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5517 y Fp(3)p eop %%Page: 4 5 4 4 bop 0 399 a Fp(Notice)46 b(that)f(in)f(the)h(ab)s(o)m(v)m(e)h (expression)d(w)m(e)j(passed)e(from)g(in)m(tegration)h(with)f(resp)s (ect)h(to)g Fm(G)50 b Fp(to)0 511 y(in)m(tegration)30 b(with)f(resp)s(ect)i(to)g Fm(G)1146 525 y Fn(0)1185 511 y Fp(.)41 b(F)-8 b(rom)31 b(this)e(w)m(e)i(get)h(that)1508 689 y Fo(Z)1577 645 y Fl(Q)1570 718 y(T)1661 689 y Fp(=)25 b Fo(C)1822 703 y Fl(T)1903 689 y Fm(Z)1976 645 y Fl(Q)1969 718 y(T)3356 689 y Fp(\(2.9\))0 867 y(with)k Fo(C)272 881 y Fl(T)352 867 y Fo(>)c Fp(0)31 b(indep)s(enden)m(t)d(of)j Fo(Q)f Fp(and)606 1101 y Fm(Z)679 1056 y Fl(Q)672 1129 y(T)764 1101 y Fp(=)25 b(exp)1014 972 y Fk(\022)1081 1101 y Fm(\000)1167 977 y Fk(Z)1258 1003 y Fl(T)1217 1183 y Fj(\000)p Fl(T)1342 977 y Fk(Z)1433 1003 y Fl(T)1393 1183 y Fj(\000)p Fl(T)1518 1101 y Fo(W)13 b Fp(\()p Fo(q)1693 1115 y Fl(s)1749 1101 y Fm(\000)20 b Fo(q)1881 1115 y Fl(t)1910 1101 y Fo(;)15 b(s)21 b Fm(\000)f Fo(t)p Fp(\))p Fo(dtds)2343 972 y Fk(\023)2425 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Fo(q)s(;)i(t)p Fp(\))25 b(=)g Fm(\000)1386 2021 y Fp(1)p 1386 2062 46 4 v 1386 2145 a(4)1456 1959 y Fk(Z)1507 2165 y Fh(R)1555 2146 y Fg(3)1619 2021 y Fm(j)8 b Fp(^)-53 b Fo(\045)p Fp(\()p Fo(k)s Fp(\))p Fm(j)1836 1988 y Fn(2)p 1619 2062 258 4 v 1697 2145 a Fm(j)p Fo(k)s Fm(j)1887 2083 y Fo(e)1929 2045 y Fj(\000j)p Fl(k)r Fj(jj)p Fl(t)p Fj(j)2171 2083 y Fp(cos\()p Fo(k)24 b Fm(\001)d Fo(q)s Fp(\))k Fo(dk)669 b Fp(\(2.11\))0 2280 y(This)29 b(implies)e(that)k(the)g(marginal)e(distribution)d Fm(N)1819 2294 y Fl(T)1904 2280 y Fp(of)31 b Fm(P)2071 2294 y Fl(T)2156 2280 y Fp(for)g(the)f(particle)g(paths)g(satis\014es) 1534 2463 y Fo(d)p Fm(N)1656 2477 y Fl(T)p 1534 2503 177 4 v 1535 2587 a Fo(d)p Fm(N)1670 2560 y Fn(0)1746 2524 y Fp(=)1852 2460 y Fm(Z)1925 2416 y Fl(Q)1918 2489 y(T)p 1852 2503 133 4 v 1858 2587 a Fm(Z)1924 2601 y Fl(T)1994 2524 y Fo(:)1292 b Fp(\(2.12\))0 2764 y(with)29 b Fm(Z)273 2778 y Fl(T)353 2764 y Fp(=)449 2690 y Fk(R)525 2764 y Fm(Z)598 2719 y Fl(Q)591 2792 y(T)658 2764 y Fo(d)p Fm(N)793 2731 y Fn(0)832 2764 y Fp(.)41 b(The)30 b Fo(t)25 b Fp(=)g(0)31 b(single)e(time)h(distribution)d(of)j Fm(N)2449 2778 y Fl(T)2534 2764 y Fp(will)e(b)s(e)i(denoted)g(b)m(y)g Ff(N)3363 2778 y Fl(T)3419 2764 y Fp(.)0 2929 y Fd(Lemma)j(2.1)46 b Fc(Ther)-5 b(e)24 b(is)f(some)h Fo(e)1155 2896 y Fj(\003)1220 2929 y