Fo(R)1474 5291 y Fk(4)1512 5279 y Fo(;)1848 5753 y Fp(21)p eop %%Page: 22 22 22 21 bop 0 390 a Fp(where)575 468 y Fo(R)638 480 y Fk(1)758 468 y Fp(:=)23 b(\()p Fo(H)i Fn(\012)18 b Fp(1)g(+)g(1)g Fn(\012)g Fo(h)h Fn(\000)f Fo(E)5 b Fp(\))1612 438 y Fl(\000)p Fk(1)1701 468 y Fp(\()p Fo(v)22 b Fn(\000)c Fo(v)1918 480 y Fi(\033)1963 468 y Fp(\)\011)2060 480 y Fi(\033)2105 468 y Fo(;)575 639 y(R)638 651 y Fk(2)758 639 y Fp(:=)23 b(\()p Fo(H)i Fn(\012)18 b Fp(1)g(+)g(1)g Fn(\012)g Fo(h)h Fn(\000)f Fo(E)5 b Fp(\))1612 609 y Fl(\000)p Fk(1)1701 639 y Fo(a)1745 609 y Fl(\003)1783 639 y Fp(\()p Fo(v)22 b Fn(\000)c Fo(v)2000 651 y Fi(\033)2045 639 y Fp(\)\()p Fo(H)2178 651 y Fi(\033)2242 639 y Fn(\012)g Fp(1)g(+)g(1)g Fn(\012)g Fo(h)g Fn(\000)g Fo(E)2821 651 y Fi(\033)2866 639 y Fp(\))2898 609 y Fl(\000)p Fk(1)2987 639 y Fo(v)3027 651 y Fi(\033)3072 639 y Fp(\011)3137 651 y Fi(\033)3182 639 y Fo(;)575 809 y(R)638 821 y Fk(3)758 809 y Fp(:=)23 b(\()p Fo(H)i Fn(\012)18 b Fp(1)g(+)g(1)g Fn(\012)g Fo(h)h Fn(\000)f Fo(E)5 b Fp(\))1612 779 y Fl(\000)p Fk(1)1701 809 y Fo(a)p Fp(\()p Fo(v)22 b Fn(\000)c Fo(v)1962 821 y Fi(\033)2007 809 y Fp(\)\()p Fo(H)2140 821 y Fi(\033)2204 809 y Fn(\012)g Fp(1)g(+)g(1)g Fn(\012)g Fo(h)g Fn(\000)g Fo(E)2783 821 y Fi(\033)2828 809 y Fp(\))2860 779 y Fl(\000)p Fk(1)2949 809 y Fo(v)2989 821 y Fi(\033)3034 809 y Fp(\011)3099 821 y Fi(\033)3144 809 y Fo(;)575 980 y(R)638 992 y Fk(4)758 980 y Fp(:=)23 b(\()p Fo(E)962 992 y Fi(\033)1025 980 y Fn(\000)c Fo(E)5 b Fp(\)\()p Fo(H)25 b Fn(\012)18 b Fp(1)g(+)g(1)g Fn(\012)g Fo(h)h Fn(\000)f Fo(E)5 b Fp(\))1950 950 y Fl(\000)p Fk(1)2039 980 y Fp(\()p Fo(H)2140 992 y Fi(\033)2204 980 y Fn(\012)18 b Fp(1)g(+)g(1)g Fn(\012)g Fo(h)g Fn(\000)g Fo(E)2783 992 y Fi(\033)2828 980 y Fp(\))2860 950 y Fl(\000)p Fk(1)2949 980 y Fo(v)2989 992 y Fi(\033)3034 980 y Fp(\011)3099 992 y Fi(\033)3144 980 y Fo(:)0 1124 y Fp(No)n(w,)27 b(b)n(y)g(Assumption)h(F,)1151 1223 y Fn(k)p Fo(R)1256 1235 y Fk(1)1293 1223 y Fn(k)22 b(\024)h Fo(c)p Fn(k)p Fo(h)1571 1189 y Fl(\000)p Fk(1)1659 1223 y Fp(\()p Fo(v)f Fn(\000)c Fo(v)1876 1235 y Fi(\033)1921 1223 y Fp(\)\(1)h(+)f Fo(K)6 b Fp(\))2238 1189 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2394 1223 y Fn(k)22 b(!)h Fp(0)p Fo(:)0 1371 y Fp(Using)k(Prop)r(osition)f (4.3,)908 1551 y Fn(k)p