Content-Type: multipart/mixed; boundary="-------------0009081319651" This is a multi-part message in MIME format. ---------------0009081319651 Content-Type: text/plain; name="00-351.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-351.comments" For the proceedings of the XIII International Congress of Mathematical Physics, London, July 18-24, 2000. ---------------0009081319651 Content-Type: text/plain; name="00-351.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-351.keywords" Bose gas, ground state, Gross-Pitaevskii, Charged Bose gas ---------------0009081319651 Content-Type: application/postscript; name="london.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="london.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: london-proc.dvi %%Pages: 21 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) 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Fs(\026)p Ft([)p Fs(u)2166 3850 y Fj(0)2166 3905 y Fr(0)2201 3885 y Ft(\()p Fs(r)r Ft(\))20 b Fp(\000)e Ft(\()p Fs(u)2487 3897 y Fr(0)2524 3885 y Ft(\()p Fs(r)r Ft(\))p Fs(=r)r Ft(\)])2763 3850 y Fr(2)2821 3885 y Ft(+)g Fs(v)s Ft(\()p Fs(r)r Ft(\))p Fp(j)p Fs(u)3121 3897 y Fr(0)3159 3885 y Ft(\()p Fs(r)r Ft(\)])3285 3850 y Fr(2)3324 3885 y Fp(g)p Fs(dr)1570 4080 y Ft(=)82 b(8)p Fs(\031)s(\026a)p Fp(j)p Fs(u)1974 4092 y Fr(0)2011 4080 y Ft(\()p Fs(R)q Ft(\))p Fp(j)2162 4045 y Fr(2)2200 4080 y Fs(=R)2306 4045 y Fr(2)2365 4080 y Fp(!)23 b Ft(8)p Fs(\031)s(\026a)83 b Ft(for)27 b Fs(R)d Fp(!)f(1)p Fs(;)-2706 b Ft(\(2.11\))456 4219 y(if)24 b Fs(u)576 4231 y Fr(0)636 4219 y Ft(is)g(normalized)e(so) h(that)h Fs(f)1451 4231 y Fr(0)1488 4219 y Ft(\()p Fs(R)q Ft(\))g Fp(!)f Ft(1)g(as)g Fs(R)h Fp(!)f(1)p Ft(.)35 b(Moreo)n(v)n(er,)22 b(for)h(p)r(ositiv)n(e)g(in)n(teraction)456 4319 y(p)r(oten)n(tials)g(the)h(scattering)e(solution)h(minimizes)h 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Fp(\031)g Ft(4)p Fs(\031)s(\026aN)9 b Ft(\()p Fs(N)14 b Fp(\000)5 b Ft(1\))p Fs(=L)1951 4787 y Fr(3)1986 4817 y Ft(,)22 b(whic)n(h)f(giv)n(es)f(\(2.8\))h(in)g(the)g(thermo)r(dynamic)456 4917 y(limit.)605 5016 y(This)30 b(simple)g(heuristics)f(is)h(far)f (from)h(a)f(rigorous)e(pro)r(of,)j(ho)n(w)n(ev)n(er,)f(esp)r(ecially)g (for)g(the)456 5116 y(lo)n(w)n(er)17 b(b)r(ound.)35 b(In)19 b(fact,)i(it)f(is)f(rather)f(remark)-5 b(able)17 b(that)j(the)f(same)g (asymptotic)g(form)n(ula)f(holds)456 5216 y(b)r(oth)34 b(for)f(`soft')h(in)n(teraction)f(p)r(oten)n(tials,)i(where)f(p)r (erturbation)f(theory)g(can)h(b)r(e)g(exp)r(ected)p eop %%Page: 5 5 5 4 bop 1089 226 a Fr(THE)29 b(BOSE)g(GAS:)g(A)g(SUBTLE)g(MANY-BOD)n(Y) h(PR)n(OBLEM)600 b(5)456 425 y Ft(to)37 b(b)r(e)i(a)e(go)r(o)r(d)h (appro)n(ximation,)g(and)g(p)r(oten)n(tials)g(lik)n(e)f(hard)h(spheres) f(where)g(this)i(is)e(not)456 525 y(so.)64 b(In)38 b(the)f(former)f (case)g(the)i(ground)e(state)h(is)g(appro)n(ximately)e(the)j(constan)n (t)e(function)456 624 y(and)30 b(the)i(energy)e(is)h Fo(mostly)i(p)l(otential)p Ft(:)45 b(According)30 b(to)h(p)r (erturbation)f(theory)g Fs(E)3084 636 y Fr(0)3122 624 y Ft(\()p Fs(N)t(;)14 b(L)p Ft(\))29 b Fp(\031)456 724 y Fs(N)9 b Ft(\()p Fs(N)31 b Fp(\000)23 b Ft(1\))p Fs(=)p Ft(\(2)p Fs(L)997 694 y Fr(3)1033 724 y Ft(\))1079 657 y Fk(R)1148 724 y Fs(v)s Ft(\()p Fp(j)o Fs(~)-41 b(x)q Fp(j)p Ft(\))p Fs(d)o(~)g(x)q Ft(.)57 b(In)35 b(particular)e(it)h(is)h Fo(indep)l(endent)h(of)f Fs(\026)p Ft(,)i(i.e.)57 b(of)34 b(Planc)n(k's)456 825 y(constan)n(t)c(and)i(mass.)48 b(Since,)33 b(ho)n(w)n(ev)n(er,)1799 758 y Fk(R)1869 825 y Fs(v)s Ft(\()p Fp(j)o Fs(~)-41 b(x)q Fp(j)p Ft(\))p Fs(d)o(~)g(x)32 b Ft(is)g(the)g(\014rst)f(Born)g(appro)n(ximation)f(to) 456 924 y(8)p Fs(\031)s(\026a)36 b Ft(\(note)h(that)f Fs(a)h Ft(dep)r(ends)g(on)f Fs(\026)p Ft(!\),)j(this)e(is)f(not)h(in)f (con\015ict)h(with)g(\(2.8\).)63 b(F)-7 b(or)36 b(`hard')456 1024 y(p)r(oten)n(tials)g(on)g(the)g(other)g(hand,)i(the)f(ground)e (state)h(is)g Fo(highly)k(c)l(orr)l(elate)l(d)p Ft(,)g(i.e.,)e(it)f(is) f(far)456 1124 y(from)31 b(b)r(eing)i(a)e(pro)r(duct)i(of)f(single)f (particle)h(states.)50 b(The)32 b(energy)g(is)g(here)f Fo(mostly)k(kinetic)p Ft(,)456 1223 y(b)r(ecause)f(the)h(w)n(a)n(v)n(e) e(function)i(is)f(v)n(ery)g(small)g(where)g(the)h(p)r(oten)n(tial)g(is) f(large.)57 b(These)34 b(t)n(w)n(o)456 1323 y(quite)27 b(di\013eren)n(t)g(regimes,)f(the)i(p)r(oten)n(tial)f(energy)e (dominated)i(one)g(and)g(the)g(kinetic)h(energy)456 1422 y(dominated)40 b(one,)k(cannot)d(b)r(e)g(distinguished)g(b)n(y)g(the)g (lo)n(w)f(densit)n(y)h(asymptotics)f(of)h(the)456 1522 y(energy)-7 b(.)59 b(Whether)35 b(they)h(b)r(eha)n(v)n(e)e(di\013eren)n (tly)i(with)g(resp)r(ect)f(to)g(other)g(phenomena,)i(e.g.,)456 1622 y(Bose-Einstein)26 b(condensation,)g(is)i(not)f(kno)n(wn)g(at)h (presen)n(t.)605 1721 y(Bogolub)r(o)n(v's)i(analysis)h([)p Fu(11)o Ft(])h(presupp)r(oses)f(the)i(existence)e(of)h(Bose-Einstein)f (conden-)456 1821 y(sation.)54 b(Nev)n(ertheless,)34 b(it)g(is)f(correct)g(\(for)g(the)h(energy\))f(for)g(the)h (one-dimensional)e(delta-)456 1921 y(function)25 b(Bose)f(gas)g([)p Fu(17)p Ft(],)h(despite)h(the)f(fact)g(that)g(there)g(is)g (\(presumably\))g(no)f(condensation)456 2020 y(in)34 b(that)f(case.)55 b(It)34 b(turns)f(out)h(that)g(BE)f(condensation)g (is)g(not)h(really)f(needed)g(in)h(order)f(to)456 2120 y(understand)26 b(the)g(energy)-7 b(.)36 b(As)26 b(w)n(e)g(shall)g (see,)h(`global')e(condensation)g(can)h(b)r(e)h(replaced)f(b)n(y)g(a) 456 2219 y(`lo)r(cal')i(condensation)f(on)h(b)r(o)n(xes)g(whose)g(size) g(is)g(indep)r(enden)n(t)h(of)g Fs(L)p Ft(.)39 b(It)29 b(is)f(this)h(crucial)e(un-)456 2319 y(derstanding)g(that)h(enables)g (us)g(to)g(pro)n(v)n(e)f(Theorem)g(1.1)g(without)i(ha)n(ving)e(to)h (decide)h(ab)r(out)456 2419 y(BE)e(condensation.)605 2518 y(An)g(imp)r(ortan)n(t)f(idea)h(of)f(Dyson)h(w)n(as)e(to)i (transform)e(the)i(hard)f(sphere)g(p)r(oten)n(tial)h(in)n(to)f(a)456 2618 y(soft)31 b(p)r(oten)n(tial)h(at)f(the)h(cost)f(of)h (sacri\014cing)e(the)i(kinetic)f(energy)-7 b(,)32 b(i.e.,)h(e\013ectiv) n(ely)e(to)h(mo)n(v)n(e)456 2718 y(from)f(one)g(regime)g(to)h(the)g (other.)48 b(W)-7 b(e)33 b(shall)e(mak)n(e)g(use)g(of)h(this)g(idea)f (in)h(our)f(pro)r(of)g(of)h(the)456 2817 y(lo)n(w)n(er)27 b(b)r(ound)j(b)r(elo)n(w.)41 b(But)29 b(\014rst)g(w)n(e)f(discuss)h (the)h(simpler)e(upp)r(er)i(b)r(ound,)f(whic)n(h)g(relies)g(on)456 2917 y(other)e(ideas)g(from)g(Dyson's)g(b)r(eautiful)i(pap)r(er)e([)p Fu(15)o Ft(].)605 3070 y Fu(2.1.)46 b(UPPER)22 b(BOUND.)41 b Ft(The)18 b(follo)n(wing)g(generalization)e(of)i(Dyson's)g(upp)r(er)h (b)r(ound)456 3170 y(holds)27 b([)p Fu(3)p Ft(],)g([)p Fu(10)p Ft(]:)605 3324 y Fh(Theorem)32 b(2.2)f Ft(\()p Fu(Upp)s(er)h(b)s(ound)p Ft(\))p Fh(.)41 b Fo(De\014ne)21 b Fs(\032)2149 3336 y Fr(1)2209 3324 y Ft(=)h(\()p Fs(N)9 b Fp(\000)p Ft(1\))p Fs(=L)2642 3293 y Fr(3)2699 3324 y Fo(and)22 b Fs(b)h Ft(=)f(\(4)p Fs(\031)s(\032)3165 3336 y Fr(1)3203 3324 y Fs(=)p Ft(3\))3319 3293 y Fj(\000)p Fr(1)p Fq(=)p Fr(3)3474 3324 y Fo(.)456 3423 y(F)-6 b(or)29 b(nonne)l(gative)h(p)l(otentials)h Fs(v)s Fo(,)f(and)g Fs(b)23 b(>)f(a)30 b Fo(the)g(gr)l(ound)f(state)h(ener)l(gy)f(of)i (\(2.1\))g(with)f(p)l(eri-)456 3523 y(o)l(dic)h(b)l(oundary)f(c)l (onditions)h(satis\014es)1116 3733 y Fs(E)1177 3745 y Fr(0)1215 3733 y Ft(\()p Fs(N)t(;)14 b(L)p Ft(\))p Fs(=)-5 b(N)32 b Fp(\024)22 b Ft(4)p Fs(\031)s(\026\032)1852 3745 y Fr(1)1889 3733 y Fs(a)1943 3672 y Ft(1)c Fp(\000)2096 3639 y Fq(a)p 2096 3653 37 4 v 2099 3700 a(b)2161 3672 y Ft(+)2244 3604 y Fk(\000)2292 3639 y Fq(a)p 2292 3653 V 2295 3700 a(b)2338 3604 y Fk(\001)2376 3622 y Fr(2)2431 3672 y Ft(+)2524 3639 y Fr(1)p 2524 3653 34 4 v 2524 3700 a(2)2581 3604 y Fk(\000)2629 3639 y Fq(a)p 2629 3653 37 4 v 2632 3700 a(b)2675 3604 y Fk(\001)2713 3622 y Fr(3)p 1943 3714 808 4 v 2191 3744 a Fk(\000)2229 