Fo(>)h Fp(0)f Fc(such)f(that)i(for)f(al)5 b(l)24 b Fm(j)p Fo(e)p Fm(j)i(\024)f Fo(e)2272 2896 y Fj(\003)2335 2929 y Fc(the)e(we)-5 b(ak)24 b(lo)-5 b(c)g(al)26 b(limit)d Fp(lim)3214 2943 y Fl(T)10 b Fj(!1)3425 2929 y Fm(N)3500 2943 y Fl(T)0 3042 y Fp(=)25 b Fm(N)42 b Fc(exists)29 b(and)h(is)f(a)g(Gibbs)g(pr)-5 b(ob)g(ability)32 b(me)-5 b(asur)g(e)30 b(on)f Fo(C)7 b Fp(\()p Fi(R)s Fo(;)15 b Fi(R)2230 3009 y Fn(3)2275 3042 y Fp(\))p Fc(.)41 b(Mor)-5 b(e)g(over,)31 b Fm(N)42 b Fc(do)-5 b(es)30 b(not)g(dep)-5 b(end)0 3155 y(on)33 b Fo(h)p Fc(,)g(i.e.)41 b(it)33 b(c)-5 b(oincides)33 b(with)h(the)f(limiting)f(me)-5 b(asur)g(e)34 b(c)-5 b(onstructe)g(d)35 b(for)e Fo(h)26 b Fm(\021)f Fp(0)p Fc(.)0 3321 y(Pr)-5 b(o)g(of)21 b Fp(:)51 b(By)36 b(using)f(cluster)g(expansion,)h(the)g(Gibbs)e(measure) i(for)f Fo(h)g Fm(\021)f Fp(0)i(can)g(b)s(e)f(sho)m(wn)g(to)h(exist)0 3433 y(for)e Fo(W)47 b Fp(at)36 b(su\016cien)m(tly)d(w)m(eak)j (couplings)c Fm(j)p Fo(e)p Fm(j)i(\024)e Fo(e)1757 3400 y Fj(\003)1797 3433 y Fp(.)53 b(Here)35 b(w)m(e)g(ha)m(v)m(e)h(an)f(in) m(teraction)f(whic)m(h)g(di\013ers)0 3546 y(from)41 b(the)i(cases)f (studied)f(in)f([4)q(])i(b)m(y)g(the)g(last)g(t)m(w)m(o)h(factors)g(in) d(\(2.10\).)78 b(Ho)m(w)m(ev)m(er,)47 b(they)42 b(app)s(ear)0 3659 y(as)d(extra)f(energies)h(coming)f(from)f(in)m(teractions)h(b)s (et)m(w)m(een)h(pieces)f(of)h(paths)f(lying)e(inside)g([)p Fm(\000)p Fo(T)8 b(;)15 b(T)e Fp(])0 3772 y(with)32 b(constan)m(t)k(b)s (oundary)31 b(conditions)i(in)f Fi(R)f(r)22 b Fp([)p Fm(\000)p Fo(T)8 b(;)15 b(T)e Fp(])34 b(w)m(eigh)m(ted)g(b)m(y)g Fo(h)p Fp(,)h(to)g(whic)m(h)d(our)i(previous)0 3885 y(argumen)m(ts)d (extend)f(directly)-8 b(.)2418 b Fb(2)141 3998 y Fp(F)-8 b(or)35 b(an)m(y)g(\014xed)f(path)g Fo(Q)f Fp(=)e Fm(f)p Fo(q)1217 4012 y Fl(t)1279 3998 y Fp(:)i Fo(t)f Fm(2)g Fi(R)r Fm(g)39 b(2)32 b Fo(C)7 b Fp(\()p Fi(R)r Fo(;)16 b Fi(R)2003 3965 y Fn(3)2049 3998 y Fp(\))35 b(consider)e(no)m(w)i(the) f(conditional)f(distri-)0 4111 y(bution)290 4119 y Fm(P)360 4067 y Fl(Q)353 4139 y(T)448 4111 y Fp(=)548 4119 y Fm(P)611 4125 y Fl(T)666 4111 y Fp(\()c Fm(\001)f(jf)p Fo(q)894 4125 y Fl(t)924 4111 y Fm(g)h Fp(=)f Fo(Q)p Fp(\))k(as)h(a)g (probabilit)m(y)c(measure)j(on)h Fo(C)7 b Fp(\()p Fi(R)r Fo(;)16 b Fi(R)2651 4078 y Fn(3)2718 4111 y Fm(\002)21 b(S)2872 4078 y Fj(0)2895 4111 y Fp(\()p Fi(R)2990 4078 y Fn(3)3036 4111 y Fp(\)\).)47 b(Since)31 b(the)0 4237 y(in)m(teraction)d(is)f(linear)f(in)h