Fo(R)1013 1563 y Fk(4)1050 1551 y Fn(k)d(\024)f(j)p Fo(E)i Fn(\000)18 b Fo(E)1454 1563 y Fi(\033)1499 1551 y Fn(jk)p Fo(h)1612 1517 y Fl(\000)p Fk(2)1701 1551 y Fo(v)1741 1563 y Fi(\033)1786 1551 y Fp(\(1)g(+)g Fo(K)6 b Fp(\))2070 1517 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2226 1551 y Fn(kk)p Fp(\(1)17 b(+)h Fo(K)6 b Fp(\))2593 1517 y Fk(1)p Fi(=)p Fk(2)2697 1551 y Fp(\011)2762 1563 y Fi(\033)2806 1551 y Fn(k)p Fo(:)0 1731 y Fp(Th)n(us,)27 b Fo(R)296 1743 y Fk(4)357 1731 y Fn(!)c Fp(0)k(b)r(ecause)g Fn(j)p Fo(E)d Fn(\000)18 b Fo(E)1091 1743 y Fi(\033)1136 1731 y Fn(j)23 b Fp(=)g Fo(o)p Fp(\()p Fo(\033)s Fp(\))p Fo(;)29 b Fn(k)p Fo(h)1566 1701 y Fl(\000)p Fk(2)1654 1731 y Fo(v)1694 1743 y Fi(\033)1739 1731 y Fp(\(1)18 b(+)g Fo(K)6 b Fp(\))2023 1701 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2179 1731 y Fn(k)23 b Fp(=)f Fo(O)r Fp(\()p Fo(\033)2478 1701 y Fl(\000)p Fk(1)2569 1731 y Fp(\),)28 b(b)n(y)f(Assumption)h(F.) 125 1831 y(Next,)715 1925 y Fo(R)778 1937 y Fk(2)898 1925 y Fp(=)23 b(\()p Fo(H)i Fn(\012)18 b Fp(1)g(+)g(1)g Fn(\012)g Fo(h)h Fn(\000)f Fo(E)5 b Fp(\))1729 1895 y Fl(\000)p Fk(1)1846 1925 y Fp(1)p Fn(\012)o Fo(h)898 2095 y Fn(\002)p Fp(1)18 b Fn(\012)g Fp(1)1148 2110 y Fk([)p Fi(\033)o(;)p Fl(1)p Fk([)1312 2095 y Fp(\()p Fo(h)p Fp(\))p Fo(h)1472 2065 y Fl(\000)p Fk(1)898 2270 y Fn(\002)p Fo(a)1007 2240 y Fl(\003)1045 2270 y Fp(\()p Fo(v)k Fn(\000)c Fo(v)1262 2282 y Fi(\033)1307 2270 y Fp(\))28 b(\(1)18 b(+)g Fo(K)6 b Fp(\))1651 2240 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1807 2270 y Fn(\012)p Fp(\(1)18 b(+)g Fo(N)9 b Fp(\))2155 2240 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2311 2270 y Fn(\012)p Fp(1)898 2444 y Fn(\002)p Fp(\(1)18 b(+)g Fo(K)6 b Fp(\))1247 2414 y Fk(1)p Fi(=)p Fk(2)1351 2444 y Fn(\012)p Fp(\(1)18 b(+)g Fo(N)9 b Fp(\))1699 2414 y Fk(1)p Fi(=)p Fk(2)1803 2444 y Fn(\012)p Fp(1)27 b(\()p Fo(H)2038 2456 y Fi(\033)2102 2444 y Fn(\012)18 b Fp(1)g(+)g(1)g Fn(\012)g Fo(h)g Fn(\000)g Fo(E)2681 2456 y Fi(\033)2726 2444 y Fp(\))2758 2414 y Fl(\000)p Fk(1)2847 2444 y Fo(v)2887 2456 y Fi(\033)2932 2444 y Fp(\011)2997 2456 y Fi(\033)3042 2444 y Fo(:)3567 2185 y Fp(\(4.19\))0 2588 y(and)871 2670 y Fo(R)934 2682 y Fk(3)1054 2670 y Fp(=)23 b(\()p Fo(H)i Fn(\012)18 b Fp(1)g(+)g(1)g Fn(\012)g Fo(h)h Fn(\000)f Fo(E)5 b