3811 y Ft(1)g Fp(\000)2382 3779 y Fq(a)p 2382 3793 37 4 v 2385 3840 a(b)2428 3744 y Fk(\001)2466 3761 y Fr(8)2761 3733 y Fs(:)-2328 b Ft(\(2.12\))456 3953 y Fo(F)-6 b(or)29 b(Dirichlet)g(b)l(oundary)h(c)l(onditions)g(the)f(estimate)g(holds)h (with)f Ft(\(const)p Fs(:)p Ft(\))p Fs(=L)2940 3923 y Fr(2)3005 3953 y Fo(adde)l(d)h(to)f(the)456 4053 y(right)h(side.)40 b(Thus)30 b(in)g(the)f(thermo)l(dynamic)j(limit)e(and)g(for)h(al)t(l)g (b)l(oundary)f(c)l(onditions)1378 4193 y Fs(e)1417 4205 y Fr(0)1454 4193 y Ft(\()p Fs(\032)p Ft(\))p 1355 4230 229 4 v 1355 4306 a(4)p Fs(\031)s(\026\032a)1617 4249 y Fp(\024)1714 4188 y Ft(1)18 b Fp(\000)g Fs(Y)1924 4158 y Fr(1)p Fq(=)p Fr(3)2047 4188 y Ft(+)g Fs(Y)2196 4158 y Fr(2)p Fq(=)p Fr(3)2319 4188 y Fp(\000)2412 4155 y Fr(1)p 2412 4169 34 4 v 2412 4216 a(2)2455 4188 y Fs(Y)p 1714 4230 808 4 v 1910 4307 a Ft(\(1)g Fp(\000)g Fs(Y)2152 4283 y Fr(1)p Fq(=)p Fr(3)2256 4307 y Ft(\))2288 4283 y Fr(8)2532 4249 y Fs(:)-2099 b Ft(\(2.13\))456 4441 y Fo(pr)l(ovide)l(d)31 b Fs(Y)42 b Ft(=)23 b(4)p Fs(\031)s(\032a)1140 4411 y Fr(3)1177 4441 y Fs(=)p Ft(3)f Fs(<)g Ft(1)p Fo(.)605 4591 y(R)l(emark.)55 b Ft(The)33 b(b)r(ound)h(\(2.12\))f(holds)g(for)g (p)r(oten)n(tials)g(with)h(in\014nite)g(range,)g(pro)n(vided)456 4691 y Fs(b)22 b(>)h(a)p Ft(.)37 b(F)-7 b(or)27 b(p)r(oten)n(tials)g (of)h(\014nite)g(range)e Fs(R)1834 4703 y Fr(0)1899 4691 y Ft(it)i(can)f(b)r(e)h(impro)n(v)n(ed)e(for)i Fs(b)22 b(>)h(R)2946 4703 y Fr(0)3011 4691 y Ft(to)1195 4901 y Fs(E)1256 4913 y Fr(0)1294 4901 y Ft(\()p Fs(N)t(;)14 b(L)p Ft(\))p Fs(=)-5 b(N)31 b Fp(\024)23 b Ft(4)p Fs(\031)s(\026\032) 1931 4913 y Fr(1)1968 4901 y Fs(a)2022 4840 y Ft(1)18 b Fp(\000)2165 4772 y Fk(\000)2213 4807 y Fq(a)p 2213 4821 37 4 v 2216 4868 a(b)2259 4772 y Fk(\001)2297 4790 y Fr(2)2353 4840 y Ft(+)2446 4807 y Fr(1)p 2446 4821 34 4 v 2446 4868 a(2)2503 4772 y Fk(\000)2551 4807 y Fq(a)p 2551 4821 37 4 v 2554 4868 a(b)2597 4772 y Fk(\001)2635 4790 y Fr(3)p 2022 4882 650 4 v 2191 4912 a Fk(\000)2229 4979 y Ft(1)g Fp(\000)2382 4947 y Fq(a)p 2382 4961 37 4 v 2385 5008 a(b)2428 4912 y Fk(\001)2466 4929 y Fr(4)2682 4901 y Fs(:)-2249 b Ft(\(2.14\))605 5116 y Fo(Pr)l(o)l(of.)40 b Ft(W)-7 b(e)28 b(\014rst)g(remark)f(that)h(the)g(exp)r(ectation)g(v) -5 b(alue)28 b(of)g(\(2.1\))g(with)g(an)n(y)g(trial)f(w)n(a)n(v)n(e)456 5216 y(function)40 b(giv)n(es)f(an)g(upp)r(er)h(b)r(ound)g(to)g(the)g (b)r(osonic)f(ground)g(state)h(energy)-7 b(,)41 b(ev)n(en)f(if)g(the)p eop %%Page: 6 6 6 5 bop 456 226 a Fr(6)1172 b(ELLIOTT)29 b(H.)g(LIEB)456 425 y Ft(trial)34 b(function)h(is)f(not)g(symmetric)g(under)h(p)r(erm)n (utations)e(of)i(the)g(v)-5 b(ariables.)56 b(The)34 b(reason)456 525 y(is)h(that)h(an)g(absolute)f(ground)g(state)g(of)h(the)g(elliptic) g(di\013eren)n(tial)g(op)r(erator)e(\(2.1\))i(\(i.e.,)i(a)456 624 y(ground)27 b(state)i(without)g(symmetry)f(requiremen)n(t\))g(is)h (a)f(nonnegativ)n(e)g(function)h(whic)n(h)g(can)456 724 y(b)r(e)f(b)r(e)h(symmetrized)f(without)h(c)n(hanging)e(the)i(energy)e (b)r(ecause)h(\(2.1\))g(is)h(symmetric)f(under)456 824 y(p)r(erm)n(utations.)36 b(In)27 b(other)f(w)n(ords,)g(the)i(absolute)e (ground)g(state)h(energy)f(is)h(the)g(same)g(as)f(the)456 923 y(b)r(osonic)h(ground)f(state)i(energy)-7 b(.)605 1023 y(F)g(ollo)n(wing)26 b([)p Fu(15)p Ft(])i(w)n(e)f(c)n(ho)r(ose)f (a)i(trial)f(function)h(of)f(the)h(follo)n(wing)f(form)956 1172 y(\011\()p Fs(x)1100 1184 y Fr(1)1137 1172 y Fs(;)14 b(:)g(:)g(:)g(;)g(x)1369 1184 y Fq(N)1432 1172 y Ft(\))23 b(=)g Fs(F)1628 1184 y Fr(1)1666 1172 y Ft(\()p Fs(x)1745 1184 y Fr(1)1783 1172 y Ft(\))18 b Fp(\001)h Fs(F)1928 1184 y Fr(2)1965 1172 y Ft(\()p Fs(x)2044 1184 y Fr(1)2082 1172 y Fs(;)14 b(x)2166 1184 y Fr(2)2204 1172 y Ft(\))g Fp(\001)g(\001)g(\001)g Fs(F)2414 1184 y Fq(N)2477 1172 y Ft(\()p Fs(x)2556 1184 y Fr(1)2594 1172 y Fs(;)g(:)g(:)g(:)g(;)g(x) 2826 1184 y Fq(N)2889 1172 y Ft(\))p Fs(:)-2488 b Ft(\(2.15\))456 1322 y(More)29 b(sp)r(eci\014cally)-7 b(,)30 b Fs(F)1165 1334 y Fr(1)1229 1322 y Fp(\021)c Ft(1)k(and)f Fs(F)1608 1334 y Fq(i)1666 1322 y Ft(dep)r(ends)h(only)g(on)f(the)i(distance)e (of)h Fs(x)2906 1334 y Fq(i)2964 1322 y Ft(to)f(its)h(nearest)456 1421 y(neigh)n(b)r(or)38 b(among)h(the)i(the)f(p)r(oin)n(ts)g Fs(x)1709 1433 y Fr(1)1746 1421 y Fs(;)14 b(:)g(:)g(:)g(;)g(x)1978 1433 y Fq(i)p Fj(\000)p Fr(1)2131 1421 y Ft(\(taking)40 b(the)g(p)r(erio)r(dic)g(b)r(oundary)f(in)n(to)456 1521 y(accoun)n(t\):)947 1670 y Fs(F)1000 1682 y Fq(i)1028 1670 y Ft(\()p Fs(x)1107 1682 y Fr(1)1145 1670 y Fs(;)14 b(:)g(:)g(:)g(;)g(x)1377 1682 y Fq(i)1405 1670 y Ft(\))23 b(=)g Fs(f)9 b Ft(\()p Fs(t)1660 1682 y Fq(i)1687 1670 y Ft(\))p Fs(;)98 b(t)1870 1682 y Fq(i)1920 1670 y Ft(=)23 b(min)14 b(\()p Fp(j)p Fs(x)2262 1682 y Fq(i)2309 1670 y Fp(\000)k Fs(x)2439 1682 y Fq(j)2475 1670 y Fp(j)p Fs(;)c(j)28 b Ft(=)22 b(1)p Fs(;)14 b(:)g(:)g(:)f(;)h(i)k Fp(\000)g Ft(1\))c Fs(;)-2695 b Ft(\(2.16\))456 1819 y(with)28 b(a)f(function)h Fs(f)36 b Ft(satisfying)1589 1969 y(0)23 b Fp(\024)f Fs(f)32 b Fp(\024)22 b Ft(1)p Fs(;)97 b(f)2113 1934 y Fj(0)2159 1969 y Fp(\025)22 b Ft(0)p Fs(:)-1855 b Ft(\(2.17\))456 2123 y(The)34 b(in)n(tuition)h(b)r (ehind)g(the)g(ansatz)e(\(2.15\))h(is)g(that)g(the)h(particles)f(are)f (inserted)h(in)n(to)g(the)456 2222 y(system)22 b(one)f(at)h(the)h (time,)h(taking)d(in)n(to)h(accoun)n(t)f(the)i(particles)e(previously)g (inserted.)35 b(While)456 2322 y(suc)n(h)29 b(a)g(w)n(a)n(v)n(e)f (function)i(cannot)g(repro)r(duce)e(all)i(correlations)d(presen)n(t)i (in)h(the)g(true)f(ground)456 2422 y(state,)45 b(it)e(turns)f(out)h(to) f(capture)f(the)i(leading)f(term)g(in)g(the)h(energy)e(for)h(dilute)h (gases.)456 2521 y(The)36 b(form)h(\(2.16\))f(is)g(computationally)g (easier)f(to)i(handle)g(than)f(an)h(ansatz)f(of)g(the)h(t)n(yp)r(e)456 2559 y Fk(Q)534 2646 y Fq(i)d(b)o(;)456 3042 y Ft(\(2.18\))456 3266 y(with)31 b Fs(f)689 3278 y Fr(0)725 3266 y Ft(\()p Fs(r)r Ft(\))f(=)d Fs(u)998 3278 y Fr(0)1035 3266 y Ft(\()p Fs(r)r Ft(\))p Fs(=r)r Ft(.)47 b(The)30 b(estimates)h(\(2.12\))e(and)i(\(2.14\))e(are) h(obtained)g(b)n(y)g(somewhat)456 3365 y(length)n(y)g(computations)g (similar)h(as)f(in)h([)p Fu(15)p Ft(],)g(but)h(making)e(use)h(of)g (\(2.11\).)46 b(F)-7 b(or)30 b(details)h(w)n(e)456 3465 y(refer)26 b(to)i([)p Fu(3)p Ft(])f(and)h([)p Fu(10)o Ft(].)605 3565 y(A)34 b(test)g(w)n(a)n(v)n(e)e(function)j(with)f(Diric) n(hlet)g(b)r(oundary)f(condition)g(ma)n(y)g(b)r(e)h(obtained)g(b)n(y) 456 3664 y(lo)r(calizing)j(the)h(w)n(a)n(v)n(e)f(function)i(\(2.15\))e (on)h(the)h(length)f(scale)f Fs(L)p Ft(.)69 b(The)38 b(energy)f(cost)h(p)r(er)456 3764 y(particle)27 b(for)g(this)h(is)f (\(const)p Fs(:)p Ft(\))p Fs(=L)1504 3734 y Fr(2)1541 3764 y Ft(.)1818 b Ff(2)605 3943 y Fu(2.2.)46 b(LO)m(WER)41 b(BOUND.)g Ft(In)36 b(the)g(b)r(eginning)g(it)g(w)n(as)e(explained)i (wh)n(y)f(the)h(lo)n(w)n(er)456 4042 y(b)r(ound)25 b(for)f(the)h(b)r (osonic)g(ground)e(state)i(energy)f(of)g(\(2.1\))h(is)g(not)g(easy)e (to)i(obtain.)36 b(The)25 b(three)456 4142 y(di\013eren)n(t)i(length)h (scales)f(for)g(b)r(osons)f(will)i(pla)n(y)f(a)g(role)g(in)h(the)g(pro) r(of)f(b)r(elo)n(w.)628 4266 y Fp(\017)41 b Ft(The)28 b(scattering)e(length)i Fs(a)p Ft(.)628 4366 y Fp(\017)41 b Ft(The)28 b(mean)f(particle)g(distance)h Fs(\032)1770 4336 y Fj(\000)p Fr(1)p Fq(=)p Fr(3)1926 4366 y Ft(.)628 4466 y Fp(\017)41 b Ft(The)c(`uncertain)n(t)n(y)e(principle)h(length')h Fs(`)2034 4478 y Fq(c)2067 4466 y Ft(,)i(de\014ned)d(b)n(y)h Fs(\026`)2633 4435 y Fj(\000)p Fr(2)2633 4486 y Fq(c)2759 4466 y Ft(=)g Fs(e)2900 4478 y Fr(0)2937 4466 y Ft(\()p Fs(\032)p Ft(\),)i(i.e.,)g Fs(`)3309 4478 y Fq(c)3380 4466 y Fp(\030)711 4568 y Ft(\()p Fs(\032a)p Ft(\))862 4538 y Fj(\000)p Fr(1)p Fq(=)p Fr(2)1019 4568 y Ft(.)605 4693 y(Our)27 b(lo)n(w)n(er)f(b)r(ound)i(for)f Fs(e)1415 4705 y