Fo(\030)938 4251 y Fl(t)968 4237 y Fp(,)1021 4245 y Fm(P)1092 4193 y Fl(Q)1084 4266 y(T)1180 4237 y Fp(is)g(itself)f(a)j(Gaussian)e(measure)h(with)f (co)m(v)-5 b(ariance)29 b(\(2.3\))h(and)d(mean)1154 4463 y Fi(E)1209 4504 y Fj(P)1269 4466 y Fe(Q)1264 4521 y(T)1326 4463 y Fp([)p Fo(\030)1391 4477 y Fl(t)1421 4463 y Fp(\()p Fo(f)10 b Fp(\)])25 b(=)1692 4339 y Fk(Z)1743 4545 y Fh(R)1791 4526 y Fg(3)1864 4439 y Fp(^)1844 4463 y Fo(f)10 b Fp(\()p Fo(k)s Fp(\))s(^)-48 b Fo(g)2065 4419 y Fl(Q;t)2062 4492 y(T)2171 4463 y Fp(\()p Fo(k)s Fp(\))p Fo(dk)926 b Fp(\(2.13\))0 4682 y(where)380 4898 y(^)-48 b Fo(g)423 4853 y Fl(Q;t)420 4926 y(T)529 4898 y Fp(\()p Fo(k)s Fp(\))84 b(=)i(^)-49 b Fo(\015)5 b Fp(\()p Fo(k)s Fp(\))22 b Fm(\000)1189 4836 y Fp(^)-54 b Fo(\045)p Fp(\()p Fo(k)s Fp(\))p 1180 4877 168 4 v 1191 4960 a(2)p Fm(j)p Fo(k)s Fm(j)1373 4774 y Fk(Z)1464 4800 y Fl(T)1424 4980 y Fj(\000)p Fl(T)1549 4898 y Fo(e)1591 4860 y Fj(\000j)p Fl(k)r Fj(jj)p Fl(t)p Fj(\000)p Fl(\034)8 b Fj(j)1902 4797 y Fk(\020)1956 4898 y Fo(e)1998 4860 y Fl(ik)r Fj(\001)p Fl(q)2113 4868 y Fe(\034)2175 4898 y Fm(\000)2267 4874 y Fp(^)2266 4898 y Fo(h)q Fp(\()p Fo(k)s Fp(\))2439 4797 y Fk(\021)2509 4898 y Fo(d\034)715 b Fp(\(2.14\))733 5169 y(=)82 b Fm(\000)975 5107 y Fp(^)-53 b Fo(\045)p Fp(\()p Fo(k)s Fp(\))p 967 5148 V 978 5231 a(2)p Fm(j)p Fo(k)s Fm(j)1160 5045 y Fk(Z)1251 5071 y Fl(T)1211 5251 y Fj(\000)p Fl(T)1336 5169 y Fo(e)1378 5131 y Fj(\000j)p Fl(k)r Fj(jj)p Fl(t)p Fj(\000)p Fl(\034)8 b Fj(j)1673 5169 y Fo(e)1715 5131 y Fl(ik)r Fj(\001)p Fl(q)1830 5139 y Fe(\034)1872 5169 y Fo(d\034)31 b Fm(\000)2099 5107 y Fp(^)-53 b Fo(\045)p Fp(\()p Fo(k)s Fp(\))2259 5083 y(^)2258 5107 y Fo(h)q Fp(\()p Fo(k)s Fp(\))p 2091 5148 342 4 v 2188 5231 a(2)p Fm(j)p Fo(k)s Fm(j)2457 5045 y Fk(Z)2508 5251 y Fj(j)p Fl(\034)8 b Fj(j)p Fl(>T)2711 5169 y Fo(e)2753 5131 y Fj(\000j)p Fl(k)r Fj(jj)p Fl(t)p Fj(\000)p Fl(\034)g Fj(j)3048 5169 y Fo(d\034)d(:)1749 5517 y Fp(4)p eop %%Page: 5 6 5 5 bop 0 399 a Fd(Lemma)33 b(2.2)46 b Fc(Ther)-5 b(e)24 b(is)f(some)h Fo(e)1155 366 y Fj(\003)1220 399 y Fo(>)h Fp(0)f Fc(such)f(that)i(for)f(al)5 b(l)24 b Fm(j)p Fo(e)p Fm(j)i(\024)f Fo(e)2272 366 y Fj(\003)2335 399 y Fc(the)e(we)-5 b(ak)25 b(lo)-5 b(c)g(al)25 b(limit)e Fp(lim)3214 413 y Fl(T)10 b Fj(!1)3425 407 y Fm(P)3488 413 y Fl(T)0 511 y Fp(=)96 519 y Fm(P)195 511 y Fc(exists)29 b(and)h(is)f(a)g(pr)-5 b(ob)g(ability)31 b(me)-5 b(asur)g(e)30 