Fp(\))1885 2640 y Fl(\000)p Fk(1)2002 2670 y Fp(\(1)18 b(+)g Fo(K)6 b Fp(\))2286 2640 y Fk(1)p Fi(=)p Fk(2)2390 2670 y Fn(\012)14 b Fp(min\()p Fo(h)2687 2640 y Fk(1)p Fi(=)p Fk(2)2792 2670 y Fo(;)g(h)p Fp(\))1054 2845 y Fn(\002)p Fp(1)k Fn(\012)g Fp(1)1304 2860 y Fk([)p Fi(\033)o(;)p Fl(1)p Fk([)1468 2845 y Fp(\()p Fo(h)p Fp(\))c(max\()p Fo(h)1829 2814 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1985 2845 y Fo(;)g(h)2070 2814 y Fl(\000)p Fk(1)2159 2845 y Fp(\))1054 3019 y Fn(\002)p Fp(\(1)k(+)g Fo(K)6 b Fp(\))1403 2989 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1559 3019 y Fo(a)p Fp(\()p Fo(v)22 b Fn(\000)c Fo(v)1820 3031 y Fi(\033)1865 3019 y Fp(\)\(1)g(+)g Fo(N)9 b Fp(\))2180 2989 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2383 3019 y Fn(\012)46 b Fp(1)1054 3194 y Fn(\002)p Fp(\(1)18 b(+)g Fo(N)9 b Fp(\))1402 3163 y Fk(1)p Fi(=)p Fk(2)1507 3194 y Fn(\012)o Fp(1)27 b(\()p Fo(H)1741 3206 y Fi(\033)1805 3194 y Fn(\012)18 b Fp(1)g(+)g(1)g Fn(\012)g Fo(h)g Fn(\000)g Fo(E)2384 3206 y Fi(\033)2429 3194 y Fp(\))2461 3163 y Fl(\000)p Fk(1)2550 3194 y Fo(v)2590 3206 y Fi(\033)2635 3194 y Fp(\011)2700 3206 y Fi(\033)2745 3194 y Fo(:)3567 2931 y Fp(\(4.20\))0 3342 y(The)30 b(\014rst)g(terms)f(of)h(\(4.19\))f(and)h (\(4.20\))f(are)g(uniformly)h(b)r(ounded,)h(and)e(using)h(Prop)r (osition)e(4.4,)i(so)f(are)g(their)h(last)0 3442 y(terms.)k(The)19 b(second)g(terms)g(are)g(b)r(ounded)g(b)n(y)g Fo(O)r Fp(\()p Fo(\033)1621 3412 y Fl(\000)p Fk(1)1712 3442 y Fp(\))p Fo(:)h Fp(The)f(third)h(terms)f(are)f(b)r(ounded)i(b)n(y)f Fn(k)p Fp(\()p Fo(v)5 b Fn(\000)r Fo(v)3166 3454 y Fi(\033)3211 3442 y Fp(\)\(1)r(+)r Fo(K)h Fp(\))3495 3412 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)3650 3442 y Fn(k)23 b Fp(=)0 3541 y Fo(o)p Fp(\()p Fo(\033)s Fp(\).)38 b(Hence)28 b(b)r(oth)g(\(4.19\))f (and)g(\(4.20\))g(are)g Fo(o)p Fp(\()p Fo(\033)1560 3511 y Fk(0)1598 3541 y Fp(\).)125 3641 y(\(3\))g(Using)h(\014rst)f(Lemma)g (4.1)g(and)h(then)g(\(2\),)f(w)n(e)h(obtain)373 3820 y(\(\011)470 3832 y Fi(\033)514 3820 y Fn(j)p Fp(d\000\()p Fo(b)p Fp(\)\011)800 3832 y Fi(\033)845 3820 y Fp(\))83 b(=)23 b(\()p Fo(A)p Fp(\011)1207 3832 y Fi(\033)1252 3820 y Fn(j)p Fp(1)18 b Fn(\012)g Fo(b)27 b(A)p Fp(\011)1608 3832 y Fi(\033)1653 3820 y Fp(\))960 3981 y Fi(o)p Fk(\()p Fi(\033)1059 