Fr(0)1452 4693 y Ft(\()p Fs(\032)p Ft(\))h(is)g(as)e(follo)n (ws.)605 4850 y Fh(Theorem)32 b(2.3)f Ft(\()p Fu(Lo)m(w)m(er)i(b)s (ound)e(in)g(the)h(thermo)s(dynamic)e(limit)p Ft(\))p Fh(.)38 b Fo(F)-6 b(or)39 b(a)g(p)l(osi-)456 4950 y(tive)30 b(p)l(otential)g Fs(v)j Fo(with)d(\014nite)g(r)l(ange)g(and)g Fs(Y)48 b Fo(smal)t(l)31 b(enough)1558 5086 y Fs(e)1597 5098 y Fr(0)1633 5086 y Ft(\()p Fs(\032)p Ft(\))p 1535 5123 229 4 v 1535 5199 a(4)p Fs(\031)s(\026\032a)1797 5142 y Fp(\025)22 b Ft(\(1)c Fp(\000)h Fs(C)h(Y)2205 5108 y Fr(1)p Fq(=)p Fr(17)2343 5142 y Ft(\))-1919 b(\(2.19\))p eop %%Page: 7 7 7 6 bop 1089 226 a Fr(THE)29 b(BOSE)g(GAS:)g(A)g(SUBTLE)g(MANY-BOD)n(Y) h(PR)n(OBLEM)600 b(7)456 425 y Fo(with)30 b Fs(C)36 b Fo(a)31 b(c)l(onstant.)38 b(If)30 b Fs(v)j Fo(do)l(es)e(not)f(have)h (\014nite)e(r)l(ange,)i(but)e(de)l(cr)l(e)l(ases)i(at)e(le)l(ast)h(as)h (fast)f(as)456 525 y Ft(1)p Fs(=r)579 495 y Fr(3+)p Fq(")732 525 y Fo(at)j(in\014nity)h(with)g(some)h Fs(")30 b(>)g Ft(0)p Fo(,)k(then)g(an)g(analo)l(gous)h(b)l(ound)e(to)h(\(2.19\))h (holds,)i(but)456 624 y(with)32 b Fs(C)39 b Fo(r)l(eplac)l(e)l(d)33 b(by)g(another)g(c)l(onstant)e(and)i(1/17)h(by)f(another)f(exp)l (onent,)h(b)l(oth)g(of)g(which)456 724 y(may)d(dep)l(end)h(on)f Fs(")p Fo(.)605 876 y Ft(It)38 b(should)g(b)r(e)h(noted)f(righ)n(t)f(a) n(w)n(a)n(y)f(that)i(the)h(error)d(term)i Fp(\000)p Fs(C)20 b(Y)2750 846 y Fr(1)p Fq(=)p Fr(17)2926 876 y Ft(in)38 b(\(2.19\))g(is)g(of)456 976 y(no)29 b(fundamen)n(tal)i(signi\014cance) e(and)h(is)f(not)i(b)r(eliev)n(ed)f(to)f(re\015ect)h(the)h(true)f (state)g(of)g(a\013airs.)456 1075 y(Presumably)-7 b(,)32 b(it)g(do)r(es)g(not)g(ev)n(en)g(ha)n(v)n(e)f(the)h(righ)n(t)g(sign.)50 b(W)-7 b(e)32 b(men)n(tion)g(in)h(passing)e(that)h Fs(C)456 1175 y Ft(can)27 b(b)r(e)h(tak)n(en)f(to)g(b)r(e)h(8)p Fs(:)p Ft(9)f([)p Fu(10)p Ft(].)605 1274 y(As)h(men)n(tioned)g(in)g (the)g(In)n(tro)r(duction)f(a)g(lo)n(w)n(er)g(b)r(ound)h(on)f Fs(E)2573 1286 y Fr(0)2611 1274 y Ft(\()p Fs(N)t(;)14 b(L)p Ft(\))28 b(for)f(\014nite)h Fs(N)37 b Ft(and)456 1374 y Fs(L)29 b Ft(is)g(of)h(imp)r(ortance)f(for)g(applications)f(to)i (inhomogeneous)e(gases,)g(and)i(in)g(fact)f(w)n(e)g(deriv)n(e)456 1474 y(\(2.19\))d(from)i(suc)n(h)f(a)g(b)r(ound.)37 b(W)-7 b(e)28 b(state)g(it)g(in)g(the)g(follo)n(wing)e(w)n(a)n(y:)605 1626 y Fh(Theorem)32 b(2.4)f Ft(\()p Fu(Lo)m(w)m(er)i(b)s(ound)e(in)g (a)i(\014nite)e(b)s(o)m(x)p Ft(\))p Fh(.)41 b Fo(F)-6 b(or)22 b(a)g(p)l(ositive)g(p)l(otential)g Fs(v)j Fo(with)456 1725 y(\014nite)f(r)l(ange)i(ther)l(e)f(is)h(a)f Fs(\016)i(>)22 b Ft(0)j Fo(such)g(that)g(the)h(the)f(gr)l(ound)g(state)g(ener)l(gy)h (of)g(\(2.1\))g(with)g(Neu-)456 1825 y(mann)j(c)l(onditions)i (satis\014es)1284 1984 y Fs(E)1345 1996 y Fr(0)1382 1984 y Ft(\()p Fs(N)t(;)14 b(L)p Ft(\))p Fs(=)-5 b(N)32 b Fp(\025)23 b Ft(4)p Fs(\031)s(\026\032a)2078 1892 y Fk(\020)2127 1984 y Ft(1)18 b Fp(\000)g Fs(C)i(Y)2415 1950 y Fr(1)p Fq(=)p Fr(17)2553 1892 y Fk(\021)456 1984 y Ft(\(2.20\))456 2159 y Fo(for)33 b(al)t(l)g Fs(N)41 b Fo(and)33 b Fs(L)f Fo(with)h Fs(Y)46 b(<)28 b(\016)35 b Fo(and)e Fs(L=a)26 b(>)i(C)2006 2129 y Fj(0)2029 2159 y Fs(Y)2096 2129 y Fj(\000)p Fr(6)p Fq(=)p Fr(17)2285 2159 y Fo(.)47 b(Her)l(e)32 b Fs(C)38 b Fo(and)33 b Fs(C)2884 2129 y Fj(0)2940 2159 y Fo(ar)l(e)g(c)l(onstants,)456 2259 y(indep)l(endent)j(of)h Fs(N)44 b Fo(and)37 b Fs(L)p Fo(.)56 b(\(Note)36 b(that)g(the)g(c)l (ondition)h(on)f Fs(L=a)f Fo(r)l(e)l(quir)l(es)g(in)h(p)l(articular)456 2361 y(that)h Fs(N)46 b Fo(must)37 b(b)l(e)h(lar)l(ge)g(enough,)i Fs(N)46 b(>)37 b Ft(\(const.\))p Fs(Y)2152 2331 y Fj(\000)p Fr(1)p Fq(=)p Fr(17)2341 2361 y Fo(.\))62 b(As)37 b(in)h(The)l(or)l(em) h(2.3)f(such)g(a)456 2461 y(b)l(ound,)28 b(but)e(p)l(ossibly)j(with)f (other)g(c)l(onstants)e(and)i(another)g(exp)l(onent)f(for)h Fs(Y)18 b Fo(,)29 b(holds)f(also)g(for)456 2561 y(p)l(otentials)i Fs(v)j Fo(of)e(in\014nite)e(r)l(ange)h(de)l(cr)l(e)l(asing)h(faster)f (than)g Ft(1)p Fs(=r)2452 2531 y Fr(3)2518 2561 y Fo(at)g(in\014nity.) 605 2713 y Ft(The)35 b(\014rst)f(step)h(in)f(the)h(pro)r(of)f(of)h (Theorem)f(2.4)f(is)i(a)f(generalization)e(of)j(a)f(lemma)h(of)456 2812 y(Dyson,)29 b(whic)n(h)h(allo)n(ws)e(us)i(to)f(replace)g Fs(v)k Ft(b)n(y)c(a)g(`soft')h(p)r(oten)n(tial,)g(at)f(the)h(cost)f(of) h(sacri\014cing)456 2912 y(kinetic)d(energy)g(and)g(increasing)f(the)i (e\013ectiv)n(e)g(range.)605 3064 y Fh(Lemma)j(2.5.)41 b Fo(L)l(et)31 b Fs(v)s Ft(\()p Fs(r)r Ft(\))c Fp(\025)f Ft(0)31 b Fo(with)h(\014nite)f(r)l(ange)h Fs(R)2272 3076 y Fr(0)2309 3064 y Fo(.)44 b(L)l(et)31 b Fs(U)9 b Ft(\()p Fs(r)r Ft(\))27 b Fp(\025)f Ft(0)31 b Fo(b)l(e)h(any)f(function)456 3167 y(satisfying)829 3100 y Fk(R)899 3167 y Fs(U)9 b Ft(\()p Fs(r)r Ft(\))p Fs(r)1107 3137 y Fr(2)1146 3167 y Fs(dr)34 b Fp(\024)d Ft(1)j Fo(and)h Fs(U)9 b Ft(\()p Fs(r)r Ft(\))33 b(=)e(0)j Fo(for)h Fs(r)g(<)c(R)2340 3179 y Fr(0)2377 3167 y Fo(.)53 b(L)l(et)34 b Fp(B)g(\032)d Fu(R)2859 3130 y Fr(3)2931 3167 y Fo(b)l(e)k(star)f(shap)l(e)l(d)456 3266 y(with)c(r)l(esp)l(e)l(ct)f(to)h Ft(0)f Fo(\(e.g.)i(c)l(onvex)f (with)g Ft(0)23 b Fp(2)g(B)s Fo(\).)38 b(Then)30 b(for)h(al)t(l)g (di\013er)l(entiable)g(functions)f Fs( )1283 3334 y Fk(Z)1329 3523 y Fj(B)1393 3380 y Fk(\002)1427 3447 y Fs(\026)p Fp(jr)p Fs( )s Fp(j)1649 3413 y Fr(2)1705 3447 y Ft(+)1798 3415 y Fr(1)p 1798 3429 34 4 v 1798 3476 a(2)1841 3447 y Fs(v)s Fp(j)p Fs( )s Fp(j)1987 3413 y Fr(2)2025 3380 y Fk(\003)2083 3447 y Fp(\025)22 b Fs(\026a)2278 3334 y Fk(Z)2324 3523 y Fj(B)2387 3447 y Fs(U)9 b Fp(j)p Fs( )s Fp(j)2556 3413 y Fr(2)2594 3447 y Fs(:)-2161 b Ft(\(2.21\))605 3651 y Fo(Pr)l(o)l(of.)50 b Ft(Actually)-7 b(,)33 b(\(2.21\))e(holds)g (with)h Fs(\026)p Fp(jr)p Fs(\036)p Ft(\()o Fs(~)-41 b(x)r Ft(\))p Fp(j)2215 3621 y Fr(2)2284 3651 y Ft(replaced)30 b(b)n(y)i(the)f(\(smaller\))h(radial)456 3751 y(kinetic)h(energy)-7 b(,)34 b Fs(\026)p Fp(j)p Fs(@)5 b(\036)p Ft(\()o Fs(~)-41 b(x)q Ft(\))p Fs(=@)5 b(r)r Fp(j)1456 3721 y Fr(2)1494 3751 y Ft(,)35 b(and)e(it)h(su\016ces)f(to)g(pro)n(v)n(e)f(the)i (analog)d(of)j(\(2.21\))e(for)h(the)456 3851 y(in)n(tegral)e(along)g (eac)n(h)g(radial)g(line)i(with)f(\014xed)h(angular)d(v)-5 b(ariables.)50 b(Along)32 b(suc)n(h)g(a)f(line)i(w)n(e)456 3950 y(write)d Fs(\036)p Ft(\()o Fs(~)-41 b(x)q Ft(\))28 b(=)f Fs(u)p Ft(\()p Fs(r)r Ft(\))p Fs(=r)34 b Ft(with)d Fs(u)p Ft(\(0\))c(=)h(0.)45 b(W)-7 b(e)31 b(consider)e(\014rst)h(the)h (sp)r(ecial)f(case)g(when)g(when)456 4050 y Fs(U)36 b Ft(is)27 b(a)h(delta-function)f(at)h(some)f(radius)g Fs(R)c Fp(\025)g Fs(R)2031 4062 y Fr(0)2068 4050 y Ft(,)28 b(i.e.,)1583 4223 y Fs(U)9 b Ft(\()p Fs(r)r Ft(\))24 b(=)1903 4167 y(1)p 1874 4204 101 4 v 1874 4280 a Fs(R)1938 4256 y Fr(2)1985 4223 y Fs(\016)s Ft(\()p Fs(r)d Fp(\000)d Fs(R)q Ft(\))p Fs(:)-1861 b Ft(\(2.22\))456 4390 y(F)-7 b(or)27 b(suc)n(h)g Fs(U)36 b Ft(the)28 b(analog)e(of)i(\(2.21\))f (along)f(the)i(radial)e(line)i(is)594 4595 y Fk(Z)677 4615 y Fq(R)727 4623 y Fg(1)641 4783 y Fr(0)764 4708 y Fp(f)p Fs(\026)p Ft([)p Fs(u)927 4674 y Fj(0)949 4708 y Ft(\()p Fs(r)r Ft(\))20 b Fp(\000)e Ft(\()p Fs(u)p Ft(\()p Fs(r)r Ft(\))p Fs(=r)r Ft(\)])1474 4674 y Fr(2)1532 4708 y Ft(+)1625 4675 y Fr(1)p 1625 4689 34 4 v 1625 4736 a(2)1668 4708 y Fs(v)s Ft(\()p Fs(r)r Ft(\))p Fp(j)p Fs(u)p Ft(\()p Fs(r)r Ft(\)])2011 4674 y Fr(2)2050 4708 y Fp(g)p Fs(dr)26 b Fp(\025)2286 4566 y Fk(\()2353 4651 y Ft(0)536 b(if)28 b Fs(R)3070 4663 y Fr(1)3131 4651 y Fs(<)22 b(R)2353 4771 y(\026a)p Fp(j)p Fs(u)p Ft(\()p Fs(R)q Ft(\))p Fp(j)2669 4741 y Fr(2)2706 4771 y Fs(=R)2812 4741 y Fr(2)2931 4771 y Ft(if)28 b Fs(R)c Fp(\024)f Fs(R)3245 4783 y Fr(1)456 4520 y Ft(\(2.23\))456 4917 y(where)k Fs(R)759 4929 y Fr(1)824 4917 y Ft(is)g(the)h(length)g(of)g(the)g (radial)e(line)i(segmen)n(t)f(in)h Fp(B)s Ft(.)36 b(The)28 b(case)f Fs(R)2879 4929 y Fr(1)2939 4917 y Fs(<)c(R)28 b Ft(is)g(trivial,)456 5016 y(b)r(ecause)22 b Fs(\026)p Fp(j)p Fs(@)5 b( )s(=@)g(r)r Fp(j)1090 4986 y Fr(2)1137 5016 y Ft(+)1221 4984 y Fr(1)p 1221 4998 V 1221 5045 a(2)1264 5016 y Fs(v)s Fp(j)p Fs( )s Fp(j)1410 4986 y Fr(2)1471 5016 y Fp(\025)22 b Ft(0.)35 b(\(Note)24 b(that)f(p)r (ositivit)n(y)g(of)g Fs(v)k Ft(is)c(used)g(here.