b(on)g Fo(C)7 b Fp(\()p Fi(R)r Fo(;)16 b Fi(R)1963 478 y Fn(3)2021 511 y Fm(\002)c(S)2166 478 y Fj(0)2188 511 y Fp(\()p Fi(R)2283 478 y Fn(3)2329 511 y Fp(\)\))p Fc(.)41 b(Mor)-5 b(e)g(over,)2892 519 y Fm(P)2992 511 y Fc(c)g(oincides)29 b(with)0 624 y(the)k(p)-5 b(ath)34 b(me)-5 b(asur)g(e)34 b(for)g Fo(h)25 b Fm(\021)g Fp(0)p Fc(.)0 811 y(Pr)-5 b(o)g(of)21 b Fp(:)40 b(By)31 b(using)e(\(2.14\))j(it)e(is)g(immediate) f(that)i(lim)1889 825 y Fl(T)10 b Fj(!1)2104 811 y Fp(^)-49 b Fo(g)2146 767 y Fl(Q;t)2143 840 y(T)2277 811 y Fp(=)28 b(^)-48 b Fo(g)2419 778 y Fl(Q;t)2555 811 y Fp(exists)30 b(and)f(coincides)h(with)0 938 y(the)j(mean)g(for)546 946 y Fm(P)617 905 y Fl(Q)709 938 y Fp(with)f Fo(h)e Fm(\021)f Fp(0.)49 b(Th)m(us)31 b(w)m(e)j(ha)m(v)m(e)g(also)f(that)2185 946 y Fm(P)2256 894 y Fl(Q)2249 967 y(T)2345 938 y Fm(!)2466 946 y(P)2536 905 y Fl(Q)2629 938 y Fp(exists.)48 b(Moreo)m(v)m(er,)36 b(since)0 1073 y Fm(P)63 1079 y Fl(T)118 1065 y Fp(\()p Fo(E)29 b Fm(\002)22 b Fo(F)13 b Fp(\))32 b(=)583 991 y Fk(R)626 1096 y Fl(F)699 1073 y Fm(P)770 1020 y Fl(Q)763 1093 y(T)830 1065 y Fp(\()p Fo(E)5 b Fp(\))p Fo(d)p Fm(N)1094 1079 y Fl(T)1150 1065 y Fp(,)35 b(for)f(all)f Fo(T)45 b(>)31 b Fp(0,)36 b(where)e Fo(E)39 b Fp(and)34 b Fo(F)47 b Fp(are)35 b(appropriate)e(measurable)0 1177 y(sets,)41 b(w)m(e)d(also)h(ha)m(v)m(e)g(the)f(claim)f(of)i(the)f(lemma.)64 b(Hence)39 b(the)f(join)m(t)g(particle-\014eld)e(path)i(measure)0 1290 y(do)s(es)29 b(not)g(dep)s(end)f(on)h Fo(h)p Fp(,)g(i.e.)41 b(on)29 b(ho)m(w)g(the)g(resh)m(u\017ing)e(of)j(the)f(free)g(\014eld)f (and)g(in)m(teraction)h(energies)0 1403 y(is)g(p)s(erformed.)2953 b Fb(2)0 1524 y Fm(P)99 1516 y Fp(and)274 1524 y Fm(P)344 1483 y Fn(0)412 1516 y Fp(are)29 b(Mark)m(o)m(v)g(pro)s(cesses)g(whose) f Fo(t)d Fp(=)g(0)j(distributions)d(w)m(e)k(denote)f(b)s(elo)m(w)g(b)m (y)g Ff(P)g Fp(resp.)40 b Ff(P)3479 1483 y Fn(0)3518 1516 y Fp(.)141 1629 y(In)23 b(the)g(mo)s(di\014ed)f(function)g(space)i (represen)m(tation)f(Nelson's)h(Hamiltonian)d(is)i Fo(H)30 b Fp(on)24 b(the)f(Hilb)s(ert)0 1742 y(space)31 b Fm(H)321 1709 y Fn(0)385 1742 y Fp(=)25 b Fo(L)543 1709 y Fn(2)583 1742 y Fp(\()p Fi(R)678 1709 y Fn(3)743 1742 y Fm(\002)20 b(S)896 1709 y Fj(0)919 1742 y Fp(\()p Fi(R)1014 1709 y Fn(3)1060 1742 y Fp(\))p Fo(;)15 b(d)p Ff(P)1240 1709 y Fn(0)1280 1742 y Fp(\))31 b(acting)g(lik)m(e)177 2002 y(\()p Fo(F)s(;)15 b(e)355 1964 y Fj(\000)p Fl(tH)504 2002 y Fo(G)p Fp(\))610 