3956 y Fh(0)1092 3981 y Fk(\))1007 4035 y Fp(=)1132 3968 y Fm(\000)1170 4035 y Fp(\()p Fo(H)e Fn(\012)18 b Fp(1)g(+)g(1)g Fn(\012)g Fo(h)h Fn(\000)f Fo(E)5 b Fp(\))1913 4005 y Fl(\000)p Fk(1)2002 4035 y Fo(v)s Fp(\011)2110 4047 y Fi(\033)2155 4035 y Fn(j)p Fp(1)18 b Fn(\012)g Fo(b)27 b Fp(\()p Fo(H)e Fn(\012)18 b Fp(1)g(+)g(1)g Fn(\012)g Fo(h)h Fn(\000)f Fo(E)5 b Fp(\))3127 4005 y Fl(\000)p Fk(1)3216 4035 y Fo(v)s Fp(\011)3324 4047 y Fi(\033)3369 3968 y Fm(\001)960 4206 y Fn(\024)23 b Fp(\()p Fo(v)s Fp(\011)1188 4218 y Fi(\033)1233 4206 y Fn(j)p Fp(1)18 b Fn(\012)g Fo(h)1447 4176 y Fl(\000)p Fk(1)1536 4206 y Fo(bh)1620 4176 y Fl(\000)p Fk(1)1736 4206 y Fo(v)s Fp(\011)1844 4218 y Fi(\033)1888 4206 y Fp(\))960 4380 y(=)23 b Fn(k)p Fo(b)1126 4350 y Fk(1)p Fi(=)p Fk(2)1229 4380 y Fo(h)1277 4350 y Fl(\000)p Fk(1)1366 4380 y Fo(v)s Fp(\011)1474 4392 y Fi(\033)1519 4380 y Fn(k)1561 4350 y Fk(2)960 4555 y Fn(\024)g Fo(c)p Fn(k)p Fo(b)1162 4524 y Fk(1)p Fi(=)p Fk(2)1265 4555 y Fo(h)1313 4524 y Fl(\000)p Fk(1)1402 4555 y Fo(v)s Fp(\(1)c(+)f Fo(K)6 b Fp(\))1730 4524 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)1886 4555 y Fn(k)1928 4524 y Fk(2)1965 4555 y Fo(:)125 4731 y Fp(Using)27 b(Assumption)h(F,)g(w)n(e)f(obtain)1182 4911 y(lim)1157 4960 y Fi(r)r Fl(!1)1336 4911 y Fn(k)p Fp(1)1420 4926 y Fk([)p Fi(r)n(;)p Fk(+)p Fl(1)p Fk([)1627 4911 y Fp(\()p Fn(j)p Fo(x)p Fn(j)p Fp(\))p Fo(h)1832 4876 y Fl(\000)p Fk(1)1922 4911 y Fo(v)s Fp(\(1)18 b(+)g Fo(K)6 b Fp(\))2249 4876 y Fl(\000)p Fk(1)p Fi(=)p Fk(2)2405 4911 y Fn(k)23 b Fp(=)g(0)p Fo(:)0 5119 y Fp(Th)n(us)k(\(4\))h(follo)n (ws)f(from)g(\(3\))h(b)n(y)f(setting)h Fo(b)22 b Fp(=)h(1)1523 5134 y Fk([)p Fi(r)n(;)p Fk(+)p Fl(1)p Fk([)1730 5119 y Fp(\()p Fn(j)p Fo(x)p Fn(j)p Fp(\).)1807 b Fe(2)125 5268 y Fp(The)27 b(follo)n(wing)g(easy)f(lemma)i(follo)n(ws)e(b)n(y)i (the)g(sp)r(ectral)f(theorem.)1848 5753 y(22)p eop %%Page: 23 23 23 22 bop 0 390 a Fj(Lemma)30 b(4.4.)40 b Fg(L)l(et)29 b Fo(B)34 b Fg(b)l(e)c(a)g(p)l(ositive)h(op)l(er)l(ator)g(and)f Fp(\(\011)p Fn(j)p Fo(B)t Fp(\011\))23 b Fn(\024)f Fo(c)p Fg(.)39 b(Then)30 b Fn(k)p Fp(1)2524 405 y Fk([)p Fi(r)n(;)p Fl(1)p Fk([)2680 390 y Fp(\()p Fo(B)t Fp(\)\011)p Fn(k)2918 360 y Fk(2)2978 390 y Fn(\024)23 b Fo(c=r)r Fg(.)0 527 y Fj(Pro)s(of)28 b(of)g(Theorem)e(2.4.)34 b Fp(Since)25 