\))35 b(If)24 b Fs(R)f Fp(\024)g Fs(R)3407 5028 y Fr(1)456 5116 y Ft(w)n(e)33 b(consider)g(the)i(in)n(tegral)e(on)h(the)g(the)h (left)f(side)g(of)g(\(2.23\))g(from)g(0)f(to)h Fs(R)h Ft(instead)f(of)g Fs(R)3407 5128 y Fr(1)456 5216 y Ft(and)d(minimize)i (it)f(under)f(the)i(b)r(oundary)e(condition)g(that)h Fs(u)p Ft(\(0\))e(=)g(0)h(and)h Fs(u)p Ft(\()p Fs(R)q Ft(\))f(is)h(a)f(\014xed)p eop %%Page: 8 8 8 7 bop 456 226 a Fr(8)1172 b(ELLIOTT)29 b(H.)g(LIEB)456 425 y Ft(constan)n(t.)58 b(Since)35 b(ev)n(erything)f(is)g(homogeneous) f(in)j Fs(u)e Ft(w)n(e)h(ma)n(y)f(normalize)g(this)h(v)-5 b(alue)35 b(to)456 525 y Fs(u)p Ft(\()p Fs(R)q Ft(\))40 b(=)h Fs(R)26 b Fp(\000)f Fs(a)p Ft(.)70 b(This)38 b(minimization)g (problem)g(leads)g(to)g(the)h(zero)e(energy)h(scattering)456 624 y(equation)29 b(\(2.6\).)45 b(Since)30 b Fs(v)k Ft(is)c(p)r(ositiv) 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Fs(U)9 b Ft(\()p Fs(R)q Ft(\))p Fs(R)1507 1815 y Fr(2)1543 1846 y Fs(dR)24 b Fp(\024)f Ft(1)k(b)n(y)h(assumption.)1001 b Ff(2)605 1945 y Ft(By)35 b(dividing)g(\003)g(for)f(giv)n(en)g(p)r (oin)n(ts)g Fs(~)-41 b(x)1826 1957 y Fr(1)1864 1945 y Fs(;)14 b(:)g(:)g(:)g(;)f(~)-41 b(x)2096 1957 y Fq(N)2194 1945 y Ft(in)n(to)35 b(V)-7 b(oronoi)34 b(cells)h Fp(B)2930 1957 y Fq(i)2992 1945 y Ft(that)g(con)n(tain)456 2045 y(all)29 b(p)r(oin)n(ts)h(closer)f(to)f Fs(~)-41 b(x)1209 2057 y Fq(i)1267 2045 y Ft(than)30 b(to)f Fs(~)-41 b(x)1614 2057 y Fq(j)1680 2045 y Ft(with)30 b Fs(j)i Fp(6)p Ft(=)26 b Fs(i)k Ft(\(these)g(cells)g(are)f(star)g(shap)r(ed)h(w.r.t.)42 b Fs(~)-41 b(x)3393 2057 y Fq(i)3421 2045 y Ft(,)456 2145 y(indeed)32 b(con)n(v)n(ex\),)g(the)h(follo)n(wing)e(corollary)e (of)j(Lemma)g(2.5)g(can)f(b)r(e)i(deriv)n(ed)e(in)i(the)f(same)456 2244 y(w)n(a)n(y)26 b(as)h(the)h(corresp)r(onding)e(Eq.)h(\(28\))g(in)h ([)p Fu(15)o Ft(].)605 2405 y Fh(Cor)n(ollar)-6 b(y)31 b(2.6.)40 b Ft(F)-7 b(or)27 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Fq(j;)12 b(j)s Fj(6)p Fr(=)p Fq(i)2181 3411 y Fp(j)o Fs(~)-41 b(x)2251 3423 y Fq(i)2297 3411 y Fp(\000)17 b Fs(~)-41 b(x)2427 3423 y Fq(j)2463 3411 y Fp(j)p Fs(:)-2053 b Ft(\(2.27\))456 3617 y(\(Note)26 b(that)h Fs(t)896 3629 y Fq(i)949 3617 y Ft(has)f(here)g(a)f(sligh)n(tly)h(di\013eren)n(t)g(meaning)g(than)g (in)g(\(2.16\),)g(where)g(it)g(denoted)456 3716 y(the)i(distance)f(to)g (the)h(nearest)f(neigh)n(b)r(or)g(among)f(the)h Fs(~)-41 b(x)2252 3728 y Fq(j)2315 3716 y Ft(with)28 b Fs(j)g Fp(\024)23 b Fs(i)18 b Fp(\000)g Ft(1.\))605 3816 y(Dyson)33 b(considers)e(in)j([)p Fu(15)o Ft(])f(a)g(one)f(parameter)g(family)h (of)g Fs(U)9 b Ft('s)32 b(that)h(is)g(essen)n(tially)f(the)456 3916 y(same)27 b(as)g(the)h(follo)n(wing)e(c)n(hoice,)h(whic)n(h)g(is)h (con)n(v)n(enien)n(t)e(for)h(the)h(presen)n(t)f(purp)r(ose:)1161 4140 y Fs(U)1218 4152 y Fq(R)1272 4140 y Ft(\()p Fs(r)r Ft(\))d(=)1487 3998 y Fk(\()1554 4083 y Ft(3\()p Fs(R)1692 4053 y Fr(3)1747 4083 y Fp(\000)18 b Fs(R)1894 4053 y 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Fp(2)f Ft(\003)2045 4806 y Fq(N)2138 4836 y Ft(with)k Fp(j)o Fs(~)-41 b(x)2399 4848 y Fq(i)2447 4836 y Fp(\000)18 b Fs(~)-41 b(x)2578 4848 y Fq(j)2613 4836 y Fp(j)27 b(\025)f Fs(R)2817 4848 y Fr(0)2880 4836 y Ft(=)g Fs(a)p Ft(,)k(b)r(ecause)g(of)456 4936 y(the)e(hard)f(core.)35 b(A)n(t)28 b(\014xed)g Fs(R)g Ft(simple)g(geometry)e(giv)n(es)1288 5135 y(inf)1182 5193 y Fr(\()o Fq(~)-33 b(x)1246 5201 y Fg(1)1278 5193 y Fq(;:::)o(;)o(~)g(x)1415 5201 y Fe(N)1468 5193 y Fr(\))1508 5135 y Fs(W)1586 5147 y Fq(R)1641 5135 y Ft(\()o Fs(~)-41 b(x)1720 5147 y Fr(1)1758 5135 y Fs(;)14 b(:)g(:)g(:)f(;)g(~)-41 b(x)1989 5147 y Fq(N)2052 5135 y Ft(\))24 b Fp(\025)2195 5018 y Fk(\022)2286 5079 y Fs(A)p 2267 5116 101 4 v 2267 5192 a(R)2331 5168 y Fr(3)2396 5135 y Fp(\000)2527 5079 y Fs(B)p 2489 5116 144 4 v 2489 5192 a(\032R)2596 5168 y Fr(6)2643 5018 y Fk(\023)456 5135 y Ft(\(2.30\))p eop %%Page: 9 9 9 8 bop 1089 226 a Fr(THE)29 b(BOSE)g(GAS:)g(A)g(SUBTLE)g(MANY-BOD)n(Y) h(PR)n(OBLEM)600 b(9)456 425 y Ft(with)33 b(certain)g(constan)n(ts)f Fs(A)i Ft(and)f Fs(B)t Ft(.)54 b(An)34 b(ev)-5 b(aluation)33 b(of)g(these)h(constan)n(ts)e(giv)n(es)g(Dyson's)456 525 y(b)r(ound)1458 687 y Fs(E)5 b Ft(\()p Fs(N)t(;)14 b(L)p Ft(\))p Fs(=)-5 b(N)32 b Fp(\025)2063 631 y Ft(1)p 1987 668 194 4 v 1987 753 a(10)2071 685 y Fp(p)p 2139 685 42 4 v 2139 753 a Ft(2)2190 687 y(4)p Fs(\031)s(\026\032a:)-1986 b Ft(\(2.31\))605 875 y(The)29 b(main)f(reason)f(this)i(metho)r(d)g(do) r(es)f(not)h(giv)n(e)f(a)g(b)r(etter)h(b)r(ound)g(is)f(that)h Fs(R)g Ft(m)n(ust)g(b)r(e)456 975 y(c)n(hosen)d(quite)h(big,)g(namely)f (of)h(the)g(order)f(of)h(the)g(mean)f(particle)h(distance)f Fs(\032)2952 944 y Fj(\000)p Fr(1)p Fq(=)p Fr(3)3109 975 y Ft(,)h(in)g(order)456 1074 y(to)g(guaran)n(tee)e(that)i(the)g (spheres)g(of)g(radius)f Fs(R)h Ft(around)g(the)g Fs(N)36 b Ft(p)r(oin)n(ts)27 b(o)n(v)n(erlap.)34 b(Otherwise)456 1174 y(the)22 b(in\014m)n(um)g(of)g Fs(W)1079 1186 y Fq(R)1156 1174 y Ft(will)g(b)r(e)g(zero.)34 b(But)22 b(large)e Fs(R)j Ft(means)e(that)h Fs(W)2562 1186 y Fq(R)2639 1174 y Ft(is)f(small.)35 b(It)22 b(should)g(also)456 1273 y(b)r(e)j(noted)h(that)f(this)h(metho)r(d)g(do)r(es)f(not)g(w)n (ork)f(for)g(p)r(oten)n(tials)h(other)g(than)h(hard)e(spheres:)35 b(If)456 1373 y Fp(j)o Fs(~)-41 b(x)526 1385 y Fq(i)571 1373 y Fp(\000)17 b Fs(~)-41 b(x)701 1385 y Fq(j)736 1373 y Fp(j)28 b Ft(is)f(allo)n(w)n(ed)e(to)j(b)r(e)f(less)g(than)g Fs(R)1787 1385 y Fr(0)1825 1373 y Ft(,)g(then)h(the)f(righ)n(t)g(side)g (of)g(\(2.29\))g(is)g(zero)f(b)r(ecause)456 1473 y Fs(U)9 b Ft(\()p Fs(r)r Ft(\))24 b(=)e(0)28 b(for)f Fs(r)e(<)e(R)1146 1485 y Fr(0)1183 1473 y Ft(.)605 1572 y(F)-7 b(or)29 b(these)g(reasons)f(w)n(e)h(tak)n(e)g(another)f(route.)42 b(W)-7 b(e)30 b(still)g(use)f(Lemma)g(2.21)f(to)i(get)f(in)n(to)456 1672 y(the)c(soft)h(p)r(oten)n(tial)f(regime,)g(but)h(w)n(e)f(do)g Fo(not)g Ft(sacri\014ce)g Fo(al)t(l)h Ft(the)g(kinetic)g(energy)e(as)h (in)g(\(2.25\).)456 1772 y(Instead)i(w)n(e)g(write,)h(for)f Fs(")22 b(>)h Ft(0)1133 1916 y Fs(H)1202 1928 y Fq(N)1288 1916 y Ft(=)g Fs("H)1484 1928 y Fq(N)1565 1916 y Ft(+)18 b(\(1)g Fp(\000)g Fs(")p Ft(\))p Fs(H)1963 1928 y Fq(N)2049 1916 y Fp(\025)23 b Fs("T)2225 1928 y Fq(N)2306 1916 y Ft(+)18 b(\(1)g Fp(\000)g Fs(")p Ft(\))p Fs(H)2704 1928 y Fq(N)456 1916 y Ft(\(2.32\))456 2065 y(with)28 b Fs(T)694 2077 y Fq(N)779 2065 y Ft(=)23 b Fp(\000)946 2002 y Fk(P)1033 2090 y Fq(i)1074 2065 y Ft(\001)1143 2077 y Fq(i)1199 2065 y Ft(and)k(use)h(\(2.25\))f(only)g(for)g(the)h (part)f(\(1)18 b Fp(\000)g Fs(")p Ft(\))p Fs(H)2691 2077 y Fq(N)2754 2065 y Ft(.)37 b(This)28 b(giv)n(es)1455 2209 y Fs(H)1524 2221 y Fq(N)1610 2209 y Fp(\025)22 b Fs("T)1785 2221 y Fq(N)1866 2209 y Ft(+)c(\(1)g Fp(\000)g Fs(")p Ft(\))p Fs(\026aW)2367 2221 y Fq(R)2422 2209 y Fs(:)-1989 b Ft(\(2.33\))456 2353 y(W)-7 b(e)25 b(consider)e(the)i(op)r (erator)e(on)i(the)g(righ)n(t)f(side)g(from)h(the)g(viewp)r(oin)n(t)g (of)f(\014rst)h(order)e(p)r(ertur-)456 2453 y(bation)k(theory)-7 b(,)27 b(with)h Fs("T)1268 2465 y Fq(N)1358 2453 y Ft(as)f(the)h(unp)r (erturb)r(ed)g(part,)f(denoted)h Fs(H)2669 2465 y Fr(0)2706 2453 y Ft(.)605 2553 y(The)h(ground)f(state)h(of)g Fs(H)1430 2565 y Fr(0)1497 2553 y Ft(in)g(a)g(b)r(o)n(x)f(of)i(side)f(length)g Fs(L)f Ft(is)h(\011)2581 2565 y Fr(0)2618 2553 y Ft(\()o Fs(~)-41 b(x)2697 2565 y Fr(1)2735 2553 y Fs(;)14 b(:)g(:)g(:)g(;)f(~) -41 b(x)2967 2565 y Fq(N)3030 2553 