2022 y Fj(H)670 2003 y Fg(0)735 2002 y Fp(=)25 b Fi(E)885 2028 y Fj(P)946 2003 y Fg(0)1000 1874 y Fk(\024)1048 2002 y Fo(F)13 b Fp(\()p Fo(q)1195 2016 y Fn(0)1235 2002 y Fo(;)i(\030)1315 2016 y Fn(0)1354 2002 y Fp(\))p Fo(G)p Fp(\()p Fo(q)1536 2016 y Fl(t)1567 2002 y Fo(;)g(\030)1647 2016 y Fl(t)1676 2002 y Fp(\))g(exp)q(\()p Fm(\000)1987 1878 y Fk(Z)2078 1905 y Fl(t)2038 2084 y Fn(0)2123 2002 y Fm(L)2186 1964 y Fl(u)2186 2024 y Fn(in)n(t)2274 2002 y Fp(\()p Fm(f)p Fo(q)2395 2016 y Fl(s)2432 2002 y Fo(;)g(\030)2512 2016 y Fl(s)2549 2002 y Fm(g)p Fp(\))p Fo(ds)p Fp(\))2754 1874 y Fk(\025)2818 2002 y Fo(;)91 b(t)25 b(>)g Fp(0)178 b(\(2.15\))0 2261 y(with)43 b(suitable)g Fo(F)58 b Fp(and)44 b Fo(G)p Fp(,)k(and)c(where)g Fm(L)1556 2228 y Fl(u)1556 2287 y Fn(in)n(t)1644 2261 y Fp(\()p Fm(f)p Fo(q)1765 2275 y Fl(t)1795 2261 y Fo(;)15 b(\030)1875 2275 y Fl(t)1905 2261 y Fm(g)p Fp(\))50 b(=)2154 2187 y Fk(R)2197 2293 y Fh(R)2245 2274 y Fg(3)2284 2261 y Fp(\()p Fo(\045)p Fp(\()p Fo(x)30 b Fm(\000)g Fo(q)2625 2275 y Fl(t)2654 2261 y Fp(\))g Fm(\000)f Fp(\()p Fo(\045)h Fm(\003)g Fo(h)p Fp(\)\()p Fo(x)p Fp(\)\))p Fo(\030)3290 2275 y Fl(t)3321 2261 y Fp(\()p Fo(x)p Fp(\))p Fo(dx)0 2373 y Fp(is)43 b(the)i(Lagrangian)f(corresp)s(onding)f(to)i(\(1.7\).) 84 b(The)44 b(b)s(ottom)h(of)f(the)h(sp)s(ectrum)e(of)i Fo(H)51 b Fp(will)42 b(b)s(e)0 2486 y(denoted)36 b(b)m(y)h Fo(E)549 2500 y Fn(0)588 2486 y Fp(.)59 b(W)-8 b(e)37 b(de\014ne)f(the)g(Euclidean)e(Hamiltonian)h Fo(H)2296 2500 y Fn(euc)2438 2486 y Fp(acting)i(on)f(the)g(Hilb)s(ert)f(space)0 2599 y Fm(H)26 b Fp(=)f Fo(L)261 2566 y Fn(2)300 2599 y Fp(\()p Fi(R)395 2566 y Fn(3)448 2599 y Fm(\002)8 b(S)589 2566 y Fj(0)612 2599 y Fp(\()p Fi(R)707 2566 y Fn(3)752 2599 y Fp(\))p Fo(;)15 b(d)p Ff(P)p Fp(\))26 b(as)e(the)h(self-adjoin)m (t)e(op)s(erator)i(generating)f(the)h(symmetric)e(con)m(tracting)0 2712 y(semigroup)29 b Fo(T)488 2726 y Fl(t)548 2712 y Fp(giv)m(en)i(b)m(y)928 2916 y(\()p Fo(F)s(;)15 b(T)1117 2930 y Fl(t)1148 2916 y Fo(G)p Fp(\))1254 2930 y Fj(H)1344 2916 y Fp(=)25 b Fi(E)1495 2936 y Fj(P)1560 2916 y Fp([)p Fo(F)13 b Fp(\()p Fo(q)1732 2930 y Fn(0)1772 2916 y Fo(;)i(\030)1852 2930 y Fn(0)1891 2916 y Fp(\))p Fo(G)p Fp(\()p Fo(\030)2072 2930 y Fl(t)2103 2916 y Fo(;)g(q)2184 2930 y Fl(t)2213 2916 y Fp(\)])p Fo(;)92 b(t)25 b(>)g Fp(0)p Fo(:)697 b Fp(\(2.16\))0 3119 y Fd(Theorem)34 b(2.3)46 b Fc(Supp)-5 b(ose)40 b Fm(j)p Fo(e)p Fm(j)d(\024)f Fo(e)1253 