b(the)f(unit)h(ball)f(in)h(an)n(y)e(Hilb)r(ert)i(space)e(is)h(w)n (eakly)f(sequen)n(tially)h(compact,)g(w)n(e)g(can)0 626 y(\014nd)29 b(a)g(sequence)f Fo(\033)630 638 y Fi(n)701 626 y Fn(!)d Fp(0)k(suc)n(h)f(that)h(\011)1314 638 y Fi(n)1384 626 y Fp(:=)c(\011)1562 638 y Fi(\033)1600 646 y Fb(n)1674 626 y Fp(con)n(v)n(erges)h(w)n(eakly)i(to)h(some)f (\011)d Fn(2)g(H)q Fp(.)41 b(It)29 b(is)g(easy)f(to)g(c)n(hec)n(k)g (that)0 726 y Fo(H)7 b Fp(\011)22 b(=)h Fo(E)5 b Fp(\011.)37 b(It)28 b(remains)e(to)i(pro)n(v)n(e)e(that)i(\011)22 b Fn(6)p Fp(=)h(0)p Fo(:)125 826 y Fp(Assume)29 b(that)i(\011)26 b(=)g(0.)44 b(W)-7 b(e)30 b(ha)n(v)n(e)f(\(\011)1344 838 y Fi(\033)1388 826 y Fn(j)p Fo(H)1480 838 y Fk(fr)1531 826 y Fp(\011)1596 838 y Fi(\033)1640 826 y Fp(\))e Fn(\024)g Fo(c;)j Fp(and,)g(using)f(\(4.17\),)h(\(\011)2649 838 y Fi(\033)2694 826 y Fn(j)p Fp(1)19 b Fn(\012)h Fo(N)39 b Fp(\011)3034 838 y Fi(\033)3078 826 y Fp(\))27 b Fn(\024)f Fo(c:)k Fp(Hence,)h(using)0 925 y(Lemma)c(4.4,)g(w)n(e)g(see)h(that)f (for)h(an)n(y)e Fo(\017)d(>)g Fp(0,)k(w)n(e)g(can)g(\014nd)h Fo(r)j Fp(suc)n(h)c(that)1344 1071 y Fn(k)p Fp(1)1428 1086 y Fk([)p Fi(r)n(;)p Fl(1)p Fk([)1584 1071 y Fp(\(1)18 b Fn(\012)g Fo(N)55 b Fp(+)18 b Fo(H)2033 1083 y Fk(fr)2084 1071 y Fp(\)\011)2181 1083 y Fi(n)2226 1071 y Fn(k)23 b(\024)f Fo(\017:)1132 b Fp(\(4.21\))125 1216 y(Using)27 b(Prop)r(osition)f(4.7)h(w)n(e)g(see)g(that)h(w)n(e)f(can)g(\014nd)h Fo(n)1865 1228 y Fk(0)1930 1216 y Fp(and)g Fo(r)i Fp(suc)n(h)d(that)h (for)f Fo(n)c(>)g(n)2864 1228 y Fk(0)1368 1362 y Fp(\(\011)1465 1374 y Fi(n)1510 1362 y Fn(j)p Fp(d\000\(1)1705 1377 y Fk([)p Fi(r)n(;)p Fk(+)p Fl(1)p Fk([)1912 1362 y Fp(\()p Fn(j)p Fo(x)p Fn(j)p Fp(\)\)\011)2166 1374 y Fi(n)2212 1362 y Fp(\))h Fo(<)e(\017:)0 1508 y Fp(Hence,)28 b(using)f(Lemma)g (4.4,)1243 1607 y Fn(k)p Fp(1)1327 1622 y Fk([1)p Fi(=)p Fk(2)p Fi(;)p Fl(1)p Fk([)1568 1540 y Fm(\000)1606 1607 y Fp(d\000\(1)1778 1622 y Fk([)p Fi(r)n(;)p Fl(1)p Fk([)1934 1607 y Fp(\()p Fn(j)p Fo(x)p Fn(j)p Fp(\)\))2123 1540 y Fm(\001)2176 1607 y Fp(\011)2241 1619 y Fi(n)2286 1607 y Fn(k)c Fo(<)f Fp(2)p Fo(\017:)1030 b Fp(\(4.22\))0 1735 y(Th)n(us,)27 b(b)n(y)h(\(4.21\))f(and)g(\(4.22\),)1417 1834 y(lim)14 b(sup)1457 1900 y Fi(n)p Fl(!1)1685 1834 y Fn(k)p Fo(C)6 b Fp(\011)1857 1846 y Fi(n)1920 1834 y Fn(\000)18 b Fp(\011)2068 1846 y Fi(n)2112 1834 y Fn(k)23 b(\024)g Fp(3)p Fo(\017;)1203 b Fp(\(4.23\))0 2002 y(where)999 2102 y Fo(C)29 b Fp(:=)23 b(1)1240 2117 y Fk([0)p Fi(;)p Fk(1)p Fi(=)p Fk(2[)1448 2034 y Fm(\000)1486 2102 y Fp(d\000\(1)1658 2117 y Fk([)p Fi(r)n(;)p Fl(1)p Fk([)1815 2102 y Fp(\()p Fn(j)p Fo(x)p Fn(j)p Fp(\)\))2004 2034 y Fm(\001)2057 2102 y Fp(1)2099 2117 y Fk([0)p Fi(;r)r Fk([)2226 2102 y Fp(\(1)18 b Fn(\012)g Fo(N)55 b Fp(+)18 b Fo(H)2675 2114 y Fk(fr)2725 2102 y Fp(\))p Fo(:)125 2229 y Fp(No)n(w,)i(using)f 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Fn(r)p Fo(h)p Fp(\()p Fo(\030)t Fp(\))h Fn(6)p Fp(=)f(0)k(for)g Fo(\030)g Fn(6)p Fp(=)c(0)p Fo(;)2413 4897 y Fp(lim)2529 4909 y Fi(\030)r Fl(!1)2711 4897 y Fo(h)p Fp(\()p Fo(\030)t Fp(\))h(=)f(+)p Fn(1)p Fo(;)2413 4997 y Fp(inf)e Fo(h)p Fp(\()p Fo(\030)t Fp(\))j(=)f Fo(h)p Fp(\(0\))g(=:)f Fo(m)h(>)g Fp(0)p Fo(;)420 5122 y Fp(\()p Fo(I)7 b Fp(1\))249 b Fo(v)26 b Fn(2)d(B)s Fp(\()p Fn(K)q Fo(;)14 b Fn(K)20 b(\012)e(Z)7 b Fp(\))p Fo(;)0 5268 y Fp(and)27 b(pro)n(v)n(e)f(the)i (follo)n(wing)f(HVZ-t)n(yp)r(e)h(theorem.)1848 5753 y(23)p eop %%Page: 24 24 24 23 bop 0 390 a Fj(Theorem)62 b(5.1.)40 b Fg(Supp)l(ose)30 b Fp(\()p Fo(H)7 b Fp(0\))p Fo(;)14 b Fp(\()p Fo(H)7 b Fp(1\))p Fo(;)14 b Fp(\()p Fo(I)7 b Fp(1\))30 b Fg(ar)l(e)g (satis\014e)l(d,)g(then)1405 533 y Fp(sp)1484 554 y Fk(ess)1570 533 y Fo(H)g Fp(=)22 b([inf)f Fo(H)k Fp(+)18 b Fo(m;)c Fp(+)p Fn(1)p Fp([)p Fo(:)125 677 y Fp(The)26 b(assumption)h(on)f(the)i (disp)r(ersion)e(relation)g Fo(h)h Fp(is)f(similar)h(to)f(ours)g 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3318 y(than)28 b(ours)e(b)r(ecause)h(it)h(uses)g(an)f(additional)g(v)n(ector)f Fo(z)31 b Fp(in)d(the)g(one)f(particle)g(space.)0 3593 y Fq(References)249 3775 y Fp([1])41 b(Arai)35 b(A.:)51 b(Ground)35 b(state)g(of)g(the)g(massless)e(Nelson)i(mo)r(del)g (without)g(infrared)g(cuto\013)g(in)g(a)f(Non-F)-7 b(o)r(c)n(k)378 3874 y(represen)n(tation,)26 b Fg(R)l(ev.)k(Math.)h(Phys.)e Fj(13)p Fp(,)e(1075)f(\(2001\).)249 4040 y([2])41 b(Arai)d(A.,)j(Hirok) -5 b(a)n(w)n(a)37 b(M.:)58 b(On)38 b(the)h(existence)f(and)g (uniqueness)g(of)g(ground)f(states)h(of)g(a)g(generalized)378 4140 y(spin-b)r(oson)27 b(mo)r(del,)h Fg(J.)i(F)-6 b(unc.)29 b(A)n(nal.)f Fj(151)p Fp(,)f(455)f(\(1997\).)249 4306 y([3])41 b(Bac)n(h)30 b(V.,)h(F)-7 b(r\177)-42 b(ohlic)n(h)30 b(J.,)h(Sigal)f(I:)h(Quan)n(tum)f(electro)r(dynamics)f(of)i(con\014ned) f(non-relativistic)f(particles,)378 4406 y Fg(A)l(dv.Math.)h Fj(137)p Fp(,)d(299)f(\(1998\).)249 4572 y([4])41 b(Derezi)r(\023)-44 b(nski)38 b(J.,)j(G)n(\023)-39 b(erard)36 b(C:)i Fg(Sc)l(attering)h (The)l(ory)i(of)f(Classic)l(al)i(and)e(Quantum)e Fo(N)9 b Fg(-Particle)40 b(Systems)p Fp(,)378 4671 y(Springer)27 b(1997.)249 4837 y([5])41 b(Derezi)r(\023)-44 b(nski)34 b(J.,)h(G)n(\023)-39 b(erard)32 b(C.:)49 b(Asymptotic)34 b(completeness)g(in)g(quan)n(tum)g(\014eld)g(theory)-7 b(.)33 b(Massiv)n(e)g(P)n(auli-)378 4937 y(Fierz)27 b(Hamiltonians,)h Fg(R)l(ev.)i(Math.)h(Phys.)e Fj(11)p Fp(,)e(383)f(\(1999\).)249 5103 y([6])41 b(Derezi)r(\023)-44 b(nski)27 b(J.,)h(G)n(\023)-39 b(erard)26 b(C.:)37 b(Sp)r(ectral)27 b(and)h(scattering)f(theory)f(of)i (spatially)f(cut-o\013)h Fo(P)12 b Fp(\()p Fo(\036)p Fp(\))3324 5115 y Fk(2)3390 5103 y Fp(Hamiltoni-)378 5203 y(ans,)27 b Fg(Comm.)k(Math.)g(Phys.)e Fj(213)p Fp(,)e(39)g(\(2000\).)1848 5753 y(25)p eop %%Page: 26 26 26 25 bop 249 390 a Fp([7])41 b(Derezi)r(\023)-44 b(nski)27 b(J.,)g(G)n(\023)-39 b(erard)25 b(C.:)37 b(Scattering)26 b(theory)h(of)g(infrared)f(div)n(ergen)n(t)g(P)n(auli-Fierz)g (Hamiltonians,)h(to)378 490 y(app)r(ear)g(in)h(Ann.)g(H.)g(P)n(oincar)n (\023)-39 b(e.)249 656 y([8])41 b(Derezi)r(\023)-44 b(nski)35 b(J.,)j(Jak)l(\024)-37 b(si)n(\023)e(c)32 b(V.:)54 b(Sp)r(ectral)35 b(theory)g(of)g(P)n(auli-Fierz)f(op)r(erators,)i Fg(J.)h(F)-6 b(unc.)37 b(A)n(nal.)f Fj(180)p Fp(,)h(243)378 756 y(\(2001\).)249 922 y([9])k(F)-7 b(r\177)-42 b(ohlic)n(h,)23 b(J.:)35 b(On)22 b(the)h(infrared)g(problem)f(in)h(a)f(mo)r(del)h(of)g(scalar)e (electrons)h(and)h(massless)e(scalar)g(b)r(osons,)378 1021 y Fg(A)n(nn.)29 b(Inst.)h(Henri)f(Poic)l(ar)n(\023)-40 b(e)36 b Fj(A19)p Fp(,)28 b(1)f(\(1973\).)208 1187 y([10])40 b(F)-7 b(r\177)-42 b(ohlic)n(h,)41 b(J,)d(Griesemer,)i(M.,)i(Sc)n (hlein,)f(B.:)59 b(Asymptotic)38 b(completeness)g(for)g(Compton)h (scattering,)378 1287 y(preprin)n(t)27 b(mp-arc)g(03-504)e(\(2003\).) 208 1453 y([11])40 b(Georgescu)c(V.,)41 b(G)n(\023)-39 b(erard)36 b(C.,)k(M\034ller)d(J.S.:)58 b(Sp)r(ectral)37 b(Theory)g(of)h(Massless)e(P)n(auli-Fierz)g(Mo)r(dels,)41 b(to)378 1553 y(app)r(ear)27 b(in)h(Comm.)f(Math.)h(Ph)n(ys.)208 1719 y([12])40 b(G)n(\023)-39 b(erard)36 b(C.:)57 b(On)37 b(the)i(existence)e(of)h(ground)e(states)i(for)f(massless)f(P)n (auli-Fierz)g(Hamiltonians,)k Fg(A)n(nn.)378 1818 y(Henri.)30 b(Poinc)l(ar)n(\023)-40 b(e.)30 b Fj(1)p Fp(,)d(443)g(\(2000\).)208 1984 y([13])40 b(G)n(\023)-39 b(erard)26 b(C.:)37 b(priv)-5 b(ate)27 b(comm)n(unication.)208 2150 y([14])40 b(Griesemer)22 b(M.,)h(Lieb)g(E.H.,)g(Loss)e(M.:)35 b(Ground)22 b(states)g(in)g (non-relativistic)f(quan)n(tum)i(electro)r(dynamics,)378 2250 y Fg(Invent.)30 b(Math.)f Fj(145)p Fp(,)e(557)f(\(2001\).)208 2416 y([15])40 b(Hainzl)35 b(C.,)i(V)-7 b(ougalter)34 b(V.,)j(V)-7 b(ugalter)34 b(S.A.:)53 b(Enhanced)34 b(binding)h(in)g (non-relativistic)f(QED,)g Fg(Comm.)378 2516 y(Math.)d(Phys.)e Fj(223)p Fp(,)e(13)g(\(2003\).)208 2682 y([16])40 b(Hel\013er,)d(B.,)i (Sj\177)-42 b(ostrand,)37 b(J.:)55 b(Equation)35 b(de)i(Sc)n(hr\177)-42 b(odinger)35 b(a)n(v)n(ec)h(c)n(hamp)g(magn)n(\023)-39 b(etique)35 b(et)i(equation)g(de)378 2781 y(Harp)r(er,)27 b Fg(Springer)j(L)l(e)l(ctur)l(e)f(Notes)h(in)g(Physics.)f Fj(345,)i Fp(118-197)24 b(\(1989\).)208 2947 y([17])40 b(Hiroshima)32 b(F.,)i(Sp)r(ohn)g(H.:)47 b(Ground)33 b(state)f(degeneracy)f(of)i(the)g(P)n(auli-Fierz)e(Hamiltonian)h (including)378 3047 y(spin,)c Fg(A)l(dv.)i(The)l(or.)i(Math.)f(Phys.)e Fj(5)p Fp(,)e(1091)f(\(2001\).)208 3213 y([18])40 b(Reed)33 b(M.,)i(Simon)e(B.:)48 b Fg(Metho)l(ds)36 b(of)f(mo)l(dern)g (Mathematic)l(al)i(Physics,)h(II)p Fp(,)33 b(Academic)g(Press,)g (London)378 3313 y(1976.)208 3479 y([19])40 b(Reed,)27 b(M.,)g(Simon,)g(B.:)37 b Fg(Metho)l(ds)30 b(of)f(Mo)l(dern)h (Mathematic)l(al)h(Physics,)g(IV)p Fp(,)26 b(Academic)h(Press,)e (London)378 3578 y(1978.)208 3744 y([20])40 b(Rosen)e(L.:)59 b(The)39 b(\()p Fo(\036)1055 3714 y Fk(2)p Fi(n)1134 3744 y Fp(\))1166 3756 y Fk(2)1242 3744 y Fp(Quan)n(tum)f(Field)h (Theory:)58 b(High)39 b(Order)e(Estimates,)k Fg(Comm.)g(Pur)l(e)e (Appl.)378 3844 y(Math.)29 b Fj(24)p Fp(,)e(417)g(\(1971\).)208 4010 y([21])40 b(Sp)r(ohn)23 b(H.:)35 b(Ground)23 b(state)g(of)f(a)h (quan)n(tum)g(particle)f(coupled)h(to)g(a)f(scalar)f(Bose)h(\014eld,)i Fg(L)l(ett.)h(Math.)i(Phys.)378 4110 y Fj(44)p Fp(,)g(9)h(\(1998\).) 1848 5753 y(26)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0407161252791--