y Ft(\))26 b Fp(\021)f Fs(L)3235 2522 y Fj(\000)p Fr(3)p Fq(N)q(=)p Fr(2)456 2652 y Ft(and)g(w)n(e)g(denote)g(exp)r(ectation)h(v)-5 b(alues)25 b(in)g(this)h(state)f(b)n(y)h Fp(h\001i)2347 2664 y Fr(0)2384 2652 y Ft(.)37 b(A)25 b(computation,)h(cf.)g(Eq.)f (\(21\))456 2752 y(in)i([)p Fu(1)p Ft(],)h(giv)n(es)570 2896 y(4)p Fs(\031)s(\032)719 2829 y Fk(\000)757 2896 y Ft(1)18 b Fp(\000)922 2863 y Fr(1)p 910 2877 59 4 v 910 2925 a Fq(N)978 2829 y Fk(\001)1099 2896 y Fp(\025)83 b(h)p Fs(W)1357 2908 y Fq(R)1412 2896 y Fp(i)1444 2908 y Fr(0)1482 2896 y Fs(=)-5 b(N)1099 3048 y Fp(\025)83 b Ft(4)p Fs(\031)s(\032)1396 2980 y Fk(\000)1434 3048 y Ft(1)18 b Fp(\000)1599 3015 y Fr(1)p 1587 3029 V 1587 3076 a Fq(N)1655 2980 y Fk(\001)c(\000)1745 3048 y Ft(1)k Fp(\000)1898 3015 y Fr(2)p Fq(R)p 1898 3029 84 4 v 1917 3076 a(L)1992 2980 y Fk(\001)2030 2996 y Fr(3)2081 2980 y Fk(\000)2119 3048 y Ft(1)g(+)g(4)p Fs(\031)s(\032)p Ft(\(1)g Fp(\000)2595 3015 y Fr(1)p 2582 3029 59 4 v 2582 3076 a Fq(N)2650 3048 y Ft(\)\()p Fs(R)2778 3013 y Fr(3)2834 3048 y Fp(\000)g Fs(R)2981 3013 y Fr(3)2980 3068 y(0)3018 3048 y Ft(\))p Fs(=)p Ft(3\))3166 2980 y Fk(\001)3204 2996 y Fj(\000)p Fr(1)3307 3048 y Fs(:)-2874 b Ft(\(2.34\))456 3200 y(The)37 b(rationale)f(b)r(ehind)i(the)g(v)-5 b(arious)36 b(factors)g(is)h(as)g(follo)n(ws:)55 b(\(1)25 b Fp(\000)2743 3167 y Fr(1)p 2730 3181 V 2730 3228 a Fq(N)2799 3200 y Ft(\))38 b(comes)e(from)h(the)456 3304 y(fact)f(that)h(the)g(n)n(um)n(b)r(er)f(of)g(pairs)g(is)g Fs(N)9 b Ft(\()p Fs(N)33 b Fp(\000)24 b Ft(1\))p Fs(=)p Ft(2)36 b(and)g(not)g Fs(N)2581 3274 y Fr(2)2618 3304 y Fs(=)p Ft(2,)i(\(1)24 b Fp(\000)g Ft(2)p Fs(R=L)p Ft(\))3186 3274 y Fr(3)3259 3304 y Ft(tak)n(es)456 3404 y(in)n(to)35 b(accoun)n(t)h(the)g(fact)g(that)g(the)h(particles)e(do)h(not)g(in)n (teract)f(b)r(ey)n(ond)h(the)g(b)r(oundary)g(of)456 3503 y(\003,)28 b(and)g(the)h(last)f(factor)g(measures)f(the)i(probabilit)n (y)e(to)i(\014nd)g(another)e(particle)h(within)h(the)456 3603 y(in)n(teraction)d(range)g(of)i(the)g(p)r(oten)n(tial)g Fs(U)1745 3615 y Fq(R)1826 3603 y Ft(for)f(a)h(giv)n(en)e(particle.)605 3703 y(The)34 b(\014rst)f(order)g(result)g(\(2.34\))g(lo)r(oks)g(at)h (\014rst)f(sigh)n(t)h(quite)g(promising,)g(for)f(if)i(w)n(e)e(let)456 3802 y Fs(L)39 b Fp(!)h(1)p Ft(,)g Fs(N)49 b Fp(!)40 b(1)e Ft(with)g Fs(\032)i Ft(=)g Fs(N)r(=L)1736 3772 y Fr(3)1810 3802 y Ft(\014xed,)g(and)e(subsequen)n(tly)f(tak)n(e)g Fs(R)k Fp(!)f(1)p Ft(,)g(then)456 3902 y Fp(h)p Fs(W)566 3914 y Fq(R)621 3902 y Fp(i)653 3914 y Fr(0)690 3902 y Fs(=)-5 b(N)34 b Ft(con)n(v)n(erges)22 b(to)j(4)p Fs(\031)s(\032)p Ft(,)g(whic)n(h)g(is)g(just)g(what)g(is)g(desired.)35 b(But)25 b(the)h(\014rst)e(order)g(result)456 4001 y(\(2.34\))g(is)h (not)f(a)h(rigorous)d(b)r(ound)k(on)e Fs(E)1728 4013 y Fr(0)1766 4001 y Ft(\()p Fs(N)t(;)14 b(L)p Ft(\),)26 b(w)n(e)e(need)h Fo(err)l(or)j(estimates)p Ft(,)e(and)e(these)h(will) 456 4101 y(dep)r(end)j(on)f Fs(")p Ft(,)h Fs(R)g Ft(and)f Fs(L)p Ft(.)605 4201 y(W)-7 b(e)19 b(no)n(w)f(recall)g Fo(T)-6 b(emple's)23 b(ine)l(quality)d Ft([)p Fu(18)o Ft(])f(for)f(the)h(exp)r(ectations)g(v)-5 b(alues)18 b(of)h(an)f(op)r(erator)456 4300 y Fs(H)29 b Ft(=)23 b Fs(H)711 4312 y Fr(0)750 4300 y Ft(+)r Fs(V)38 b Ft(in)19 b(the)h(ground)e(state)h Fp(h\001i)1681 4312 y Fr(0)1738 4300 y Ft(of)h Fs(H)1894 4312 y Fr(0)1931 4300 y Ft(.)34 b(It)20 b(is)f(a)g(simple)g(consequence)f(of)i(the)f(op)r(erator)456 4400 y(inequalit)n(y)1534 4544 y(\()p Fs(H)25 b Fp(\000)18 b Fs(E)1804 4556 y Fr(0)1842 4544 y Ft(\)\()p Fs(H)25 b Fp(\000)18 b Fs(E)2144 4556 y Fr(1)2182 4544 y Ft(\))23 b Fp(\025)g Ft(0)-1911 b(\(2.35\))456 4689 y(for)27 b(the)h(t)n(w)n(o)e (lo)n(w)n(est)h(eigen)n(v)-5 b(alues,)26 b Fs(E)1645 4701 y Fr(0)1706 4689 y Fs(<)d(E)1855 4701 y Fr(1)1892 4689 y Ft(,)28 b(of)g Fs(H)34 b Ft(and)27 b(reads)1449 4881 y Fs(E)1510 4893 y Fr(0)1571 4881 y Fp(\025)22 b(h)p Fs(H)7 b Fp(i)1798 4893 y Fr(0)1854 4881 y Fp(\000)1947 4825 y(h)p Fs(H)2055 4795 y Fr(2)2092 4825 y Fp(i)2124 4837 y Fr(0)2180 4825 y Fp(\000)18 b(h)p Fs(H)7 b Fp(i)2403 4795 y Fr(2)2403 4846 y(0)p 1947 4862 494 4 v 2005 4938 a Fs(E)2066 4950 y Fr(1)2122 4938 y Fp(\000)18 b(h)p Fs(H)7 b Fp(i)2345 4950 y Fr(0)456 4881 y Ft(\(2.36\))456 5098 y(pro)n(vided)30 b Fs(E)862 5110 y Fr(1)921 5098 y Fp(\000)20 b(h)p Fs(H)7 b Fp(i)1146 5110 y Fr(0)1213 5098 y Fs(>)29 b Ft(0.)48 b(F)-7 b(urthermore,)32 b(if)g Fs(V)48 b Fp(\025)29 b Ft(0)i(w)n(e)h(ma)n(y)e(use)i Fs(E)2792 5110 y Fr(1)2859 5098 y Fp(\025)d Fs(E)3019 5055 y Fr(\(0\))3014 5121 y(1)3108 5098 y Ft(=)i(second)456 5216 y(lo)n(w)n(est)26 b(eigen)n(v)-5 b(alue)27 b(of)g Fs(H)1265 5228 y Fr(0)1330 5216 y Ft(and)h(replace)e Fs(E)1834 5228 y Fr(1)1899 5216 y Ft(in)i(\(2.36\))f(b)n(y)h Fs(E)2418 5172 y Fr(\(0\))2413 5238 y(1)2507 5216 y Ft(.)p eop %%Page: 10 10 10 9 bop 456 226 a Fr(10)1139 b(ELLIOTT)29 b(H.)g(LIEB)605 425 y Ft(F)-7 b(rom)27 b(\(2.34\))g(and)g(\(2.36\))g(w)n(e)h(get)f(the) h(estimate)1285 552 y Fs(E)1346 564 y Fr(0)1383 552 y Ft(\()p Fs(N)t(;)14 b(L)p Ft(\))p 1285 589 328 4 v 1411 665 a Fs(N)1645 608 y Fp(\025)23 b Ft(4)p Fs(\031)s(\026a\032)14 b Ft(\(1)k Fp(\000)g(E)7 b Ft(\()p Fs(\032;)14 b(L;)g(R)q(;)g(")p Ft(\)\))-2156 b(\(2.37\))456 766 y(with)465 905 y(1)18 b Fp(\000)g(E)7 b Ft(\()p Fs(\032;)14 b(L;)g(R)q(;)g(")p Ft(\))83 b(=)f(\(1)19 b Fp(\000)f Fs(")p Ft(\))1528 813 y Fk(\020)1577 905 y Ft(1)g Fp(\000)1770 873 y Fr(1)p 1730 887 113 4 v 1730 934 a Fq(\032L)1810 917 y Fg(3)1852 813 y Fk(\021)1916 838 y(\000)1954 905 y Ft(1)g Fp(\000)2107 873 y Fr(2)p Fq(R)p 2107 887 84 4 v 2126 934 a(L)2200 838 y Fk(\001)2238 854 y Fr(3)2289 838 y Fk(\000)2327 905 y Ft(1)g(+)2480 873 y Fr(4)p Fq(\031)p 2480 887 75 4 v 2501 934 a Fr(3)2564 905 y Fs(\032)p Ft(\(1)g Fp(\000)2805 873 y Fr(1)p 2792 887 59 4 v 2792 934 a Fq(N)2861 905 y Ft(\)\()p Fs(R)2989 871 y Fr(3)3045 905 y Fp(\000)g Fs(R)3192 871 y Fr(3)3191 926 y(0)3229 905 y Ft(\)\))3293 838 y Fk(\001)3332 854 y Fj(\000)p Fr(1)1120 1138 y Fp(\002)1267 996 y Fk( )1333 1138 y Ft(1)g Fp(\000)1564 1076 y Fs(\026a)1658 1009 y Fk(\000)1696 1076 y Fp(h)p Fs(W)1818 1046 y Fr(2)1806 1099 y Fq(R)1861 1076 y Fp(i)1893 1088 y Fr(0)1949 1076 y Fp(\000)g(h)p Fs(W)2142 1088 y Fq(R)2197 1076 y Fp(i)2229 1046 y Fr(2)2229 1097 y(0)2266 1009 y Fk(\001)p 1486 1119 897 4 v 1486 1215 a Fp(h)p Fs(W)1596 1227 y Fq(R)1651 1215 y Fp(i)1683 1227 y Fr(0)1721 1148 y Fk(\000)1759 1215 y Fs(E)1825 1172 y Fr(\(0\))1820 1238 y(1)1932 1215 y Fp(\000)g Fs(\026a)p Fp(h)p Fs(W)2219 1227 y Fq(R)2274 1215 y Fp(i)2306 1227 y Fr(0)2344 1148 y Fk(\001)2392 996 y(!)2472 1138 y Fs(:)-2039 b Ft(\(2.38\))456 1350 y(T)-7 b(o)27 b(ev)-5 b(aluate)27 b(this)h(further)f(one)h(ma)n(y)e (use)i(the)g(estimates)f(\(2.34\))g(and)g(the)h(b)r(ound)1478 1528 y Fp(h)p Fs(W)1600 1493 y Fr(2)1588 1548 y Fq(R)1643 1528 y Fp(i)1675 1540 y Fr(0)1735 1528 y Fp(\024)23 b Ft(3)1988 1472 y Fs(N)p 1875 1509 304 4 v 1875 1585 a(R)1939 1561 y Fr(3)1994 1585 y Fp(\000)18 b Fs(R)2141 1556 y Fr(3)2140 1607 y(0)2188 1528 y Fp(h)p Fs(W)2298 1540 y Fq(R)2353 1528 y Fp(i)2385 1540 y Fr(0)456 1528 y Ft(\(2.39\))456 1718 y(whic)n(h)31 b(follo)n(ws)g(from)h Fs(U)1241 1688 y Fr(2)1232 1741 y Fq(R)1316 1718 y Ft(=)e(3\()p Fs(R)1549 1688 y Fr(3)1604 1718 y Fp(\000)18 b Fs(R)1751 1688 y Fr(3)1750 1739 y(0)1788 1718 y Ft(\))1820 1688 y Fj(\000)p Fr(1)1909 1718 y Fs(U)1966 1730 y Fq(R)2052 1718 y Ft(together)31 b(with)i(the)f(Cauc)n(h)n(y-Sc)n(h)n(w)n(arz)d(in-)456 1818 y(equalit)n(y)-7 b(.)57 b(A)35 b(glance)f(at)g(the)h(form)f(of)h (the)g(error)e(term)h(rev)n(eals,)h(ho)n(w)n(ev)n(er,)f(that)h(it)g(is) g Fo(not)456 1917 y Ft(p)r(ossible)j(here)g(to)h(tak)n(e)f(the)h (thermo)r(dynamic)g(limit)g Fs(L)i Fp(!)h(1)d Ft(with)g Fs(\032)g Ft(\014xed:)59 b(W)-7 b(e)40 b(ha)n(v)n(e)456 2028 y Fs(E)522 1985 y Fr(\(0\))517 2051 y(1)644 2028 y Ft(=)33 b Fs("\031)s(\026=L)980 1998 y Fr(2)1051 2028 y Ft(\(this)h(is)g(the)g(kinetic)g(energy)f(of)h(a)f Fo(single)i Ft(particle)e(in)i(the)f(\014rst)f(excited)456 2146 y(state)22 b(in)h(the)h(b)r(o)n(x\),)f(and)g(the)g(factor)f Fs(E)1689 2103 y Fr(\(0\))1684 2168 y(1)1787 2146 y Fp(\000)9 b Fs(\026a)p Fp(h)p Fs(W)2065 2158 y Fq(R)2120 2146 y Fp(i)2152 2158 y Fr(0)2212 2146 y Ft(in)23 b(the)g(denominator)f(in)h (\(2.38\))f(is,)i(up)456 2245 y(to)h(unimp)r(ortan)n(t)g(constan)n(ts)f (and)h(lo)n(w)n(er)f(order)g(terms,)h Fp(\030)e Ft(\()p Fs("L)2459 2215 y Fj(\000)p Fr(2)2561 2245 y Fp(\000)14 b Fs(a\032)2727 2215 y Fr(2)2763 2245 y Fs(L)2820 2215 y Fr(3)2857 2245 y Ft(\).)37 b(Hence)25 b(the)h(de-)456 2345 y(nominator)i(ev)n(en)n(tually)g(b)r(ecomes)h(negativ)n(e)f(and)g (T)-7 b(emple's)30 b(inequalit)n(y)e(lo)r(oses)g(its)i(v)-5 b(alidit)n(y)456 2445 y(if)28 b Fs(L)f Ft(is)g(large)g(enough.)605 2544 y(As)j(a)g(w)n(a)n(y)f(out)h(of)h(this)f(dilemma)g(w)n(e)g(divide) h(the)f(big)g(b)r(o)n(x)g(\003)g(in)n(to)g(cubic)g Fo(c)l(el)t(ls)h Ft(of)g(side)456 2644 y(length)i Fs(`)h Ft(that)g(is)g(k)n(ept)f Fo(\014xe)l(d)h Ft(as)f Fs(L)g Fp(!)g(1)p Ft(.)56 b(The)33 b(n)n(um)n(b)r(er)h(of)g(cells,)h Fs(L)2776 2614 y Fr(3)2813 2644 y Fs(=`)2890 2614 y Fr(3)2926 2644 y Ft(,)g(on)f(the)g(other)456 2743 y(hand,)j(increases)e(with)h Fs(L)p Ft(.)61 b(The)35 b Fs(N)45 b Ft(particles)35 b(are)g(distributed)h(among)e(these)i (cells,)i(and)456 2843 y(w)n(e)c(use)g(\(2.38\),)h(with)g Fs(L)f Ft(replaced)g(b)n(y)g Fs(`)p Ft(,)i Fs(N)43 b Ft(b)n(y)35 b(the)f(particle)g(n)n(um)n(b)r(er,)i Fs(n)p Ft(,)g(in)f(a)f(cell)h(and)456 2943 y Fs(\032)h Ft(b)n(y)g Fs(n=`)786 2913 y Fr(3)822 2943 y Ft(,)i(to)e(estimate)h(the)f(energy)f (in)i(eac)n(h)e(cell)h(with)h Fo(Neumann)e Ft(conditions)h(on)g(the)456 3042 y(b)r(oundary)-7 b(.)34 b(This)24 b(b)r(oundary)e(condition)h (leads)g(to)g(lo)n(w)n(er)f(energy)g(than)i(an)n(y)e(other)h(b)r (oundary)456 3142 y(condition.)36 b(F)-7 b(or)26 b(eac)n(h)g (distribution)h(of)f(the)h(particles)f(w)n(e)g(add)h(the)g(con)n (tributions)f(from)g(the)456 3242 y(cells,)35 b(neglecting)e(in)n (teractions)f(across)g(b)r(oundaries.)54 b(Since)34 b Fs(v)j Fp(\025)c Ft(0)g(b)n(y)g(assumption,)i(this)456 3341 y(can)h(only)f(lo)n(w)n(er)g(the)i(energy)-7 b(.)62 b(Finally)-7 b(,)38 b(w)n(e)e(minimize)h(o)n(v)n(er)e(all)h(p)r (ossible)g(c)n(hoices)f(of)h(the)456 3441 y(particle)25 b(n)n(um)n(b)r(ers)h(for)g(the)g(v)-5 b(arious)25 b(cells)h(adding)g (up)g(to)g Fs(N)9 b Ft(.)37 b(The)26 b(energy)f(obtained)h(in)g(this) 456 3540 y(w)n(a)n(y)21 b(is)i(a)g(lo)n(w)n(er)e(b)r(ound)i(to)g Fs(E)1384 3552 y Fr(0)1421 3540 y Ft(\()p Fs(N)t(;)14 b(L)p Ft(\),)24 b(b)r(ecause)f(w)n(e)f(are)g(e\013ectiv)n(ely)h(allo)n (wing)f(discon)n(tin)n(uous)456 3640 y(test)27 b(functions)h(for)f(the) h(quadratic)f(form)g(giv)n(en)g(b)n(y)g Fs(H)2214 3652 y Fq(N)2277 3640 y Ft(.)605 3740 y(In)h(mathematical)f(terms,)g(the)h (cell)g(metho)r(d)g(leads)f(to)1223 3887 y Fs(E)1284 3899 y Fr(0)1322 3887 y Ft(\()p Fs(N)t(;)14 b(L)p Ft(\))p Fs(=)-5 b(N)32 b Fp(\025)22 b Ft(\()p Fs(\032`)1884 3852 y Fr(3)1921 3887 y Ft(\))1953 3852 y Fj(\000)p Fr(1)2057 3887 y Ft(inf)2174 3808 y Fk(X)2171 3986 y Fq(n)p Fj(\025)p Fr(0)2311 3887 y Fs(c)2347 3899 y Fq(n)2392 3887 y Fs(E)2453 3899 y Fr(0)2491 3887 y Ft(\()p Fs(n;)14 b(`)p Ft(\))-2221 b(\(2.40\))456 4111 y(where)24 b(the)i(in\014m)n(um)g(is)g(o)n(v)n(er)d (all)i(c)n(hoices)g(of)g(co)r(e\016cien)n(ts)g Fs(c)2351 4123 y Fq(n)2419 4111 y Fp(\025)e Ft(0)i(\(relativ)n(e)f(n)n(um)n(b)r (er)h(of)h(cells)456 4211 y(con)n(taining)g(exactly)h Fs(n)h Ft(particles\),)f(satisfying)g(the)h(constrain)n(ts)1405 4283 y Fk(X)1402 4461 y Fq(n)p Fj(\025)p Fr(0)1542 4362 y Fs(c)1578 4374 y Fq(n)1646 4362 y Ft(=)23 b(1)p Fs(;)1981 4283 y Fk(X)1978 4461 y Fq(n)p Fj(\025)p Fr(0)2118 4362 y Fs(c)2154 4374 y Fq(n)2199 4362 y Fs(n)g Ft(=)g Fs(\032`)2438 4328 y Fr(3)2475 4362 y Fs(:)-2042 b Ft(\(2.41\))605 4582 y(The)24 b(minimization)f(problem)g(for)g(the)h(distributions)f (of)g(the)h(particles)f(among)f(the)i(cells)456 4682 y(w)n(ould)31 b(b)r(e)g(easy)g(if)h(w)n(e)f(knew)g(that)h(the)g(ground) e(state)h(energy)g Fs(E)2583 4694 y Fr(0)2620 4682 y Ft(\()p Fs(n;)14 b(`)p Ft(\))31 b(\(or)g(a)g(go)r(o)r(d)g(lo)n(w)n(er) 456 4781 y(b)r(ound)26 b(to)f(it\))i(w)n(ere)e(con)n(v)n(ex)f(in)i Fs(n)p Ft(.)36 b(Then)26 b(w)n(e)g(could)g(immediately)f(conclude)h (that)g(it)g(is)g(b)r(est)456 4881 y(to)36 b(ha)n(v)n(e)g(the)h (particles)f(as)g(ev)n(enly)g(distributed)h(among)e(the)j(b)r(o)n(xes)d (as)h(p)r(ossible,)j(i.e.,)g Fs(c)3399 4893 y Fq(n)456 4980 y Ft(w)n(ould)27 b(b)r(e)g(zero)g(except)g(for)g(the)h Fs(n)f Ft(equal)g(to)g(the)h(in)n(teger)f(closest)f(to)h Fs(\032`)2773 4950 y Fr(3)2810 4980 y Ft(.)37 b(This)27 b(w)n(ould)g(giv)n(e)1296 5102 y Fs(E)1357 5114 y Fr(0)1394 5102 y Ft(\()p Fs(N)t(;)14 b(L)p Ft(\))p 1296 5140 328 4 v 1422 5216 a Fs(N)1656 5159 y Fp(\025)23 b Ft(4)p Fs(\031)s(\026a\032)14 b Ft(\(1)k Fp(\000)g(E)7 b Ft(\()p Fs(\032;)14 b(`;)g(R)q(;)g(")p Ft(\)\))-2145 b(\(2.42\))p eop %%Page: 11 11 11 10 bop 1089 226 a Fr(THE)29 b(BOSE)g(GAS:)g(A)g(SUBTLE)g(MANY-BOD)n (Y)h(PR)n(OBLEM)567 b(11)456 425 y Ft(i.e.,)29 b(replacemen)n(t)e(of)i Fs(L)f Ft(in)g(\(2.37\))g(b)n(y)g Fs(`)p Ft(,)h(whic)n(h)f(is)h(indep)r (enden)n(t)g(of)f Fs(L)p Ft(.)39 b(The)29 b(blo)n(w)f(up)h(of)f Fp(E)456 525 y Ft(for)f Fs(L)22 b Fp(!)h(1)28 b Ft(w)n(ould)f(th)n(us)h (b)r(e)g(a)n(v)n(oided.)605 624 y(Since)i(con)n(v)n(exit)n(y)e(of)h Fs(E)1352 636 y Fr(0)1390 624 y Ft(\()p Fs(n;)14 b(`)p Ft(\))29 b(is)h(not)g(kno)n(wn)e(\(except)i(in)g(the)g(thermo)r (dynamic)f(limit\))456 724 y(w)n(e)h(m)n(ust)h(resort)f(to)h(other)f (means)h(to)g(sho)n(w)f(that)h Fs(n)e Ft(=)f Fs(O)r Ft(\()p Fs(\032`)2457 694 y Fr(3)2495 724 y Ft(\))j(in)g(all)g(b)r(o)n(xes.)46 b(The)31 b(rescue)456 824 y(comes)c(from)g Fo(sup)l(er)l(additivity)i Ft(of)f Fs(E)1603 836 y Fr(0)1641 824 y Ft(\()p Fs(n;)14 b(`)p Ft(\),)27 b(i.e.,)h(the)g(prop)r(ert)n(y)1318 964 y Fs(E)1379 976 y Fr(0)1417 964 y Ft(\()p Fs(n)18 b Ft(+)g Fs(n)1650 930 y Fj(0)1674 964 y Fs(;)c(`)p Ft(\))22 b Fp(\025)h Fs(E)1949 976 y Fr(0)1987 964 y Ft(\()p Fs(n;)14 b(`)p Ft(\))k(+)g Fs(E)2335 976 y Fr(0)2372 964 y Ft(\()p Fs(n)2454 930 y Fj(0)2478 964 y Fs(;)c(`)p Ft(\))-2126 b(\(2.43\))456 1105 y(whic)n(h)28 b(follo)n(ws)g(immediately)g(from)h Fs(v)e Fp(\025)e Ft(0)j(b)n(y)g(dropping)g(the)h(in)n(teractions)e(b)r (et)n(w)n(een)i(the)g Fs(n)456 1205 y Ft(particles)f(and)h(the)g Fs(n)1147 1175 y Fj(0)1199 1205 y Ft(particles.)41 b(The)29 b(b)r(ound)g(\(2.43\))f(implies)i(in)f(particular)e(that)j(for)e(an)n (y)456 1305 y Fs(n;)14 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Ft(\))2811 916 y Fr(3)2847 940 y Ft(\))2890 766 y Fk(\023)456 883 y Ft(\(2.52\))456 1097 y(with)28 b Fs(Y)41 b Ft(=)23 b(4)p Fs(\031)s(\032a)1001 1067 y Fr(3)1038 1097 y Fs(=)p Ft(3)k(as)g(b)r(efore.)36 b(Also,)27 b(the)h(factor)1264 1189 y Fk(\022)1325 1306 y Ft(1)18 b Fp(\000)1514 1249 y Ft(1)p 1478 1286 115 4 v 1478 1363 a Fs(\032`)1556 1339 y Fr(3)1602 1189 y Fk(\023)1686 1306 y Ft(=)23 b(\(1)18 b Fp(\000)g Ft(\(const)p Fs(:)p Ft(\))p Fs(Y)2293 1271 y Fj(\000)p Fr(1)2382 1306 y Ft(\()p Fs(a=`)p Ft(\))2567 1271 y Fr(3)2604 1306 y Ft(\))-2180 b(\(2.53\))456 1519 y(in)31 b(\(2.50\))g(\(whic)n(h)h(is)g(the)f(ratio) g(b)r(et)n(w)n(een)h Fs(n)p Ft(\()p Fs(n)21 b Fp(\000)f Ft(1\))32 b(and)f Fs(n)2395 1489 y Fr(2)2432 1519 y Ft(\))h(m)n(ust)g (not)f(b)r(e)h(b)r(e)g(forgotten.)456 1619 y(W)-7 b(e)28 b(no)n(w)f(mak)n(e)f(the)i(ansatz)1177 1777 y Fs(")23 b Fp(\030)f Fs(Y)1393 1743 y Fq(\013)1440 1777 y Fs(;)97 b(a=`)22 b Fp(\030)h Fs(Y)1857 1743 y Fq(\014)1902 1777 y Fs(;)97 b Ft(\()p Fs(R)2118 1743 y Fr(3)2174 1777 y Fp(\000)18 b Fs(R)2321 1743 y Fr(3)2320 1798 y(0)2358 1777 y Ft(\))p Fs(=`)2467 1743 y Fr(3)2526 1777 y Fp(\030)23 b Fs(Y)2681 1743 y Fq(\015)456 1777 y Ft(\(2.54\))456 1936 y(with)i(exp)r(onen)n(ts)g Fs(\013)p Ft(,)h Fs(\014)j Ft(and)c Fs(\015)k Ft(that)c(w)n(e)g(c)n(ho)r(ose)f(in)h(an)f(optimal)h (w)n(a)n(y)-7 b(.)35 b(The)25 b(conditions)f(to)h(b)r(e)456 2036 y(met)j(are)e(as)h(follo)n(ws:)628 2165 y Fp(\017)41 b Fs(")p Ft(\()p Fs(a=`)p Ft(\))935 2135 y Fr(2)985 2165 y Fp(\000)14 b Ft(\(const)p Fs(:)o Ft(\))p Fs(Y)1407 2135 y Fr(2)1444 2165 y Ft(\()p Fs(`=a)p Ft(\))1629 2135 y Fr(3)1689 2165 y Fs(>)23 b Ft(0.)36 b(This)25 b(holds)f(for)h(all)g (small)g(enough)f Fs(Y)19 b Ft(,)26 b(pro)n(vided)711 2264 y Fs(\013)19 b Ft(+)f(5)p Fs(\014)27 b(<)c Ft(2)k(whic)n(h)g (follo)n(ws)g(from)g(the)h(conditions)f(b)r(elo)n(w.)628 2364 y Fp(\017)41 b Fs(\013)24 b(>)e Ft(0)27 b(in)h(order)f(that)g Fs(")c Fp(!)g Ft(0)k(for)h Fs(Y)41 b Fp(!)23 b Ft(0.)628 2464 y Fp(\017)41 b Ft(3)p Fs(\014)22 b Fp(\000)c Ft(1)23 b Fs(>)g Ft(0)k(in)h(order)e(that)i Fs(Y)1687 2434 y Fj(\000)p Fr(1)1777 2464 y Ft(\()p Fs(a=`)p Ft(\))1962 2434 y Fr(3)2021 2464 y Fp(!)c Ft(0)j(for)g(for)g Fs(Y)42 b Fp(!)23 b Ft(0.)628 2563 y Fp(\017)41 b Ft(1)18 b Fp(\000)g Ft(3)p Fs(\014)k Ft(+)c Fs(\015)28 b(>)23 b Ft(0)k(in)h(order)e(that)i Fs(Y)19 b Ft(\()p Fs(`=a)p Ft(\))2022 2533 y Fr(3)2058 2563 y Ft(\()p Fs(R)2154 2533 y Fr(3)2210 2563 y Fp(\000)f Fs(R)2357 2533 y Fr(3)2356 2584 y(0)2394 2563 y Ft(\))p Fs(=`)2503 2533 y Fr(3)2563 2563 y Fp(!)23 b Ft(0)k(for)g(for)g Fs(Y)42 b Fp(!)23 b Ft(0.)628 2663 y Fp(\017)41 b Ft(1)18 b Fp(\000)g Fs(\013)h Fp(\000)f Ft(2)p Fs(\014)k Fp(\000)c Fs(\015)28 b(>)23 b Ft(0)k(to)g(con)n(trol)g(the)h(last)f(factor)g(in)g (\(2.52\).)456 2792 y(T)-7 b(aking)1339 2950 y Fs(\013)23 b Ft(=)g(1)p Fs(=)p Ft(17)p Fs(;)95 b(\014)27 b Ft(=)c(6)p Fs(=)p Ft(17)p Fs(;)95 b(\015)28 b Ft(=)22 b(3)p Fs(=)p Ft(17)-2107 b(\(2.55\))456 3109 y(all)27 b(these)h(conditions)f(are)f (satis\014ed,)i(and)1027 3268 y Fs(\013)c Ft(=)e(3)p Fs(\014)h Fp(\000)18 b Ft(1)k(=)h(1)18 b Fp(\000)g Ft(3)p Fs(\014)k Ft(+)c Fs(\015)28 b Ft(=)22 b(1)c Fp(\000)g Fs(\013)h Fp(\000)f Ft(2)p Fs(\014)k Fp(\000)d Fs(\015)27 b Ft(=)c(1)p Fs(=)p Ft(17)p Fs(:)-2419 b Ft(\(2.56\))456 3443 y(It)22 b(is)f(also)g(clear)g(that)h(2)p Fs(R)q(=`)f Fp(\030)i Fs(Y)1504 3413 y Fq(\015)t(=)p Fr(3)1636 3443 y Ft(=)g Fs(Y)1791 3413 y Fr(1)p Fq(=)p Fr(17)1928 3443 y Ft(,)g(up)f(to)g(higher)f(order)f(terms.)35 b(This)22 b(completes)456 3542 y(the)28 b(pro)r(of)g(of)g(Theorems)g(3.1)f(and)h (3.2,)g(for)g(the)h(case)e(of)h(p)r(oten)n(tials)g(with)h(\014nite)g (range.)38 b(By)456 3642 y(optimizing)21 b(the)g(prop)r(ortionalit)n(y) e(constan)n(ts)i(in)g(\(2.54\))f(one)h(can)g(sho)n(w)f(that)i Fs(C)29 b Ft(=)23 b(8)p Fs(:)p Ft(9)d(is)h(p)r(os-)456 3742 y(sible)28 b(in)i(Theorem)e(1.1)f([)p Fu(10)p Ft(].)41 b(The)29 b(extension)f(to)h(p)r(oten)n(tials)f(of)h(in\014nite)h(range) d(decreasing)456 3841 y(faster)d(than)g(1)p Fs(=r)995 3811 y Fr(3)1057 3841 y Ft(at)h(in\014nit)n(y)g(is)f(obtained)h(b)n(y)f (appro)n(ximation)f(b)n(y)h(\014nite)h(range)f(p)r(oten)n(tials,)456 3941 y(con)n(trolling)29 b(the)i(c)n(hange)f(of)h(the)g(scattering)f (length)g(as)h(the)g(cut-o\013)g(is)f(remo)n(v)n(ed.)45 b(See)31 b(Ap-)456 4040 y(p)r(endix)e(B)g(in)g([)p Fu(3)p Ft(])g(for)g(details.)40 b(A)30 b(slo)n(w)n(er)d(decrease)g(than)j(1)p Fs(=r)2457 4010 y Fr(3)2523 4040 y Ft(implies)f(in\014nite)h (scattering)456 4140 y(length.)2677 b Ff(2)605 4240 y Ft(The)28 b(exp)r(onen)n(ts)f(\(2.55\))g(mean)g(in)h(particular)e(that) 1349 4406 y Fs(a)d Fp(\034)g Fs(R)h Fp(\034)f Fs(\032)1758 4371 y Fj(\000)p Fr(1)p Fq(=)p Fr(3)1937 4406 y Fp(\034)g Fs(`)g Fp(\034)g Ft(\()p Fs(\032a)p Ft(\))2358 4371 y Fj(\000)p Fr(1)p Fq(=)p Fr(2)2514 4406 y Fs(;)-2081 b Ft(\(2.57\))456 4580 y(whereas)28 b(Dyson's)h(metho)r(d)h(required)f Fs(R)e Fp(\030)f Fs(\032)1941 4550 y Fj(\000)p Fr(1)p Fq(=)p Fr(3)2127 4580 y Ft(as)i(already)h(explained.)42 b(The)30 b(condition)456 4680 y Fs(\032)499 4650 y Fj(\000)p Fr(1)p Fq(=)p Fr(3)678 4680 y Fp(\034)23 b Fs(`)18 b Ft(is)h(required)f(in)h(order)e(to)i(ha)n(v)n(e)f(man)n(y)g(particles)g (in)h(eac)n(h)f(b)r(o)n(x)g(and)h(th)n(us)g Fs(n)p Ft(\()p Fs(n)q Fp(\000)q Ft(1\))j Fp(\031)456 4783 y Fs(n)506 4753 y Fr(2)543 4783 y Ft(.)60 b(The)35 b(condition)h Fs(`)f Fp(\034)h Ft(\()p Fs(\032a)p Ft(\))1517 4753 y Fj(\000)p Fr(1)p Fq(=)p Fr(2)1709 4783 y Ft(is)g(necessary)d(for)i(a)g (sp)r(ectral)g(gap)g(gap)f Fp(\035)i Fs(e)3195 4795 y Fr(0)3232 4783 y Ft(\()p Fs(\032)p Ft(\))g(in)456 4882 y(T)-7 b(emple's)34 b(inequalit)n(y)-7 b(.)55 b(It)34 b(is)g(also)f(clear)g(that)h(this)g(c)n(hoice)f(of)h Fs(`)f Ft(w)n(ould)h(lead)f(to)h(a)f(far)h(to)r(o)456 4982 y(big)28 b(energy)g(and)g(no)h(b)r(ound)g(for)f Fs(e)1565 4994 y Fr(0)1602 4982 y Ft(\()p Fs(\032)p Ft(\))i(if)f(w)n(e) f(had)h(c)n(hosen)f(Diric)n(hlet)h(instead)f(of)h(Neumann)456 5082 y(b)r(oundary)d(conditions)i(for)f(the)h(cells.)36 b(But)28 b(with)g(the)g(latter)f(the)h(metho)r(d)h(w)n(orks!)p eop %%Page: 13 13 13 12 bop 1089 226 a Fr(THE)29 b(BOSE)g(GAS:)g(A)g(SUBTLE)g(MANY-BOD)n (Y)h(PR)n(OBLEM)567 b(13)1179 425 y Fu(3.)47 b(THE)32 b(DILUTE)h(BOSE)e(GAS)h(IN)g(2D)605 575 y Ft(The)e(t)n(w)n (o-dimensional)d(theory)-7 b(,)30 b(in)f(con)n(trast,)g(b)r(egan)g(to)g (receiv)n(e)f(atten)n(tion)i(only)f(rela-)456 674 y(tiv)n(ely)h(late.) 44 b(The)30 b(\014rst)g(deriv)-5 b(ation)30 b(of)g(the)g(asymptotic)g (form)n(ula)f(w)n(as,)h(to)g(our)g(kno)n(wledge,)456 774 y(done)d(b)n(y)g(Sc)n(hic)n(k)g([)p Fu(19)p Ft(],)h(as)e(late)i(as) f(1971!)35 b(He)28 b(found)1623 1029 y Fs(e)1662 1041 y Fr(0)1698 1029 y Ft(\()p Fs(\032)p Ft(\))c Fp(')1993 972 y Ft(4)p Fs(\031)s(\026\032)p 1927 1010 318 4 v 1927 1086 a Fp(j)14 b Ft(ln\()p Fs(\032a)2152 1062 y Fr(2)2189 1086 y Ft(\))p Fp(j)2254 1029 y Fs(:)-1821 b Ft(\(3.1\))605 1214 y(The)39 b(scattering)f(length)h Fs(a)h Ft(in)f(\(3.1\))g(is)g (de\014ned)g(using)g(the)h(zero)e(energy)g(scattering)456 1314 y(equation)18 b(\(2.3\))g(but)i(instead)f(of)f Fs( )s Ft(\()p Fs(r)r Ft(\))25 b Fp(\031)e Ft(1)q Fp(\000)q Fs(a=r)d Ft(w)n(e)e(no)n(w)g(imp)r(ose)h(the)g(asymptotic)g(condition) 456 1413 y Fs( )s Ft(\()p Fs(r)r Ft(\))24 b Fp(\031)f Ft(ln\()p Fs(r)r(=a)p Ft(\).)38 b(This)27 b(is)h(explained)f(in)h(the)g (app)r(endix)g(to)f([)p Fu(2)p Ft(].)605 1513 y(Note)g(that)f(the)h (answ)n(er)e(could)h(not)h(p)r(ossibly)f(b)r(e)h Fs(e)2235 1525 y Fr(0)2272 1513 y Ft(\()p Fs(\032)p Ft(\))c Fp(')g Ft(4)p Fs(\031)s(\026\032a)j Ft(b)r(ecause)g(that)h(w)n(ould)456 1613 y(b)r(e)35 b(dimensionally)f(wrong.)57 b(But)35 b Fs(e)1621 1625 y Fr(0)1658 1613 y Ft(\()p Fs(\032)p Ft(\))g(m)n(ust)g(essen)n(tially)f(b)r(e)h(prop)r(ortional)e(to)i Fs(\032)p Ft(,)i(whic)n(h)456 1712 y(lea)n(v)n(es)29 b(no)h(ro)r(om)f(for)h(an)h Fs(a)f Ft(dep)r(endence)h(|)g(whic)n(h)f (is)h(ridiculous!)45 b(It)30 b(turns)h(out)f(that)h(this)456 1812 y(dep)r(endence)d(comes)f(ab)r(out)g(in)h(the)g(ln\()p Fs(\032a)1811 1782 y Fr(2)1849 1812 y Ft(\))f(factor.)605 1911 y(One)f(of)g(the)g(in)n(triguing)g(facts)g(ab)r(out)g(\(3.1\))g (is)g(that)g(the)h(energy)e(for)g Fs(N)35 b Ft(particles)26 b(is)g Fo(not)456 2011 y(e)l(qual)i Ft(to)g Fs(N)9 b Ft(\()p Fs(N)27 b Fp(\000)18 b Ft(1\))p Fs(=)p Ft(2)27 b(times)h(the)h(energy)e(for)g(t)n(w)n(o)g(particles)g(in)h(the)h(lo)n (w)e(densit)n(y)h(limit)g(|)456 2111 y(as)c(is)h(the)g(case)f(in)h (three)f(dimensions.)36 b(The)25 b(latter)f(quan)n(tit)n(y)-7 b(,)25 b Fs(E)2507 2123 y Fr(0)2545 2111 y Ft(\(2)p Fs(;)14 b(L)p Ft(\),)25 b(is,)g(asymptotically)-7 b(,)456 2227 y(for)34 b(large)f Fs(L)p Ft(,)k(equal)d(to)h(8)p Fs(\031)s(\026L)1451 2197 y Fj(\000)p Fr(2)1553 2160 y Fk(\002)1588 2227 y Ft(ln\()p Fs(L)1746 2197 y Fr(2)1783 2227 y Fs(=a)1869 2197 y Fr(2)1906 2227 y Ft(\))1938 2160 y Fk(\003)1973 2177 y Fj(\000)p Fr(1)2062 2227 y Ft(.)59 b(Th)n(us,)36 b(if)g(the)f Fs(N)9 b Ft(\()p Fs(N)32 b Fp(\000)23 b Ft(1\))p Fs(=)p Ft(2)34 b(rule)g(w)n(ere)456 2327 y(to)39 b(apply)f(in)i(2D,)f(\(3.1\))g(w)n(ould)g(ha)n(v)n(e)f(to)h(b)r(e)g (replaced)f(b)n(y)h(the)h(m)n(uc)n(h)f(smaller)f(quan)n(tit)n(y)456 2443 y(4)p Fs(\031)s(\026\032)655 2376 y Fk(\002)689 2443 y Ft(ln\()p Fs(L)847 2413 y Fr(2)884 2443 y Fs(=a)970 2413 y Fr(2)1007 2443 y Ft(\))1039 2376 y Fk(\003)1074 2393 y Fj(\000)p Fr(1)1163 2443 y Ft(.)49 b(In)31 b(other)g(w)n(ords,)g Fs(L)p Ft(,)i(whic)n(h)e(tends)h(to)f Fp(1)h Ft(in)g(the)g(thermo)r (dynamic)456 2554 y(limit,)c(has)e(to)h(b)r(e)h(replaced)e(b)n(y)h(the) h(mean)f(particle)f(separation,)g Fs(\032)2611 2524 y Fj(\000)p Fr(1)p Fq(=)p Fr(2)2795 2554 y Ft(in)h(the)h(logarithmic)456 2654 y(factor.)67 b(V)-7 b(arious)36 b(p)r(o)r(etic)j(form)n(ulations)e (of)g(this)i(curious)e(fact)h(ha)n(v)n(e)e(b)r(een)j(giv)n(en,)h(but)e (it)456 2754 y(remains)26 b(true)i(that)g(the)g(non-linearit)n(y)e(is)i (something)f(that)h(do)r(es)g(not)f(o)r(ccur)g(in)h(more)f(than)456 2853 y(t)n(w)n(o-dimensions)g(and)i(its)g(precise)g(nature)f(is)h (hardly)g(ob)n(vious,)f(ph)n(ysically)-7 b(.)41 b(This)29 b(anomaly)456 2953 y(is)e(the)h(main)g(reason)e(that)h(the)h(2D)g (result)f(is)h(not)f(a)h(trivial)f(extension)g(of)g([)p Fu(1)p Ft(].)605 3053 y(The)g(pro)r(of)f(of)h(\(3.1\))g(is)g(in)g([)p Fu(2)p Ft(].)37 b(The)27 b(\(relativ)n(e\))f(error)f(terms)i(to)f (\(3.1\))h(giv)n(ein)f(in)i([)p Fu(2)o Ft(])f(are)456 3155 y Fp(j)14 b Ft(ln\()p Fs(\032a)681 3125 y Fr(2)718 3155 y Ft(\))p Fp(j)773 3125 y Fj(\000)p Fr(1)890 3155 y Ft(for)27 b(the)h(upp)r(er)g(b)r(ound)g(and)f Fp(j)14 b Ft(ln)q(\()p Fs(\032a)2041 3125 y Fr(2)2078 3155 y Ft(\))p Fp(j)2133 3125 y Fj(\000)p Fr(1)p Fq(=)p Fr(5)605 3255 y Ft(T)-7 b(o)28 b(pro)n(v)n(e)e(\(3.1\))i(the)g(essen)n(tial)g (new)g(step)g(is)g(to)g(mo)r(dify)g(Dyson's)g(lemma)g(for)f(2D.)h(The) 456 3355 y(rest)f(of)g(the)h(pro)r(of)f(parallels)f(that)i(for)f(3D.)h (The)g(2D)f(v)n(ersion)f(of)i(Lemma)f(2.5)g(is)h([)p Fu(2)o Ft(]:)605 3505 y Fh(Lemma)j(3.1.)41 b Fo(L)l(et)c Fs(v)s Ft(\()p Fs(r)r Ft(\))h Fp(\025)e Ft(0)h Fo(with)h(\014nite)f(r)l (ange)g Fs(R)2322 3517 y Fr(0)2360 3505 y Fo(.)61 b(L)l(et)37 b Fs(U)9 b Ft(\()p Fs(r)r Ft(\))38 b Fp(\025)e Ft(0)h Fo(b)l(e)g(any)h(func-)456 3605 y(tion)d(satisfying)1008 3538 y Fk(R)1078 3605 y Fs(U)9 b Ft(\()p Fs(r)r Ft(\))14 b(ln)q(\()p Fs(r)r(=a)p Ft(\))p Fs(r)r(dr)38 b Fp(\024)33 b Ft(1)i Fo(and)h Fs(U)9 b Ft(\()p Fs(r)r Ft(\))35 b(=)e(0)i Fo(for)i Fs(r)f(<)d(R)2771 3617 y Fr(0)2809 3605 y Fo(.)56 b(L)l(et)35 b Fp(B)h(\032)d Fu(R)3300 3575 y Fr(3)3372 3605 y Fo(b)l(e)456 3705 y(star)g(shap)l(e)l(d)h(with)g(r)l(esp)l(e)l (ct)f(to)g Ft(0)g Fo(\(e.g.)i(c)l(onvex)e(with)h Ft(0)28 b Fp(2)i(B)s Fo(\).)48 b(Then)34 b(for)g(al)t(l)h(di\013er)l(entiable) 456 3804 y(functions)29 b Fs( )1305 3866 y Fk(Z)1351 4054 y Fj(B)1415 3912 y Fk(\002)1449 3979 y Fs(\026)p Fp(jr)p Fs( )s Fp(j)1671 3945 y Fr(2)1727 3979 y Ft(+)1820 3946 y Fr(1)p 1820 3960 34 4 v 1820 4007 a(2)1863 3979 y Fs(v)s Fp(j)p Fs( )s Fp(j)2009 3945 y Fr(2)2047 3912 y Fk(\003)2105 3979 y Fp(\025)22 b Fs(\026)2256 3866 y Fk(Z)2302 4054 y Fj(B)2365 3979 y Fs(U)9 b Fp(j)p Fs( )s Fp(j)2534 3945 y Fr(2)2572 3979 y Fs(:)-2139 b Ft(\(3.2\))1100 4321 y Fu(4.)47 b(BOSE-EINSTEIN)31 b(CONDENSA)-8 b(TION)605 4471 y Ft(Let)19 b(us)f(commen)n(t)g(v)n(ery)f(brie\015y)h(on)g(the)h (notion)f(of)h(Bose-Einstein)d(condensation)i(\(BEC\).)456 4570 y(Giv)n(en)28 b(the)g(normalized)g(ground)f(state)h(w)n(a)n(v)n(e) f(function)i(\011)2347 4582 y Fr(0)2384 4570 y Ft(\()o Fs(~)-41 b(x)2463 4582 y Fr(1)2501 4570 y Fs(;)14 b(:)g(:)g(:)f(;)g(~) -41 b(x)2732 4582 y Fq(N)2796 4570 y Ft(\))28 b(w)n(e)g(can)g(form)g (the)456 4670 y(one-b)r(o)r(dy)c(densit)n(y)i(matrix)e(whic)n(h)i(is)f (an)g(op)r(erator)f(on)h Fs(L)2300 4640 y Fr(2)2336 4670 y Ft(\()p Fu(R)2440 4640 y Fq(n)2485 4670 y Ft(\))h(\()p Fs(n)d Ft(=)g(2)i(or)f(3\))i(giv)n(en)e(b)n(y)h(the)456 4770 y(k)n(ernel)698 4928 y Fs(\015)5 b Ft(\()o Fs(~)-41 b(x)q(;)27 b(~)-42 b(y)s Ft(\))23 b(=)g Fs(N)1153 4815 y Fk(Z)1199 5004 y Fr(BO)n(X)1343 4987 y Fe(N)5 b Fd(\000)p Fg(1)1488 4928 y Ft(\011)1553 4940 y Fr(0)1590 4928 y Ft(\()o Fs(~)-41 b(x;)27 b(~)-41 b(x)1767 4940 y Fr(2)1805 4928 y Fs(;)14 b(:)g(:)g(:)g(;)f(~)-41 b(x)2037 4940 y Fq(N)2100 4928 y Ft(\)\011)2197 4940 y Fr(0)2234 4928 y Ft(\()o Fs(~)g(y)s(;)27 b(~)-41 b(x)2408 4940 y Fr(2)2445 4928 y Fs(;)14 b(:)g(:)g(:)g(;)f(~)-41 b(x)2677 4940 y Fq(N)2740 4928 y Ft(\))p Fs(d)o(~)g(x)2862 4940 y Fr(2)2914 4928 y Fp(\001)14 b(\001)g(\001)g Fs(d)o(~)-41 b(x)3115 4940 y Fq(N)3179 4928 y Fs(:)456 5116 y Ft(Then)679 5049 y Fk(R)748 5116 y Fs(\015)5 b Ft(\()o Fs(~)-41 b(x;)27 b(~)-41 b(x)p Ft(\))p Fs(d)o(~)g(x)35 b Ft(=)e(T)-7 b(race)o(\()p Fs(\015)5 b Ft(\))33 b(=)h Fs(N)9 b Ft(.)55 b(BEC)33 b(is)h(the)g(assertion)f(that)h(this)g(op)r(erator)e(has)456 5216 y(an)g(eigen)n(v)-5 b(alue)32 b(of)g(order)f Fs(N)9 b Ft(.)52 b(Since)33 b Fs(\015)k Ft(is)32 b(a)h(p)r(ositiv)n(e)f(k)n (ernel)f(and,)j(hop)r(efully)-7 b(,)35 b(translation)p eop %%Page: 14 14 14 13 bop 456 226 a Fr(14)1139 b(ELLIOTT)29 b(H.)g(LIEB)456 425 y Ft(in)n(v)-5 b(arian)n(t)33 b(in)i(the)f(thermo)r(dynamic)h (limit,)i(the)d(eigenfunction)h(b)r(elonging)f(to)g(the)h(largest)456 525 y(eigen)n(v)-5 b(alue)25 b(m)n(ust)h(b)r(e)g(the)g(constan)n(t)g (function)g(\(v)n(olume\))2287 495 y Fj(\000)p Fr(1)p Fq(=)p Fr(2)2443 525 y Ft(.)37 b(Therefore,)25 b(another)g(w)n(a)n(y)g (to)456 624 y(sa)n(y)h(that)i(BEC)f(exists)g(is)h(that)1468 697 y Fk(Z)51 b(Z)1662 810 y Fs(\015)5 b Ft(\()o Fs(~)-41 b(x;)27 b(~)-41 b(y)r Ft(\))p Fs(d)o(~)g(x)q(d)o(~)g(y)26 b Ft(=)d(O\()p Fs(N)9 b Ft(\))p Fs(:)605 996 y Ft(Unfortunately)-7 b(,)25 b(this)f(is)f(something)h(that)g(is)f(frequen)n(tly)h(in)n(v)n (ok)n(ed)e(but)i(nev)n(er)f(pro)n(v)n(ed)f(|)456 1095 y(except)30 b(for)f(one)h(sp)r(ecial)g(case:)41 b(hard)30 b(core)f(b)r(osons)g(on)h(a)g(lattice)g(at)g(half-\014lling)g(\(i.e.,)i Fs(N)k Ft(=)456 1195 y(half)27 b(the)h(n)n(um)n(b)r(er)g(of)f(lattice)h (sites\).)37 b(The)28 b(pro)r(of)f(is)g(in)h([)p Fu(20)o Ft(].)605 1295 y(The)c(problem)g(remains)g(op)r(en)g(after)g(ab)r(out)h (70)e(y)n(ears.)34 b(It)25 b(is)f(not)h(at)f(all)g(clear)f(that)i(BEC) 456 1394 y(is)31 b(essen)n(tial)g(for)g(sup)r(er\015uidit)n(y)-7 b(,)34 b(as)d(frequen)n(tly)g(claimed.)49 b(Our)31 b(construction)g(in) h(section)g(2)456 1494 y(sho)n(ws)21 b(that)h(BEC)g(exists)f(on)h(a)g (length)g(scale)g(of)g(order)f Fs(\032)2244 1464 y Fj(\000)p Fr(1)p Fq(=)p Fr(3)2400 1494 y Fs(Y)2467 1464 y Fj(\000)p Fr(1)p Fq(=)p Fr(17)2678 1494 y Ft(whic)n(h,)j(unfortunately)-7 b(,)456 1593 y(is)27 b(not)h(a)f(`thermo)r(dynamic')g(length)h(lik)n(e) f(v)n(olume)2061 1563 y Fr(1)p Fq(=)p Fr(3)2166 1593 y Ft(.)568 1801 y Fu(5.)47 b(GR)m(OSS-PIT)-8 b(AEVSKI)s(I)33 b(EQUA)-8 b(TION)32 b(F)m(OR)f(TRAPPED)i(BOSONS)605 1950 y Ft(In)c(the)g(recen)n(t)f(exp)r(erimen)n(ts)g(on)h(Bose)e (condensation,)h(the)h(particles)f(ha)n(v)n(e)g(to)g(b)r(e)h(con-)456 2050 y(\014ned)34 b(in)f(a)g(cold)h(`trap')f(instead)g(of)h(a)f(`b)r(o) n(x')g(and)h(w)n(e)f(are)f(certainly)h(not)h(at)f(the)h(`thermo-)456 2149 y(dynamic)e(limit'.)53 b(F)-7 b(or)32 b(a)g(`trap')g(w)n(e)h(add)f (a)g Fo(slow)t(ly)j Ft(v)-5 b(arying)31 b(con\014ning)h(p)r(oten)n (tial)h Fs(V)19 b Ft(,)34 b(with)456 2249 y Fs(V)18 b Ft(\()o Fs(~)-41 b(x)q Ft(\))24 b Fp(!)f(1)k Ft(as)g Fp(j)o Fs(~)-41 b(x)q Fp(j)23 b(!)g(1)28 b Ft(.)37 b(The)27 b(Hamiltonian)h(b)r(ecomes)1115 2478 y Fs(H)i Ft(=)1332 2374 y Fq(N)1302 2399 y Fk(X)1308 2576 y Fq(i)p Fr(=1)1435 2478 y Fp(\000)p Fs(\026)p Ft(\001)1619 2490 y Fq(i)1665 2478 y Ft(+)18 b Fs(V)h Ft(\()o Fs(~)-41 b(x)1894 2490 y Fq(i)1922 2478 y Ft(\))19 b(+)2147 2399 y Fk(X)2056 2577 y Fr(1)p Fj(\024)p Fq(i