3086 y Fj(\003)1331 3119 y Fc(with)j(some)g Fo(e)1814 3086 y Fj(\003)1890 3119 y Fo(>)d Fp(0)p Fc(.)60 b(Then)39 b Fo(H)45 b Fc(has)39 b(a)g(unique)f(strictly)h(p)-5 b(osi-)0 3232 y(tive)36 b(gr)-5 b(ound)37 b(state)g Fp(\011)775 3246 y Fn(0)850 3232 y Fc(c)-5 b(orr)g(esp)g(onding)39 b(to)e(the)f(lowest)h(eigenvalue)f Fo(E)2479 3246 y Fn(0)2518 3232 y Fc(,)h(and)g Fo(H)29 b Fm(\000)23 b Fo(E)3029 3246 y Fn(0)3104 3232 y Fc(is)36 b(unitarily)0 3345 y(e)-5 b(quivalent)33 b(with)g Fo(H)696 3359 y Fn(euc)802 3345 y Fc(.)0 3532 y(Pr)-5 b(o)g(of)21 b Fp(:)38 b(F)-8 b(rom)25 b([5)q(])g(w)m(e)h(kno)m(w)f(that)g(the)h(existence)f(of)g(a)h (strictly)e(p)s(ositiv)m(e)g(ground)g(state)i(of)f Fo(H)32 b Fp(implies)0 3645 y(that)37 b Fo(H)31 b Fm(\000)24 b Fo(E)472 3659 y Fn(0)547 3645 y Fp(and)36 b Fo(H)806 3659 y Fn(euc)948 3645 y Fp(are)g(unitarily)e(equiv)-5 b(alen)m(t.)58 b(Moreo)m(v)m(er,)40 b(b)m(y)c(Theorem)g(4.1)h(of)f(the) h(same)0 3758 y(reference)e(w)m(e)g(also)f(kno)m(w)g(that)h(for)f(pro)m (ving)g(existence)g(of)h(the)f(ground)g(state)h(it)f(su\016ces)g(to)h (sho)m(w)0 3871 y(that)1172 4035 y(lim)15 b(inf)1202 4095 y Fl(T)10 b Fj(!1)1439 3911 y Fk(Z)1545 3907 y(\022)1622 3973 y Fo(d)p Ff(P)1727 3987 y Fl(T)p 1622 4014 161 4 v 1630 4097 a Fo(d)p Ff(P)1735 4071 y Fn(0)1792 3907 y Fk(\023)1859 3929 y Fn(1)p Fl(=)p Fn(2)1984 4035 y Fo(d)p Ff(P)2089 3997 y Fn(0)2180 4035 y Fo(>)50 b Fp(0)p Fo(:)940 b Fp(\(2.17\))0 4246 y(W)-8 b(e)32 b(ha)m(v)m(e)141 4435 y Fo(d)p Ff(P)246 4391 y Fl(Q)246 4463 y(T)p 141 4478 166 4 v 169 4561 a Fo(d)p Ff(G)316 4499 y Fp(\()p Fo(\030)t Fp(\))26 b(=)f(exp)706 4371 y Fk(\022)773 4375 y(Z)823 4581 y Fh(R)871 4562 y Fg(3)910 4499 y Fp(\()954 4475 y(^)945 4499 y Fo(\030)t Fp(\()p Fo(k)s Fp(\))c Fm(\000)j Fp(^)-49 b Fo(\015)5 b Fp(\()p Fo(k)s Fp(\)\))j(^)-53 b Fo(\045)r Fp(\()p Fo(k)s Fp(\))p Fo(dk)1710 4375 y Fk(Z)1801 4401 y Fl(T)1761 4581 y Fj(\000)p Fl(T)1871 4499 y Fp(\()p Fo(e)1948 4461 y Fl(ik)r Fj(\001)p Fl(q)2063 4469 y Fe(t)2114 4499 y Fm(\000)2206 4475 y Fp(^)2205 4499 y Fo(h)p Fp(\()p Fo(k)s Fp(\)\))p Fo(e)2454 4461 y Fj(\000j)p Fl(k)r Fj(jj)p Fl(t)p Fj(j)2658 4499 y Fo(dt)2738 4371 y Fk(\023)2820 4499 y Fm(\002)297 4764 y Fp(exp)451 4636 y Fk(\022)518 4764 y Fm(\000)599 4703 y Fp(1)p 599 4743 46 4 v 599 4827 a(4)669 4641 y Fk(Z)720 4847 y Fh(R)768 4828 y Fg(3)822 4764 y Fo(dk)934 4641 y Fk(Z)1025 4667 y Fl(T)985 4847 y Fj(\000)p Fl(T)1110 4641 y Fk(Z)1201 4667 y Fl(T)1160 4847 y Fj(\000)p Fl(T)1295 4703 y Fm(j)8 b Fp(^)-53 b Fo(\045)q Fp(\()p Fo(k)s Fp(\))p Fm(j)1513 4670 y Fn(2)p 1295 4743 258 4 v 1374 4827 a Fm(j)p Fo(k)s Fm(j)1563 4764 y Fp(\()p Fo(e)1640 4727 y Fl(ik)r Fj(\001)p Fl(q)1755 4735 y Fe(t)1807 4764 y Fm(\000)1899 4740 y Fp(^)1898 4764 y Fo(h)p Fp(\()p Fo(k)s Fp(\)\)\()p Fo(e)2182 4727 y Fj(\000)p Fl(ik)r Fj(\001)p Fl(q)2352 4735 y Fe(s)2410 4764 y Fm(\000)2502 4740 y Fp(^)2501 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y(that)k(the)f(subtraction)g(of)974 5244 y(^)973 5268 y Fo(h)h Fp(regularizes)e(the)i(in)m(teraction)f(at)h (small)e Fo(k)k Fp(and)d(mak)m(es)h Fo(H)3163 5235 y Fn(ren)3156 5295 y(N)3305 5268 y Fp(a)f(w)m(ell)1749 5517 y(7)p eop %%Page: 8 9 8 8 bop 0 399 a Fp(de\014ned)35 b(Hamiltonian)g(ha)m(ving)h(a)h(ground) e(state)j(in)d Fo(L)1946 366 y Fn(2)1985 399 y Fp(\()p Fi(R)2080 366 y Fn(3)2126 399 y Fo(;)15 b(dq)s Fp(\))25 b Fm(\012)f(F)9 b Fp(.)59 b(In)36 b(an)g(op)s(erator)h(theoretic)0 511 y(con)m(text,)d(Arai)d(prop)s(osed)f Fo(H)1027 478 y Fn(ren)1020 538 y(N)1161 511 y Fp(of)i(the)g(form)f(\(3.5\))i(for)e (the)h(sp)s(ecial)e(case)2648 487 y(^)2647 511 y Fo(h)e Fm(\021)f Fp(1)32 b(and)f(pro)m(v)m(ed)h(that)0 624 y(under)d(similar)f (assumptions)g(on)i(the)h(p)s(oten)m(tial)f(as)h(ours)e Fo(H)2132 591 y Fn(ren)2125 651 y(N)2265 624 y Fp(has)h(a)h(ground)e (state)j([1].)141 737 y(Ob)m(viously)-8 b(,)41 b(for)f(obtaining)f(the) h(ground)f(state)j(exp)s(ectation)e(of)h(an)f(op)s(erator)g Fo(A)g Fp(of)h(ph)m(ysical)0 850 y(in)m(terest,)46 b(one)d(can)g(use)g Fo(H)978 817 y Fn(ren)971 877 y(N)1123 850 y Fp(only)f(in)f (conjunction)h(with)f(the)i(transformation)f Fo(A)k Fm(7!)g(V)7 b Fo(A)p Fm(V)3423 817 y Fj(\000)p Fn(1)3518 850 y Fp(.)0 963 y(Th)m(us)43 b(exp)s(ectations)h(of)g Fo(A)g Fp(obtained)f(in)f (the)i(standard)f(w)m(a)m(y)i(b)m(y)f(\014rst)f(in)m(tro)s(ducing)e (the)j(cut-o\013)0 1076 y(Hamiltonian,)d(taking)e(ground)g(state)i(exp) s(ectations)g(with)d(it,)k(and)d(subsequen)m(tly)f(remo)m(ving)i(the)0 1189 y(cut-o\013)31 b(are)g(iden)m(tical)f(with)f(computing)g(a)m(v)m (erages)k(of)e Fm(V)7 b Fo(A)p Fm(V)2120 1156 y Fj(\000)p Fn(1)2245 1189 y Fp(b)m(y)30 b(using)f Fo(H)2692 1156 y Fn(ren)2685 1216 y(N)2795 1189 y Fp(.)41 b(As)30 b(pro)m(v)m(ed)h(ab) s(o)m(v)m(e,)0 1302 y(the)g(infrared)d(limit)g(is)i(th)m(us)g (correctly)g(tak)m(en)i(care)f(of.)141 1533 y Fd(Ac)m(kno)m(wledgmen)m (ts:)41 b Fp(R.A.M.)32 b(thanks)e(Zen)m(trum)g(Mathematik)h(of)g(T)-8 b(ec)m(hnisc)m(he)31 b(Univ)m(ersit\177)-45 b(at)0 1646 y(M)s(\177)d(unc)m(hen)37 b(for)g(w)m(arm)h(hospitalit)m(y)e(and)h (\014nancial)f(supp)s(ort.)60 b(He)38 b(also)g(thanks)f(the)g(Russian)f (F)-8 b(un-)0 1759 y(damen)m(tal)36 b(Researc)m(h)g(F)-8 b(oundation)35 b(\(gran)m(ts)h(99-01-00284)k(and)35 b(00-01-00271\))41 b(and)34 b(CRDF)i(\(gran)m(t)0 1872 y(NRM)31 b(1-2085\))i(for)d (\014nancial)f(supp)s(ort.)0 2158 y Fv(References)45 2361 y Fp([1])47 b(Arai,)42 b(A.:)60 b(Ground)39 b(state)i(of)g(the)f (massless)f(Nelson)h(mo)s(del)e(without)h(infrared)f(cuto\013)j(in)e(a) 187 2474 y(non-F)-8 b(o)s(c)m(k)32 b(represen)m(tation,)e(to)h(app)s (ear)f Fc(R)-5 b(ev.)33 b(Math.)g(Phys.)d Fp(\(2001\))45 2661 y([2])47 b(Arai,)30 b(A.,)g(Hirok)-5 b(a)m(w)m(a,)32 b(M.)e(and)g(Hiroshima,)e(F.:)42 b(On)29 b(the)h(absence)h(of)f(eigen)m (v)m(ectors)i(of)e(Hamil-)187 2774 y(tonians)d(in)g(a)i(class)f(of)g (massless)f(quan)m(tum)h(\014eld)e(mo)s(dels)h(without)g(infrared)g (cuto\013,)i Fc(J.)h(F)-7 b(unct.)187 2887 y(A)n(nal.)30 b Fd(168)p Fp(,)h(470-497)j(\(1999\))45 3075 y([3])47 b(F)-8 b(r\177)-45 b(ohlic)m(h,)42 b(J.:)60 b(On)39 b(the)h(infrared)e (problem)g(in)h(a)h(mo)s(del)f(of)h(scalar)g(electrons)h(and)e (massless)187 3188 y(scalar)30 b(b)s(osons,)g Fc(A)n(nn.)i(Inst.)h (Henri)f(Poinc)-5 b(ar)n(\023)-44 b(e)31 b Fd(19)p Fp(,)g(1-103)i (\(1973\))45 3375 y([4])47 b(L})-45 b(orinczi,)34 b(J.)g(and)g(Minlos,) g(R.A.:)50 b(Gibbs)32 b(measures)j(for)f(Bro)m(wnian)f(paths)h(under)f (the)i(e\013ect)187 3488 y(of)30 b(an)h(external)f(and)g(a)g(small)f (pair)g(p)s(oten)m(tial,)i(to)g(app)s(ear)e(in)g Fc(J.)k(Stat.)g(Phys.) e Fp(\(2001\))45 3676 y([5])47 b(L})-45 b(orinczi,)31 b(J.,)i(Minlos,)e(R.A.)h(and)g(Sp)s(ohn,)e(H.:)44 b(The)32 b(infrared)e(b)s(eha)m(viour)g(in)h(Nelson's)g(mo)s(del)187 3789 y(of)38 b(a)g(quan)m(tum)f(particle)g(coupled)g(to)h(a)g(massless) f(scalar)h(\014eld,)g(submitted)e(for)i(publication)187 3902 y(\(2001\))45 4089 y([6])47 b(Nelson,)40 b(E.:)57 b(In)m(teraction)40 b(of)e(nonrelativistic)f(particles)g(with)g(a)i (quan)m(tized)g(scalar)f(\014eld,)i Fc(J.)187 4202 y(Math.)33 b(Phys.)e Fd(5)p Fp(,)f(1190-1197)35 b(\(1964\))45 4390 y([7])47 b(Obata,)31 b(N.:)41 b Fc(White)33 b(Noise)f(Calculus)h(and)h (F)-7 b(o)i(ck)33 b(Sp)-5 b(ac)g(e)p Fp(,)32 b(Springer,)c(1994)1749 5517 y(8)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF