Content-Type: multipart/mixed; boundary="-------------0512241756256" This is a multi-part message in MIME format. ---------------0512241756256 Content-Type: text/plain; name="05-438.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-438.keywords" wave guide, bound state, trapped mode, two-particle, enhanced binding ---------------0512241756256 Content-Type: application/postscript; name="L05b.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="L05b.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: L05b.dvi %%CreationDate: Fri Dec 23 10:33:45 2005 %%Pages: 8 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: c:\programme\miktex\miktex\bin\DVIPS.EXE L05b %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2005.12.23:1033 %%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 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! 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(op)s(erator)g(is)g(purely)h(con)m(tin)m(uous)h(and)386 1156 y(con)m(tains)g(ev)m(ery)h(real)e(n)m(um)m(b)s(er)h(ab)s(o)m(v)m (e)h(some)e(threshold,)i(whic)m(h)f(is)f(the)h(lo)m(w)m(est)386 1272 y(eigen)m(v)-5 b(alue)46 b(of)h(the)g(Laplace)f(op)s(erator)g(on)h (the)g(cross)h(section)f(of)g(\012.)86 b(It)47 b(is)386 1389 y(kno)m(wn)42 b(that)g(geometrical)c(p)s(erturbations)j(lik)m(e)g (b)s(ending)g(the)g(tub)s(e)h(or)f(lo)s(cal)386 1505 y(deformations)27 b(of)i(the)g(b)s(oundary)g(can)g(giv)m(e)g(rise)g(to) f(eigen)m(v)-5 b(alues)29 b(of)f Fo(\000)p Fr(\001)i(b)s(elo)m(w)386 1621 y(this)e(threshold.)43 b(In)29 b(analogy)e(to)i(the)g(case)h(of)e (the)h(Sc)m(hr\177)-49 b(odinger)29 b(op)s(erator)f(with)386 1737 y(a)e(w)m(eak)h(attractiv)m(e)f(p)s(oten)m(tial,)f(w)m(e)i(ask)g (the)f(follo)m(wing)e(question:)40 b(Do)s(es)26 b(a)g(w)m(a)m(v)m(e)386 1853 y(guide)35 b(exist)h(that)f(do)s(esn't)h(ha)m(v)m(e)h(a)f(b)s (ound)f(state)h(for)f(one)h(particle,)f(but)h(that)386 1970 y(do)s(es)d(ha)m(v)m(e)h(a)e(b)s(ound)h(state)g(for)f(a)g(system)i (of)e(t)m(w)m(o)h(in)m(teracting)f(particles?)504 2086 y(This)c(question)g(is)f(not)h(so)g(easy)h(to)e(answ)m(er)i(b)m(y)g(ph) m(ysical)f(in)m(tuition,)e(b)s(ecause)386 2202 y(the)44 b(existence)h(or)e(non-existence)i(of)e(geometrically)d(b)s(ound)k (states)g(for)f(one)386 2318 y(particle)35 b(do)s(esn't)j(dep)s(end)g (on)e(the)h(mass)g(of)f(the)i(particle)d(in)h(question.)56 b(This)386 2435 y(means)27 b(that)g(the)g(in)m(tuitiv)m(e)f(`double)h (mass)g(argumen)m(t')f(for)h(t)m(w)m(o)g(particles)g(in)f(an)386 2551 y(attractiv)m(e)39 b(p)s(oten)m(tial)e(do)s(esn't)i(apply)f(to)h (this)f(situation.)61 b(Despite)38 b(that,)j(w)m(e)386 2667 y(will)28 b(sho)m(w)k(in)e(the)h(follo)m(wing)c(t)m(w)m(o)32 b(sections)f(that)f(the)h(answ)m(er)h(to)e(the)h(question)386 2783 y(ab)s(o)m(v)m(e)i(is)f(`y)m(es'.)488 3063 y(2.)49 b Fq(Tw)n(o-p)-7 b(ar)g(ticle)38 b(bound)g(st)-7 b(a)g(tes)37 b(in)h(def)n(ormed)g(w)-9 b(a)g(ve)38 b(guides)504 3237 y Fr(W)-8 b(e)33 b(assume)g(our)g(w)m(a)m(v)m(e)h(guide)e(to)g(b)s(e)h (the)g(domain)e(\012)d Fo(\032)g Fm(R)2717 3201 y Fn(2)2795 3237 y Fr(giv)m(en)33 b(b)m(y)1362 3423 y(\012)28 b(=)f Fo(f)p Fr(\()p Fp(x;)17 b(y)t Fr(\))27 b(:)h Fo(j)p Fp(y)t Fo(j)e Fp(<)h(f)11 b Fr(\()p Fp(x)p Fr(\))p Fo(g)386 3609 y Fr(where)1200 3850 y Fp(f)g Fr(\()p Fp(x)p Fr(\))28 b(=)1521 3646 y Fj(8)1521 3735 y(<)1521 3915 y(:)1661 3693 y Fn(1)p 1661 3709 36 4 v 1661 3767 a(2)1888 3732 y Fr(for)k Fo(j)p Fp(x)p Fo(j)c Fp(>)f(L=)p Fr(2)p Fp(;)1651 3849 y(\017=)p Fr(2)100 b(for)32 b Fp(x)c Fr(=)g Fo(\006)17 b Fp(L=)p Fr(2)p Fp(;)1651 3965 y(h=)p Fr(2)83 b(for)32 b Fo(j)p Fp(x)p Fo(j)c Fp(<)f(L=)p Fr(2)386 4131 y(with)36 b Fp(L)d(>)h Fr(0)p Fp(;)17 b(h)33 b(>)g Fr(1)j(and)g(0)d Fp(<)g(\017)h(<)f Fr(1.)54 b(W)-8 b(e)36 b(imp)s(ose)f(Diric)m(hlet)f (conditions)h(on)386 4247 y Fp(@)5 b Fr(\012,)37 b(whic)m(h)f(includes) g(the)g(`barriers')f(at)g Fp(x)e Fr(=)g Fo(\006)p Fp(L=)p Fr(2.)52 b(Our)35 b(geometry)h(can)f(b)s(e)386 4363 y(in)m(terpreted)40 b(as)g(a)f(ca)m(vit)m(y)h(of)f(length)g Fp(L)h Fr(and)f(width)h Fp(h)f Fr(coupled)h(w)m(eakly)g(\(if)e Fp(\017)386 4480 y Fr(is)h(small\))e(to)i(t)m(w)m(o)h(semi-in\014nite)d(straigh)m(t)i(w) m(a)m(v)m(e)i(guides.)64 b(W)-8 b(e)40 b(c)m(ho)s(ose)g(to)f(set)386 4596 y Fp(m)i Fr(=)639 4557 y Fn(1)p 639 4573 V 639 4630 a(2)684 4596 y Fr(,)i(suc)m(h)f(that)e(the)h(one-particle)e (Hamiltonian)e(is)j(simply)f Fp(H)2992 4611 y Fn(1)3072 4596 y Fr(=)i Fo(\000)p Fr(\001.)386 4712 y(Then)34 b(standard)f (argumen)m(ts)f(imply)f(that)1476 4898 y Fp(\033)1531 4913 y Fn(ess)1623 4898 y Fr(\()p Fp(H)1742 4913 y Fn(1)1781 4898 y Fr(\))d(=)f([)p Fp(\031)2036 4856 y Fn(2)2076 4898 y Fp(;)17 b Fo(1)p Fr(\))p Fp(:)386 5083 y Fr(Eigen)m(v)-5 b(alues)30 b(ma)m(y)g(o)s(ccur)g(dep)s(ending)g(on)g(the)h(c)m(hoice)f (of)g(the)g(parameters)g Fp(L;)17 b(h)386 5199 y Fr(and)33 b Fp(\017)p Fr(,)g(but)f(w)m(e)i(will)c(sho)m(w:)386 5390 y Fu(Lemma)43 b(2.1.)i Fi(If)39 b Fp(L)1181 5354 y Fh(\000)p Fn(2)1301 5390 y Fr(+)26 b Fp(h)1459 5354 y Fh(\000)p Fn(2)1590 5390 y Fp(>)36 b Fr(1)k Fi(then)f(for)h(smal)5 b(l)39 b(enough)g Fp(\017)h Fi(ther)-5 b(e)40 b(ar)-5 b(e)39 b(no)386 5506 y(eigenvalues)44 b(of)g Fp(H)1112 5521 y Fn(1)1196 5506 y Fi(b)-5 b(elow)44 b Fp(\031)1524 5470 y Fn(2)1564 5506 y Fi(,)j(i.e.,)g(in)d(this)h(c)-5 b(ase)44 b(the)h(wave)f(guide)h(has)f(no)386 5623 y(one-p)-5 b(article)34 b(b)-5 b(ound)34 b(states.)p eop %%Page: 3 3 3 2 bop 1006 259 a Fk(TW)n(O-P)-6 b(AR)g(TICLE)32 b(BINDING)g(IN)h(A)g (W)-9 b(A)g(VE)33 b(GUIDE)581 b(3)504 459 y Fr(On)45 b(the)h(other)f(hand,)k(w)m(e)d(consider)g(a)f(system)h(of)e(t)m(w)m(o) i(b)s(osons)g(of)f(mass)386 575 y Fp(m)c Fr(=)638 536 y Fn(1)p 638 552 36 4 v 638 609 a(2)684 575 y Fr(,)h(whic)m(h)f(in)m (teract)f(via)f(a)h(harmonic)f(p)s(oten)m(tial.)65 b(The)41 b(corresp)s(onding)386 691 y(Hamiltonian)29 b(is)676 924 y Fp(H)757 939 y Fn(2)824 924 y Fr(=)f Fo(\000)p Fp(@)1061 883 y Fn(2)1056 949 y Fl(x)1096 958 y Fg(1)1157 924 y Fo(\000)23 b Fp(@)1313 883 y Fn(2)1308 949 y Fl(y)1343 958 y Fg(1)1404 924 y Fo(\000)g Fp(@)1560 883 y Fn(2)1555 949 y Fl(x)1595 958 y Fg(2)1656 924 y Fo(\000)g Fp(@)1812 883 y Fn(2)1807 949 y Fl(y)1842 958 y Fg(2)1903 924 y Fr(+)f Fp(\013)q Fr(\()p Fp(x)2157 939 y Fn(1)2219 924 y Fo(\000)g Fp(x)2373 939 y Fn(2)2413 924 y Fr(\))2451 883 y Fn(2)2513 924 y Fr(+)g Fp(\013)q Fr(\()p Fp(y)2760 939 y Fn(1)2820 924 y Fo(\000)h Fp(y)2968 939 y Fn(2)3007 924 y Fr(\))3045 883 y Fn(2)386 1166 y Fr(in)28 b Fp(L)562 1130 y Fn(2)602 1166 y Fr(\(\012)15 b Fo(\002)g Fr(\012\),)31 b(where)f Fp(x)1316 1181 y Fl(i)1374 1166 y Fr(and)f Fp(y)1608 1181 y Fl(i)1665 1166 y Fr(are)g(the)g(particle)f(co)s (ordinates)g(and)h Fp(\013)g(>)e Fr(0)i(is)386 1282 y(the)34 b(in)m(teraction)e(strength.)47 b(Again)33 b(w)m(e)h(imp)s(ose)f(Diric) m(hlet)e(conditions)i(on)g(the)386 1398 y(b)s(oundary)g(of)f(\012.)386 1616 y Fu(Lemma)37 b(2.2.)386 1732 y Fi(a\))d(F)-7 b(or)34 b(any)h(choic)-5 b(e)34 b(of)h Fp(L)g Fi(,)g Fp(h)g Fi(and)f Fp(\017)h Fi(one)g(has)1294 1977 y Fp(\033)1349 1992 y Fn(ess)1441 1977 y Fr(\()p Fp(H)1560 1992 y Fn(2)1599 1977 y Fr(\))27 b Fo(\032)i Fr([)1797 1890 y Fo(p)p 1880 1890 112 4 v 87 x Fr(2)p Fp(\013)22 b Fr(+)g(2)p Fp(\031)2219 1936 y Fn(2)2258 1977 y Fp(;)17 b Fo(1)p Fr(\))p Fp(:)386 2210 y Fi(b\))33 b(Ther)-5 b(e)33 b(is)g(a)g(choic)-5 b(e)33 b(of)g(the)g(c)-5 b(onstants)33 b Fp(L)h Fi(and)f Fp(h)g Fi(with)h Fp(L)2616 2174 y Fh(\000)p Fn(2)2729 2210 y Fr(+)19 b Fp(h)2880 2174 y Fh(\000)p Fn(2)3002 2210 y Fp(>)28 b Fr(1)33 b Fi(such)386 2327 y(that)1389 2519 y Fr(inf)23 b Fp(\033)t Fr(\()p Fp(H)1702 2534 y Fn(2)1741 2519 y Fr(\))k Fp(<)1910 2432 y Fo(p)p 1993 2432 V 87 x Fr(2)p Fp(\013)c Fr(+)f(2)p Fp(\031)2333 2478 y Fn(2)386 2732 y Fi(for)35 b(every)f Fp(\017)28 b(>)g Fr(0)p Fi(,)35 b(i.e.,)f(the)h(op)-5 b(er)g(ator)34 b Fp(H)1909 2747 y Fn(2)1983 2732 y Fi(has)h(a)f(b)-5 b(ound)35 b(state.)504 2949 y Fr(F)-8 b(rom)30 b(the)h(ab)s(o)m(v)m(e)h (lemmata)d(w)m(e)j(conclude)g(that)f(a)g(w)m(a)m(v)m(e)i(guide)e (exists)h(that)386 3066 y(has)47 b(no)g(b)s(ound)g(state)g(for)g(one)g (particle,)i(but)e(do)s(es)h(ha)m(v)m(e)g(a)e(geometrically)386 3182 y(induced)33 b(b)s(ound)g(state)g(for)f(t)m(w)m(o)h(in)m (teracting)f(particles.)504 3298 y(A)43 b(remark)g(on)f(the)i(ph)m (ysical)e(in)m(terpretation)g(of)h(this)f(e\013ect)i(is)e(in)g(order.) 386 3414 y(As)37 b(men)m(tioned)g(ab)s(o)m(v)m(e,)i(the)e(argumen)m(t)f (of)h(t)m(w)m(o)g(particles)f(acting)g(lik)m(e)g(one)h(of)386 3531 y(the)d(double)e(mass)i(do)s(esn't)f(apply)g(to)g(geometrically)e (induced)i(b)s(ound)h(states,)386 3647 y(since)f(their)e(existence)j (is)e(mass-indep)s(enden)m(t.)44 b(T)-8 b(o)32 b(gain)f(a)h(ph)m (ysical)g(in)m(tuition)386 3763 y(for)26 b(our)g(results)g(an)m(yw)m(a) m(y)-8 b(,)30 b(w)m(e)d(note)f(that)g(a)g(b)s(ound)g(state)h(in)e(a)h (w)m(a)m(v)m(e)i(guide)e(with)386 3879 y(bulges)j(can)g(b)s(e)g(seen)h (as)f(a)g(trade-o\013)f(b)s(et)m(w)m(een)j(reduced)f(kinetic)e(energy)i (in)e(the)386 3995 y(transv)m(erse)39 b(direction)d(\(due)i(to)f(the)h (enlarged)f(cross-section\))h(and)f(increased)386 4112 y(kinetic)d(energy)i(in)f(the)g(longitudinal)d(direction)h(\(due)j(to)f (the)g(lo)s(calization)c(of)386 4228 y(the)36 b(particle\).)52 b(Consider)37 b(no)m(w)f(t)m(w)m(o)h(particles)e(that)g(attract)h(eac)m (h)h(other)f(and)386 4344 y(that)f(w)m(ould)f(in)g(free)i(space)f(form) f(a)g(`molecule')g(with)g(an)h(a)m(v)m(erage)h(distance)f Fp(d)386 4460 y Fr(b)s(et)m(w)m(een)e(them.)43 b(Assume)32 b(for)f(the)h(case)g(of)e(our)i(w)m(a)m(v)m(e)h(guide)e(\012)g(that)g Fp(d)g Fr(is)g(con-)386 4577 y(siderably)25 b(bigger)g(than)h(the)g(ca) m(vit)m(y)h(width)f Fp(h)p Fr(,)h(but)f(considerably)g(smaller)e(than) 386 4693 y(the)29 b(ca)m(vit)m(y)g(length)e Fp(L)p Fr(.)43 b(This)28 b(means)h(that)f(in)f(their)h(transv)m(erse)i(mo)m(v)m(emen)m (t)f(the)386 4809 y(t)m(w)m(o)h(particles)e(act)h(rather)g(as)g(if)f (they)i(w)m(ere)g(indep)s(enden)m(t)g(of)f(eac)m(h)h(other,)g(th)m(us) 386 4925 y(receiving)g(t)m(wice)h(the)f(energy)i(decrease)g(from)d(the) i(enlarged)f(cross-section.)43 b(In)386 5042 y(longitudinal)28 b(direction,)i(on)h(the)h(other)f(hand,)h(the)g(t)m(w)m(o)g(particles)e (in)h(the)g(ca)m(v-)386 5158 y(it)m(y)38 b(b)s(eha)m(v)m(e)i(lik)m(e)d (one)i(particle)d(of)i(the)h(double)f(mass,)h(suc)m(h)h(that)e(the)h (energy)386 5274 y(increase)e(due)h(to)e(longitudinal)e(lo)s (calization)f(is)j(only)g(half)g(of)h(what)g(it)f(w)m(ould)386 5390 y(b)s(e)h(for)g(one)g(particle)f(alone.)56 b(It)38 b(follo)m(ws)e(that)g(the)i(energy)g(trade-o\013)f(is)f(more)386 5506 y(`fa)m(v)m(orable')d(for)f(the)h(system)h(of)f(t)m(w)m(o)g(in)m (teracting)f(particles)g(than)h(for)f(a)h(single)386 5623 y(one.)p eop %%Page: 4 4 4 3 bop 386 259 a Fk(4)1151 b(HELMUT)26 b(LINDE)618 459 y Fr(3.)48 b Fq(Tw)n(o-p)-7 b(ar)g(ticle)38 b(bound)g(st)-7 b(a)g(tes)37 b(ca)n(used)g(by)h(Neumann)1417 575 y(bound)n(ar)-7 b(y)39 b(conditions)504 749 y Fr(If)i(one)g(in)m(tro)s(duces)g(Neumann) g(b)s(oundary)g(conditions,)h(an)e(e\013ect)i(similar)386 866 y(to)36 b(the)h(one)g(describ)s(ed)h(ab)s(o)m(v)m(e)g(happ)s(ens)f (ev)m(en)i(for)d(particles)g(in)g(only)g(one)h(di-)386 982 y(mension:)j(Be)28 b Fp(H)1022 997 y Fn(3)1089 982 y Fr(=)g Fo(\000)p Fp(@)1326 946 y Fn(2)1321 1006 y Fl(x)1394 982 y Fr(in)f Fp(L)1569 946 y Fn(2)1609 982 y Fr(\()p Fm(R)1713 946 y Fn(+)1778 982 y Fr(\))h(with)f(a)h(Neumann)f(condition) g(at)g Fp(x)h Fr(=)g(0.)386 1098 y(Then)39 b(it)d(is)h(clear)h(that)f Fp(\033)1361 1113 y Fl(ess)1464 1098 y Fr(\()p Fp(H)1583 1113 y Fn(3)1622 1098 y Fr(\))f(=)g Fm(R)1874 1062 y Fn(+)1977 1098 y Fr(and)i Fp(H)2253 1113 y Fn(3)2329 1098 y Fr(has)g(no)g(eigen)m(v)-5 b(alues.)59 b(The)386 1214 y(corresp)s(onding)46 b(t)m(w)m(o-particle)g(Hamiltonian)d(though) k(turns)g(out)g(to)f(ha)m(v)m(e)i(a)386 1330 y(b)s(ound)33 b(state,)g(if)e(the)i(in)m(teraction)f(p)s(oten)m(tial)f(is)h(purely)g (attractiv)m(e:)386 1520 y Fu(Lemma)38 b(3.1.)43 b Fi(L)-5 b(et)36 b Fp(V)51 b Fr(:)30 b Fm(R)1401 1484 y Fn(+)1496 1520 y Fo(!)f Fm(R)47 b Fi(b)-5 b(e)36 b(a)g(strictly)g(incr)-5 b(e)g(asing)35 b(function.)48 b(Then)386 1637 y(the)35 b(op)-5 b(er)g(ator)1204 1780 y Fp(H)1285 1795 y Fn(4)1352 1780 y Fr(=)28 b Fo(\000)p Fp(@)1589 1739 y Fn(2)1584 1805 y Fl(x)1624 1814 y Fg(1)1685 1780 y Fo(\000)23 b Fp(@)1841 1739 y Fn(2)1836 1805 y Fl(x)1876 1814 y Fg(2)1937 1780 y Fr(+)f Fp(V)g Fr(\()p Fo(j)p Fp(x)2235 1795 y Fn(1)2296 1780 y Fo(\000)h Fp(x)2451 1795 y Fn(2)2491 1780 y Fo(j)p Fr(\))386 1949 y Fi(on)35 b Fp(L)592 1913 y Fn(2)632 1949 y Fr(\()p Fm(R)736 1913 y Fn(+)824 1949 y Fo(\002)23 b Fm(R)990 1913 y Fn(+)1055 1949 y Fr(\))35 b Fi(with)h(Neumann)f(b)-5 b(oundary)35 b(c)-5 b(onditions)35 b(at)g Fp(x)2844 1964 y Fn(1)2913 1949 y Fr(=)29 b(0)36 b Fi(and)f(at)386 2065 y Fp(x)441 2080 y Fn(2)509 2065 y Fr(=)29 b(0)35 b Fi(has)f(a)i(b)-5 b(ound)34 b(state,)i(i.e.,)f(an)g (eigenvalue)f(b)-5 b(elow)34 b(the)i(lower)e(thr)-5 b(eshold)386 2181 y(of)34 b(the)h(essential)f(sp)-5 b(e)g(ctrum.)504 2371 y Fr(In)32 b(view)g(of)f(Lemma)f(3.1)h(it)f(is)i(no)f(surprise)h (that)f(w)m(a)m(v)m(e)j(guides)d(exist)h(whic)m(h)386 2488 y(ha)m(v)m(e)42 b(no)g(one-particle)d(b)s(ound)j(states,)i(but)d (whic)m(h)h(do)f(ha)m(v)m(e)i(a)e(t)m(w)m(o-particle)386 2604 y(b)s(ound)49 b(state)f(induced)h(b)m(y)h(mixed)e(b)s(oundary)g (conditions.)91 b(Omitting)45 b(the)386 2720 y(pro)s(of,)j(w)m(e)f (only)f(men)m(tion)f(the)h(simple)f(example)g(of)h(a)f(straigh)m(t)h (tub)s(e)g(with)386 2836 y(Diric)m(hlet)26 b(b)s(oundary)i(conditions)g (on)g(the)g(edge)h(and)f(an)g(additional)e(Neumann)386 2953 y(condition)31 b(imp)s(osed)h(on)g(one)h(cross-section.)1251 3228 y(4.)49 b Fq(Pr)n(oofs)38 b(of)g(the)f(resul)-7 b(ts)386 3403 y Fi(Pr)i(o)g(of)35 b(of)f(L)-5 b(emma)34 b(2.1.)41 b Fr(W)-8 b(e)40 b(in)m(tro)s(duce)g(the)g(op)s(erator)2525 3377 y(~)2499 3403 y Fp(H)2580 3418 y Fn(1)2620 3403 y Fr(,)h(whic)m(h)g(w)m(e)f(de\014ne)386 3519 y(to)35 b(b)s(e)h(the)g(Laplace)f(op)s(erator)f(on)i(\012)g(with)f(Diric)m (hlet)e(conditions)h(on)i Fp(@)5 b Fr(\012)37 b(and)386 3635 y(additional)30 b(Neumann)i(conditions)g(on)g(the)h(set)1171 3819 y Fo(f)p Fr(\()p Fp(x;)17 b(y)t Fr(\))27 b(:)h Fp(x)g Fr(=)f Fo(\006)p Fp(L=)p Fr(2)33 b(and)g Fo(j)p Fp(y)t Fo(j)26 b Fp(<)i Fo(j)p Fp(\017)p Fo(jg)p Fp(:)386 4012 y Fr(T)-8 b(o)33 b(pro)m(v)m(e)g(Lemma)f(2.1)g(it)f(is)h(then)i (su\016cien)m(t)f(to)g(sho)m(w)g(that)2723 3987 y(~)2698 4012 y Fp(H)2779 4027 y Fn(1)2851 4012 y Fr(has)g(no)f(sp)s(ec-)386 4128 y(trum)c(b)s(elo)m(w)i Fp(\031)959 4092 y Fn(2)998 4128 y Fr(.)42 b(With)29 b(the)g(in)m(tro)s(duction)f(of)h(the)h(new)g (b)s(oundary)g(conditions)386 4244 y(w)m(e)50 b(ha)m(v)m(e)h(cut)f (\012)f(in)m(to)g(three)h(separate)g(domains)e(\012)2447 4208 y Fn(+)2506 4244 y Fp(=)p Fr(\012)2625 4208 y Fh(\000)2734 4244 y Fr(\(the)h(semi-strip)386 4362 y(in)c(p)s(ositiv)m(e)g(/)h (negativ)m(e)g Fp(x)p Fr(-direction\))e(and)i(\012)2198 4326 y Fn(0)2284 4362 y Fr(\(the)g(rectangle)f(\()p Fo(\000)3058 4323 y Fl(L)p 3058 4339 48 4 v 3064 4396 a Fn(2)3117 4362 y Fp(;)3170 4323 y Fl(L)p 3170 4339 V 3176 4396 a Fn(2)3228 4362 y Fr(\))31 b Fo(\002)386 4496 y Fr(\()p Fo(\000)511 4457 y Fl(h)p 511 4473 41 4 v 514 4530 a Fn(2)562 4496 y Fp(;)616 4457 y Fl(h)p 616 4473 V 619 4530 a Fn(2)666 4496 y Fr(\)\).)50 b(Th)m(us)1094 4471 y(~)1069 4496 y Fp(H)1150 4511 y Fn(1)1224 4496 y Fr(is)34 b(the)h(orthogonal)e(sum)i(of)g(the)g(Laplace)f(op)s(erators)h(on)386 4618 y(\012)456 4582 y Fn(+)515 4618 y Fr(,)e(\012)645 4582 y Fh(\000)737 4618 y Fr(and)g(\012)997 4582 y Fn(0)1069 4618 y Fr(\(sub)5 b(ject)34 b(to)f(appropriate)e(b)s(oundary)i (conditions\),)f(and)971 4810 y Fp(\033)t Fr(\()1093 4785 y(~)1068 4810 y Fp(H)1149 4825 y Fn(1)1188 4810 y Fr(\))c(=)f Fp(\033)t Fr(\()p Fo(\000)p Fr(\001)1612 4827 y 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Fr(+)23 b Fp(L)2079 5582 y Fh(\000)p Fn(2)2173 5623 y Fr(\))28 b Fp(>)f(\031)2401 5582 y Fn(2)2441 5623 y Fp(:)p eop %%Page: 5 5 5 4 bop 1006 259 a Fk(TW)n(O-P)-6 b(AR)g(TICLE)32 b(BINDING)g(IN)h(A)g (W)-9 b(A)g(VE)33 b(GUIDE)581 b(5)386 459 y Fr(W)-8 b(e)33 b(can)h(no)m(w)g(apply)f(a)f(theorem)h(of)g(Gadyl'shin)f([9])h(to)g (see)h(that)f Fp(\025)p Fr(\()p Fp(\017)p Fr(\))23 b Fo(\000)g Fp(\025)p Fr(\(0\))386 575 y(is)36 b(of)f(order)i Fp(\017)900 539 y Fn(2)940 575 y Fr(,)g(i.e.,)g(for)e(small)f(enough)j Fp(\017)d(>)g Fr(0)i(w)m(e)h(ha)m(v)m(e)h(inf)22 b Fp(\033)t Fr(\()p Fo(\000)p Fr(\001)2977 592 y Fn(\012)3028 573 y Fg(0)3068 575 y Fr(\))34 b Fp(>)f(\031)3308 539 y Fn(2)3348 575 y Fr(.)386 691 y(Altogether)f(this)g(means)h(that)1025 876 y(inf)23 b Fp(\033)t Fr(\()p Fp(H)1338 891 y Fn(1)1377 876 y Fr(\))28 b Fo(\025)g Fr(inf)22 b Fp(\033)t Fr(\()1805 851 y(~)1779 876 y Fp(H)1860 891 y Fn(1)1900 876 y Fr(\))27 b(=)h Fp(\031)2128 835 y Fn(2)2265 876 y Fr(for)k(small)e Fp(\017:)3297 1059 y Fe(\003)386 1256 y Fi(Pr)-5 b(o)g(of)35 b(of)f(L)-5 b(emma)34 b(2.2,)g(p)-5 b(art)35 b(a\).)41 b Fr(Using)33 b(the)g(cen)m(ter)h(of)e(mass)g(co)s(ordinates)386 1487 y(\(1\))543 b Fp(u)27 b Fr(=)1250 1419 y(1)p 1250 1464 49 4 v 1250 1555 a(2)1309 1487 y(\()p Fp(x)1402 1502 y Fn(2)1464 1487 y Fr(+)22 b Fp(x)1617 1502 y Fn(1)1657 1487 y Fr(\))97 b(and)h Fp(w)30 b Fr(=)2261 1419 y(1)p 2261 1464 V 2261 1555 a(2)2320 1487 y(\()p Fp(x)2413 1502 y Fn(2)2474 1487 y Fo(\000)23 b Fp(x)2629 1502 y Fn(1)2669 1487 y Fr(\))386 1698 y(w)m(e)34 b(rewrite)e Fp(H)941 1713 y Fn(2)1013 1698 y Fr(in)g(the)h(form)386 1919 y(\(2\))244 b Fp(H)836 1934 y Fn(2)903 1919 y Fr(=)27 b Fo(\000)1093 1852 y Fr(1)p 1093 1896 V 1093 1988 a(2)1153 1919 y Fp(@)1209 1878 y Fn(2)1204 1944 y Fl(u)1271 1919 y Fo(\000)1381 1852 y Fr(1)p 1381 1896 V 1381 1988 a(2)1439 1919 y Fp(@)1495 1878 y Fn(2)1490 1944 y Fl(w)1570 1919 y Fr(+)22 b(4)p Fp(\013)q(w)1853 1878 y Fn(2)1913 1919 y Fo(\000)h Fp(@)2069 1878 y Fn(2)2064 1944 y 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Fp(H)1695 2704 y Fn(2)1767 2689 y Fr(in)m(to)f(the)h(orthogonal)d(sum)688 2848 y(~)663 2874 y Fp(H)744 2889 y Fn(2)811 2874 y Fr(=)38 b Fp(H)1006 2889 y Fn(2)1045 2874 y Fo(j)1073 2903 y Fh(fj)p Fl(w)r Fh(j)p Fl(>\014)s Fh(g)1359 2874 y Fo(\010)33 b Fp(H)1550 2889 y Fn(2)1589 2874 y Fo(j)1616 2903 y Fh(fj)p Fl(w)r Fh(j)p Fl(<\014)s(;)p Fh(j)p Fl(u)p Fh(j)p Fl(<\014)s Fn(+)2106 2875 y Fd(L)p 2104 2887 43 3 v 2110 2929 a Fg(2)2156 2903 y Fh(g)2217 2874 y Fo(\010)g Fp(H)2408 2889 y Fn(2)2447 2874 y Fo(j)2475 2903 y Fh(fj)p Fl(w)r Fh(j)p Fl(<\014)s(;)p Fh(j)p Fl(u)p Fh(j)p Fl(>\014)s Fn(+)2965 2875 y Fd(L)p 2962 2887 V 2968 2929 a Fg(2)3014 2903 y Fh(g)3070 2874 y Fp(:)504 3082 y Fr(The)40 b(sp)s(ectrum)g(of)49 b Fp(H)1353 3097 y Fn(2)1392 3082 y Fo(j)1420 3111 y Fh(fj)p Fl(w)r Fh(j)p Fl(>\014)s Fh(g)1724 3082 y Fr(can)39 b(b)s(e)h(estimated)f(from)f(b)s(elo)m(w)h(b)m(y)h(4)p Fp(\013)q(\014)3336 3046 y Fn(2)386 3211 y Fr(and)k(the)h(sp)s(ectrum)f (of)54 b Fp(H)1418 3226 y Fn(2)1457 3211 y Fo(j)1485 3240 y Fh(fj)p Fl(w)r Fh(j)p Fl(<\014)s(;)p Fh(j)p Fl(u)p Fh(j)p Fl(<\014)s Fn(+)1975 3213 y Fd(L)p 1972 3225 V 1978 3266 a Fg(2)2024 3240 y Fh(g)2108 3211 y Fr(is)43 b(discrete.)79 b(By)45 b(separation)e(of)386 3349 y(v)-5 b(ariables)33 b(the)h(sp)s(ectrum)h(of)43 b Fp(H)1592 3364 y Fn(2)1632 3349 y Fo(j)1659 3378 y Fh(fj)p Fl(w)r Fh(j)p Fl(<\014)s(;)p Fh(j)p Fl(u)p Fh(j)p Fl(>\014)s Fn(+)2149 3351 y Fd(L)p 2147 3363 V 2153 3404 a Fg(2)2199 3378 y Fh(g)2272 3349 y Fr(if)33 b(found)h(to)g(b)s(e)h(purely)f(con-) 386 3481 y(tin)m(uous)d(and)h(its)f(lo)m(w)m(er)g(threshold)h(is)f (equal)g(to)g(the)h(lo)m(w)m(est)g(eigen)m(v)-5 b(alue)31 b(of)g(the)386 3598 y(`transv)m(ersal')i(op)s(erator)917 3823 y Fp(H)998 3838 y Fl(t)1055 3823 y Fr(=)27 b Fo(\000)1245 3756 y Fr(1)p 1245 3800 49 4 v 1245 3891 a(2)1305 3823 y Fp(@)1361 3782 y Fn(2)1356 3848 y Fl(w)1435 3823 y Fr(+)22 b(4)p Fp(\013)q(w)1718 3782 y Fn(2)1778 3823 y Fo(\000)h Fp(@)1934 3782 y Fn(2)1929 3848 y Fl(y)1964 3857 y Fg(1)2025 3823 y Fo(\000)g Fp(@)2181 3782 y Fn(2)2176 3848 y Fl(y)2211 3857 y Fg(2)2272 3823 y Fr(+)f Fp(\013)q Fr(\()p Fp(y)2519 3838 y Fn(2)2580 3823 y Fo(\000)g Fp(y)2727 3838 y Fn(1)2766 3823 y Fr(\))2804 3782 y Fn(2)386 4045 y Fr(on)42 b Fp(L)597 4008 y Fn(2)637 4045 y Fr(\(\()p Fo(\000)p Fp(\014)6 b(;)17 b(\014)6 b Fr(\))27 b Fo(\002)i Fr(\()p Fo(\000)p Fr(1)p Fp(=)p Fr(2)p Fp(;)17 b Fr(1)p Fp(=)p Fr(2\))1618 4008 y Fn(2)1656 4045 y Fr(\))42 b(with)g(Neumann)g (conditions)f(at)g Fo(j)p Fp(w)s Fo(j)i Fr(=)g Fp(\014)386 4161 y Fr(and)36 b(Diric)m(hlet)f(conditions)g(at)h Fo(j)p Fp(y)1655 4176 y Fn(1)1694 4161 y Fo(j)d Fr(=)h(1)p Fp(=)p Fr(2)i(and)g Fo(j)p Fp(y)2317 4176 y Fn(2)2356 4161 y Fo(j)e Fr(=)g(1)p Fp(=)p Fr(2.)54 b(Neglecting)35 b(the)386 4277 y(p)s(ositiv)m(e)h(p)s(oten)m(tial)f(term)h Fp(\013)q Fr(\()p Fp(y)1553 4292 y Fn(2)1616 4277 y Fo(\000)25 b Fp(y)1766 4292 y Fn(1)1805 4277 y Fr(\))1843 4241 y Fn(2)1883 4277 y Fr(,)37 b(w)m(e)h(see)g(that)e(the)h(lo)m(w)m(est)g 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(describ)s(e)h(our)f(w)m(a)m(v)m(e)i(guide.)57 b(W)-8 b(e)38 b(de\014ne)g(the)386 5318 y(domain)744 5292 y(~)733 5318 y(\012)33 b(as)g(the)g(set)g(of)f(all)e(\()p Fp(x)1615 5333 y Fn(1)1655 5318 y Fp(;)17 b(y)1747 5333 y Fn(1)1786 5318 y Fp(;)g(x)1885 5333 y Fn(2)1924 5318 y Fp(;)g(y)2016 5333 y Fn(2)2055 5318 y Fr(\))33 b(that)f(satisfy)h(the)g(conditions) 646 5561 y Fp(u)27 b Fo(2)823 5421 y Fj(\022)897 5561 y Fo(\000)984 5494 y Fr(3)p Fp(L)p 984 5538 116 4 v 1017 5630 a Fr(8)1109 5561 y Fp(;)1163 5494 y Fr(3)p Fp(L)p 1163 5538 V 1196 5630 a Fr(8)1288 5421 y Fj(\023)1378 5561 y Fp(;)114 b(w)30 b Fo(2)1713 5421 y Fj(\022)1787 5561 y Fo(\000)1874 5494 y Fp(L)p 1874 5538 67 4 v 1883 5630 a Fr(8)1950 5561 y Fp(;)2004 5494 y(L)p 2004 5538 V 2013 5630 a Fr(8)2080 5421 y Fj(\023)2170 5561 y Fp(;)115 b(y)2360 5576 y Fn(1)2399 5561 y Fp(;)17 b(y)2491 5576 y Fn(2)2557 5561 y Fo(2)2651 5421 y Fj(\022)2724 5561 y Fo(\000)2811 5494 y Fp(h)p 2811 5538 57 4 v 2815 5630 a Fr(2)2878 5561 y Fp(;)2932 5494 y(h)p 2932 5538 V 2936 5630 a Fr(2)2998 5421 y Fj(\023)3088 5561 y Fp(;)p eop %%Page: 6 6 6 5 bop 386 259 a Fk(6)1151 b(HELMUT)26 b(LINDE)386 459 y Fr(using)37 b(the)h(co)s(ordinates)f Fp(u)g Fr(and)h Fp(w)h Fr(as)f(de\014ned)h(in)e(\(1\).)58 b(One)37 b(can)h(c)m(hec)m(k) i(that)397 558 y(~)386 583 y(\012)28 b Fo(\032)g Fr(\012.)44 b(W)-8 b(e)33 b(no)m(w)g(de\014ne)h(the)f(test)g(function)f(\011)h(b)m (y)758 816 y(\011)28 b(=)965 676 y Fj(\022)1038 816 y Fr(cos)1196 749 y(4)p Fp(\031)t(u)p 1196 793 164 4 v 1220 885 a Fr(3)p Fp(L)1369 676 y Fj(\023)1464 816 y Fo(\001)22 b Fr(\()p Fp(\036)p Fr(\()p Fp(w)s Fr(\))f Fo(\000)i Fp(C)7 b Fr(\))22 b Fo(\001)2067 706 y Fj(\020)2126 816 y Fr(cos)2283 749 y Fp(\031)t(y)2390 764 y Fn(1)p 2283 793 146 4 v 2328 885 a Fp(h)2439 706 y Fj(\021)2521 816 y Fo(\001)2571 706 y Fj(\020)2630 816 y Fr(cos)2787 749 y Fp(\031)t(y)2894 764 y Fn(2)p 2787 793 V 2832 885 a Fp(h)2943 706 y Fj(\021)386 1071 y Fr(on)532 1046 y(~)521 1071 y(\012)33 b(and)g(\011)28 b(=)f(0)32 b(on)h(\012)p Fo(n)1369 1046 y Fr(~)1358 1071 y(\012)q(,)f(setting)1104 1268 y Fp(\036)p Fr(\()p Fp(w)s Fr(\))27 b(=)g Fp(e)1486 1227 y Fh(\000)1541 1172 y(p)p 1600 1172 81 3 v 55 x Fn(2)p Fl(\013)q(w)1734 1203 y Fg(2)1869 1268 y Fr(and)98 b Fp(C)35 b Fr(=)27 b Fp(\036)p Fr(\()p Fp(L=)p Fr(8\))p Fp(:)386 1445 y Fr(The)e(function)f(\011)h(is)e(con)m(tin)m(uous)j(and) e(di\013eren)m(tiable)f(and)i(satis\014es)g(the)g(b)s(ound-)386 1561 y(ary)31 b(conditions)g(of)f Fp(H)1211 1576 y Fn(2)1251 1561 y Fr(.)43 b(Using)30 b(the)i(form)e(\(2\))h(of)g Fp(H)2336 1576 y Fn(2)2406 1561 y Fr(and)g(the)h(min-max)d(prin-)386 1677 y(ciple)j([11)o(])h(w)m(e)g(ha)m(v)m(e)687 1929 y(inf)22 b Fp(\033)t Fr(\()p Fp(H)999 1944 y Fn(2)1038 1929 y Fr(\))83 b Fo(\024)1330 1846 y(h)p Fr(\011)p Fp(;)17 b(H)1570 1861 y Fn(2)1608 1846 y Fr(\011)p Fo(i)1723 1873 y Fl(L)1771 1854 y Fg(2)1806 1873 y Fn(\()1841 1856 y(~)1833 1873 y(\012\))p 1330 1906 586 4 v 1510 1997 a Fo(jj)p Fr(\011)p Fo(jj)1698 1968 y 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Fr(\()p Fp(w)s Fr(\))k Fo(\000)i Fp(C)7 b Fr(\))2483 2569 y Fn(2)2539 2598 y Fr(d)p Fp(w)3047 2505 y(:)386 2788 y Fr(The)33 b(last)f(term)g(can)h(b)s(e)g(estimated)f (from)f(ab)s(o)m(v)m(e)j(b)m(y)396 2900 y Fj(R)462 2926 y Fl(L=)p Fn(8)443 3015 y Fh(\000)p Fl(L=)p Fn(8)621 2980 y Fr(\()669 2941 y Fn(1)p 669 2957 36 4 v 669 3014 a(2)714 2980 y Fp(\036)772 2944 y Fh(0)795 2980 y Fr(\()p Fp(w)s Fr(\))944 2944 y Fn(2)1005 2980 y Fr(+)22 b(4)p Fp(\013)q(w)1288 2944 y Fn(2)1326 2980 y Fp(\036)p Fr(\()p Fp(w)s Fr(\))1533 2944 y Fn(2)1571 2980 y Fr(\))17 b(d)p Fp(w)p 396 3047 1357 4 v 503 3083 a Fj(R)569 3109 y Fl(L=)p Fn(8)550 3197 y Fh(\000)p Fl(L=)p Fn(8)727 3163 y Fr(\()p Fp(\036)823 3134 y Fn(2)863 3163 y Fr(\()p Fp(w)s Fr(\))k Fo(\000)i Fr(2)p Fp(C)7 b(\036)p Fr(\()p Fp(w)s Fr(\)\))17 b(d)p Fp(w)1845 3070 y Fr(=)2630 2920 y Fo(p)p 2713 2920 112 4 v 83 x Fr(2)p Fp(\013)p 2014 3047 1427 4 v 2014 3163 a Fr(1)22 b Fo(\000)h Fr(2)p Fp(C)7 b Fo(jj)p Fp(\036)p Fo(jj)2481 3122 y Fh(\000)p Fn(2)2481 3194 y([)p Fh(\000)p Fl(L=)p Fn(8)p Fl(;L=)p Fn(8])2851 3083 y Fj(R)2917 3109 y Fl(L=)p Fn(8)2898 3197 y Fh(\000)p Fl(L=)p Fn(8)3092 3163 y Fp(\036)p Fr(\()p Fp(w)s Fr(\))17 b(d)p Fp(w)1845 3353 y Fr(=)2004 3266 y Fo(p)p 2087 3266 112 4 v 87 x Fr(2)p Fp(\013)23 b Fr(+)f Fo(O)s Fr(\()p Fp(e)2484 3312 y Fh(\000)2539 3257 y(p)p 2598 3257 36 3 v 55 x Fn(2)p Fl(L=)p Fn(64)2791 3353 y Fr(\))-2367 b(\(5\))386 3531 y(for)29 b(large)f Fp(L)p Fr(,)j(where)g(in)e(the)h(last)f(step)h(w)m (e)h(ha)m(v)m(e)g(used)f(that)g Fp(\013)e Fr(=)g Fp(L)2882 3494 y Fh(\000)p Fn(2)3006 3531 y Fr(and)i(th)m(us)386 3655 y Fp(C)35 b Fr(=)27 b Fp(e)639 3619 y Fh(\000)694 3564 y(p)p 753 3564 V 55 x Fn(2)p Fl(L)836 3596 y Fg(2)871 3619 y Fl(=)p Fn(64)981 3655 y Fr(.)43 b(Putting)32 b(\(5\))h(in)m(to)e (\(4\))h(and)h(using)f(\(3\))h(yields)519 3896 y(inf)23 b Fp(\033)t Fr(\()p Fp(H)832 3911 y Fn(2)871 3896 y Fr(\))28 b Fp(<)1040 3809 y Fo(p)p 1123 3809 112 4 v 87 x Fr(2)p Fp(\013)23 b Fr(+)f(2)p Fp(M)10 b(\031)1567 3855 y Fn(2)1629 3896 y Fo(\000)1739 3828 y Fr(10)p Fp(\031)1896 3792 y Fn(2)p 1739 3873 196 4 v 1759 3964 a Fr(9)p Fp(L)1874 3935 y Fn(2)1967 3896 y Fr(+)2190 3828 y Fp(\031)2249 3792 y Fn(2)2310 3828 y Fo(\000)23 b Fr(6)p 2075 3873 500 4 v 2075 3964 a(9)p Fp(\031)2183 3935 y Fn(2)2222 3964 y Fr(\()p Fp(L)2326 3935 y Fn(2)2388 3964 y Fo(\000)f Fr(1\))2606 3896 y(+)g Fo(O)s Fr(\()p Fp(e)2869 3855 y Fh(\000)2924 3800 y(p)p 2983 3800 36 3 v 55 x Fn(2)p Fl(L=)p Fn(64)3176 3896 y Fr(\))p Fp(:)386 4128 y Fr(If)34 b(w)m(e)h(c)m(hose)g Fp(L)f Fr(su\016cien)m(tly)g(large)f(and)h Fp(M)45 b Fr(su\016cien)m(tly)34 b(close)g(to)f(one,)i(w)m(e)g(get)386 4244 y(for)d(an)m(y)h Fp(\017)28 b(>)g Fr(0)1375 4387 y(inf)23 b Fp(\033)t Fr(\()p Fp(H)1688 4402 y Fn(2)1727 4387 y Fr(\))28 b Fp(<)1896 4300 y Fo(p)p 1979 4300 112 4 v 87 x Fr(2)p Fp(\013)23 b Fr(+)f(2)p Fp(\031)2319 4346 y Fn(2)2358 4387 y Fp(;)386 4544 y Fr(pro)m(ving)32 b(part)h(b\))f(of)g(Lemma)f(2.2.)1612 b Fe(\003)386 4735 y Fi(Pr)-5 b(o)g(of)35 b(of)f(L)-5 b(emma)34 b(3.1.)41 b Fr(Using)22 b(the)i(cen)m(ter)g(of)e(mass)h(co)s(ordinates)f(from)f (\(1\))i(and)386 4851 y(a)k(similar)d(argumen)m(t)j(as)g(in)f(the)i (pro)s(of)e(of)h(Lemma)f(2.2,)i(part)f(a\),)h(one)f(can)h(sho)m(w)386 4967 y(that)1492 5104 y Fp(\033)1547 5119 y Fl(ess)1649 5104 y Fr(\()p Fp(H)1768 5119 y Fn(3)1807 5104 y Fr(\))g(=)g([)p Fp(\025;)17 b Fo(1)p Fr(\))p Fp(;)386 5260 y Fr(where)37 b Fp(\025)e Fr(is)g(the)h(lo)m(w)m(est)g(eigen)m(v)-5 b(alue)35 b(of)g(the)h(op)s(erator)f Fp(h)e Fr(=)g Fo(\000)2766 5221 y Fn(1)p 2766 5237 36 4 v 2766 5295 a(2)2811 5260 y Fp(@)2867 5224 y Fn(2)2862 5285 y Fl(w)2944 5260 y Fr(+)24 b Fp(V)d Fr(\()p Fo(j)p Fr(2)p Fp(w)s Fo(j)p Fr(\))386 5382 y(on)32 b Fp(L)587 5346 y Fn(2)627 5382 y Fr(\()p Fm(R)5 b Fr(\).)49 b(It)33 b(remains)f(to)g(sho)m(w)i(that)e Fp(H)1968 5397 y Fn(3)2040 5382 y Fr(has)h(an)f(eigen)m(v)-5 b(alue)32 b(b)s(elo)m(w)h Fp(\025)p Fr(.)504 5498 y(Consider)i(the)g(p) s(oten)m(tial)e Fp(V)21 b Fr(\()p Fo(j)p Fp(x)1696 5513 y Fn(1)1759 5498 y Fo(\000)j Fp(x)1915 5513 y Fn(2)1955 5498 y Fo(j)p Fr(\))34 b(as)h(a)g(function)f(in)f(the)i(\014rst)h (quad-)386 5623 y(ran)m(t)g(of)g(the)g(cartesian)g(\()p Fp(x)1393 5638 y Fn(1)1433 5623 y Fp(;)17 b(x)1532 5638 y Fn(2)1572 5623 y Fr(\))36 b(co)s(ordinate)f(system.)55 b(Let)2694 5598 y(~)2679 5623 y Fp(V)22 b Fr(\()p Fp(x)2851 5638 y Fn(1)2890 5623 y Fp(;)17 b(x)2989 5638 y Fn(2)3029 5623 y Fr(\))36 b(b)s(e)g(the)p eop %%Page: 7 7 7 6 bop 1006 259 a Fk(TW)n(O-P)-6 b(AR)g(TICLE)32 b(BINDING)g(IN)h(A)g (W)-9 b(A)g(VE)33 b(GUIDE)581 b(7)386 459 y Fr(p)s(oten)m(tial)28 b(de\014ned)k(on)e(the)g(whole)g Fp(x)1756 474 y Fn(1)1796 459 y Fr(-)p Fp(x)1884 474 y Fn(2)1924 459 y Fr(-plane)f(that)h(can)g (b)s(e)g(obtained)g(b)m(y)h(re-)386 575 y(\015ecting)d Fp(V)21 b Fr(\()p Fo(j)p Fp(x)922 590 y Fn(1)974 575 y Fo(\000)12 b Fp(x)1118 590 y Fn(2)1159 575 y Fo(j)p Fr(\))27 b(\014rst)h(on)g(the)g Fp(x)1797 590 y Fn(1)1837 575 y Fr(-axis)f(and)h(then)g(the)g(resulting)f(function)386 691 y(on)32 b(the)h Fp(x)744 706 y Fn(2)784 691 y Fr(-axis.)43 b(Because)34 b Fp(V)54 b Fr(is)32 b(a)h(strictly)e(increasing)h (function,)g(w)m(e)i(ha)m(v)m(e)386 852 y(\(6\))1378 827 y(~)1363 852 y Fp(V)22 b Fr(\()p Fp(x)1535 867 y Fn(1)1575 852 y Fp(;)17 b(y)1667 867 y Fn(1)1705 852 y Fr(\))28 b Fo(\024)g Fp(V)22 b Fr(\()p Fo(j)p Fp(x)2076 867 y Fn(1)2137 852 y Fo(\000)h Fp(x)2292 867 y Fn(2)2332 852 y Fo(j)p Fr(\))386 1010 y(on)46 b(the)g(whole)g(plane)g(with)g (equalit)m(y)f(only)h(in)f(the)i(\014rst)f(and)h(in)e(the)h(third)386 1126 y(quadran)m(t.)e(Let)1296 1221 y(~)1271 1247 y Fp(H)1352 1262 y Fn(3)1419 1247 y Fr(=)27 b Fo(\000)p Fp(@)1655 1205 y Fn(2)1650 1271 y Fl(x)1690 1280 y Fg(1)1752 1247 y Fo(\000)c Fp(@)1908 1205 y Fn(2)1903 1271 y Fl(x)1943 1280 y Fg(2)2004 1247 y Fr(+)2117 1221 y(~)2102 1247 y Fp(V)e Fr(\()p Fp(x)2273 1262 y Fn(1)2313 1247 y Fp(;)c(x)2412 1262 y Fn(2)2452 1247 y Fr(\))386 1402 y(in)36 b Fp(L)570 1365 y Fn(2)609 1402 y Fr(\()p Fm(R)713 1365 y Fn(2)759 1402 y Fr(\).)54 b(Then,)39 b(b)m(y)e(symmetry)g(of)1893 1376 y(~)1879 1402 y Fp(V)57 b Fr(and)37 b(b)m(y)g(the)g(b)s(oundary)g (conditions)386 1526 y(of)32 b Fp(H)578 1541 y Fn(3)617 1526 y Fr(,)h(the)g(lo)m(w)m(est)g(eigen)m(v)-5 b(alue)32 b(of)1741 1501 y(~)1715 1526 y Fp(H)1796 1541 y Fn(3)1868 1526 y Fr(is)g(also)g(the)h(lo)m(w)m(est)g(eigen)m(v)-5 b(alue)32 b(of)g Fp(H)3281 1541 y Fn(3)3320 1526 y Fr(.)504 1642 y(In)h(the)g(cen)m(ter)h(of)e(mass)h(co)s(ordinates)f(\(1\))g(one) h(has)1058 1824 y(~)1033 1849 y Fp(H)1114 1864 y Fn(3)1181 1849 y Fr(=)27 b Fo(\000)1371 1782 y Fr(1)p 1371 1826 49 4 v 1371 1917 a(2)1430 1849 y Fp(@)1486 1808 y Fn(2)1481 1874 y Fl(u)1549 1849 y Fo(\000)1659 1782 y Fr(1)p 1659 1826 V 1659 1917 a(2)1717 1849 y Fp(@)1773 1808 y Fn(2)1768 1874 y Fl(w)1848 1849 y Fr(+)22 b Fp(V)f Fr(\()p Fo(j)p Fr(2)p Fp(w)s Fo(j)p Fr(\))g(+)h Fp(D)s Fr(\()p Fp(u;)17 b(w)s Fr(\))386 2059 y(with)31 b Fp(D)s Fr(\()p Fp(x)784 2074 y Fn(1)824 2059 y Fp(;)17 b(x)923 2074 y Fn(2)962 2059 y Fr(\))28 b(=)1146 2034 y(~)1132 2059 y Fp(V)21 b Fr(\()p Fp(x)1303 2074 y Fn(1)1343 2059 y Fp(;)c(y)1435 2074 y Fn(1)1474 2059 y Fr(\))j Fo(\000)h Fp(V)h Fr(\()p Fo(j)p Fp(x)1830 2074 y Fn(1)1890 2059 y Fo(\000)f Fp(x)2043 2074 y Fn(2)2083 2059 y Fo(j)p Fr(\).)43 b(Because)33 b(of)e(\(6\))h(the)g(function)386 2175 y Fp(D)43 b Fr(is)c(zero)h(in)g (the)g(\014rst)g(and)h(third)e(quadran)m(t)h(and)h(negativ)m(e)f(in)f (the)h(second)386 2291 y(and)f(fourth.)63 b(W)-8 b(e)40 b(call)d Fp(\036)p Fr(\()p Fp(w)s Fr(\))h(the)i(\(normalized\))d(lo)m (w)m(est)i(eigenfunction)g(of)g Fp(h)386 2408 y Fr(and)33 b(de\014ne)g(the)g(test)h(function)1254 2566 y(\011)28 b(=)f Fp(\036)p Fr(\()p Fp(w)s Fr(\))p Fp(e)1713 2525 y Fh(\000)p Fl(\017)p Fh(j)p Fl(u)p Fh(j)1978 2566 y Fr(\(with)16 b Fp(\017)28 b(>)g Fr(0\))p Fp(:)386 2724 y Fr(Then)34 b(the)f(min-max)d(principle)h([11)o(])i(tells)f(us)h(that) 1011 2935 y(inf)22 b Fp(\033)t Fr(\()1268 2909 y(~)1242 2935 y Fp(H)1323 2950 y Fn(3)1363 2935 y Fr(\))83 b Fo(\024)g Fp(\017)1683 2893 y Fn(2)1745 2935 y Fr(+)22 b Fp(\025)g Fr(+)g Fp(\017)2076 2799 y Fj(Z)2192 2935 y Fr(\011)2268 2893 y Fn(2)2308 2935 y Fp(D)d Fr(d)p Fp(x)2517 2950 y Fn(1)2574 2935 y Fr(d)p Fp(x)2683 2950 y Fn(2)2723 2935 y Fp(:)386 3140 y Fr(The)43 b(in)m(tegral)e(in)h(the)h(last)e (summand)h(is)g(negativ)m(e)h(and)f(its)g(absolute)g(v)-5 b(alue)386 3256 y(increases)36 b(when)g Fp(\017)f Fr(go)s(es)f(to)h (zero.)50 b(Consequen)m(tly)-8 b(,)38 b(for)c(some)h(small)d(enough)j Fp(\017)386 3375 y Fr(w)m(e)f(ha)m(v)m(e)f(inf)23 b Fp(\033)t Fr(\()1011 3350 y(~)986 3375 y Fp(H)1067 3390 y Fn(3)1106 3375 y Fr(\))28 b Fp(<)f(\025)p Fr(,)33 b(whic)m(h)g(pro)m(v)m(es)h (Lemma)e(3.1.)824 b Fe(\003)1413 3576 y Fq(A)m(ckno)n(wledgments)504 3750 y Fr(It)39 b(is)g(a)g(pleasure)g(for)g(me)g(to)f(thank)i(Rafael)e (Benguria)g(and)h(P)m(a)m(v)m(el)h(Exner)386 3867 y(for)32 b(their)g(in)m(terest)h(in)f(this)g(w)m(ork)i(and)e(their)g(helpful)g (commen)m(ts.)1590 4068 y Fq(References)427 4225 y Ft([1])42 b(W.)28 b(Bulla,)h(F.)f(Gesztesy)-7 b(,)29 b(W.)g(Renger,)e(B.)i (Simon:)38 b Fc(We)l(akly)32 b(c)l(ouple)l(d)f(b)l(ound)f(states)h(in) 557 4325 y(qantum)d(wave)i(guides)p Ft(,)e(Pro)r(ceedings)d(of)i(the)g (American)f(Mathematical)h(So)r(ciet)n(y)-7 b(,)26 b Fb(125)557 4425 y Ft(No.)h(5)g(\(1997\))g(1487{1495)427 4524 y([2])42 b(P)-7 b(.)28 b(Duclos,)h(P)-7 b(.)29 b(Exner:)39 b Fc(Curvatur)l(e-induc)l(e)l(d)30 b(b)l(ound)h(states)g(in)g(quantum)f (wave)i(guides)557 4624 y(in)d(two)h(and)h(thr)l(e)l(e)e(dimensions)p Ft(,)g(Rev.)f(Math.)f(Ph)n(ys.)g Fb(7)g Ft(\(1995\),)g(73{102)427 4723 y([3])42 b(P)-7 b(.)28 b(Exner,)h(H.)g(Linde,)g(T.)g(W)-7 b(eidl:)41 b Fc(Lieb-Thirring)33 b(ine)l(qualities)f(for)g(ge)l (ometric)l(al)t(ly)h(in-)557 4823 y(duc)l(e)l(d)d(b)l(ound)f(states)p Ft(,)f(Lett.)g(Math.)g(Ph)n(ys.)e Fb(70)h Ft(\(2004\),)g(83{95)427 4925 y([4])42 b(P)-7 b(.)19 b(Exner,)h(P)-7 b(.)1003 4904 y(\024)1001 4925 y(Seba,)21 b(M.)f(T)-7 b(ater,)20 b(D.)g(V)-7 b(an)n(\024)-39 b(ek:)32 b Fc(Bound)23 b(states)e(and)i(sc) l(attering)f(in)h(quantum)557 5025 y(wave)34 b(guides)g(c)l(ouple)l(d)g (later)l(al)t(ly)h(thr)l(ough)f(a)g(b)l(oundary)g(window)p Ft(,)g(J.)e(Math.)g(Ph)n(ys.)e Fb(37)557 5125 y Ft(\(1996\),)c (4867{4887)427 5224 y([5])42 b(P)-7 b(.)34 b(Exner,)i(S.A.)f(V)-7 b(ugalter:)51 b Fc(Asymptotic)37 b(estimates)g(for)g(b)l(ound)f(states) g(in)h(quantum)557 5324 y(wave)24 b(guides)h(c)l(ouple)l(d)g(later)l (al)t(ly)h(thr)l(ough)e(a)g(narr)l(ow)h(window)p Ft(,)f(Ann.)e(Inst.)f (H.)h(P)n(oincar)n(\023)-39 b(e:)557 5423 y(Ph)n(ys.)26 b(th)n(\023)-39 b(eor.)26 b Fb(65)h Ft(\(1996\),)g(109{123)427 5523 y([6])42 b(P)-7 b(.)22 b(Exner,)g(S.A.)h(V)-7 b(ugalter:)33 b Fc(Bound-state)25 b(asymptotic)h(estimates)f(for)h(window-c)l(ouple)l (d)557 5623 y(Dirichlet)k(strips)h(and)f(layers)p Ft(,)f(J.)e(Ph)n(ys)g (A30)g(\(1997\),)f(7863{7878)p eop %%Page: 8 8 8 7 bop 386 259 a Fk(8)1151 b(HELMUT)26 b(LINDE)427 459 y Ft([7])42 b(P)-7 b(.)33 b(Exner,)i(S.A.)g(V)-7 b(ugalter:)49 b Fc(On)35 b(the)h(numb)l(er)f(of)i(p)l(articles)g(that)f(a)g(curve)l (d)g(quantum)557 558 y(wave)l(guide)31 b(c)l(an)f(bind)p Ft(,)e(J.)f(Math.)h(Ph)n(ys.)f Fb(40)g Ft(\(1999\),)f(4630{4638)427 658 y([8])42 b(P)-7 b(.)35 b(Exner,)h(V.A.)g(Zagrebno)n(v:)50 b Fc(Bose-Einstein)38 b(c)l(ondensation)g(in)e(ge)l(ometric)l(al)t(ly)j (de-)557 758 y(forme)l(d)30 b(tub)l(es)p Ft(,)d(J.)h(Ph)n(ys.)e Fb(A38)i Ft(\(2005\),)e(463{470)427 857 y([9])42 b(R.)e(R.)h (Gadyl'shin:)62 b Fc(Perturb)l(ation)42 b(of)g(the)g(L)l(aplacian)h(sp) l(e)l(ctrum)e(when)h(ther)l(e)f(is)h(a)557 957 y(change)29 b(in)g(the)g(typ)l(e)g(of)g(b)l(oundary)h(c)l(ondition)f(on)g(smal)t(l) h(p)l(art)e(of)i(the)f(b)l(oundary)p Ft(,)e(Com-)557 1056 y(put.)h(Math.)g(Math.)f(Ph)n(ys.)g Fb(36)g Ft(No.)h(7)f (\(1996\),)f(889{898)386 1156 y([10])41 b(Y.)26 b(Nogami,)f(F.M.)h(T)-7 b(o)n(y)n(ama,)24 b(Y.P)-7 b(.)26 b(V)-7 b(arshni:)35 b Fc(Stability)29 b(of)f(two-ele)l(ctr)l(on)g(b)l(ound)g(states)557 1256 y(in)h(a)i(mo)l(del)f(quantum)f(wir)l(e)p Ft(,)f(Ph)n(ys.)f(Lett.) h Fb(A207)f Ft(\(1995\),)f(355{361)386 1355 y([11])41 b(M.)19 b(Reed,)j(B.)d(Simon:)33 b Fc(Mathematic)l(al)24 b(Physics)p Ft(,)f(V)-7 b(ol.)20 b(IV,)g(Academic)f(Press)f(Inc.)i (\(1978\))386 1455 y([12])41 b(R.L.)29 b(Sc)n(h)n(ult,)i(D.G.)f(Ra)n(v) n(enhall,)f(H.W.)h(Wyld:)41 b Fc(Quantum)30 b(b)l(ound)i(states)f(in)h (a)g(classi-)557 1555 y(c)l(al)t(ly)f(unb)l(ound)e(system)g(of)i(cr)l (osse)l(d)f(wir)l(es)p Ft(,)e(Ph)n(ys.)f(Rev.)h Fb(B39)f Ft(\(1989\),)f(R5476)504 1672 y Fc(E-mail)31 b(addr)l(ess)7 b Ft(:)38 b Fa(Helmut.Linde@gm)o(x.d)o(e)504 1841 y Fs(Dep)-6 b(ar)g(tment)41 b(of)e(Physics,)h(Pontificia)f(Universid)n(ad)g(Ca)-6 b(t)2750 1834 y(\023)2747 1841 y(olica)39 b(de)h(Chile)386 1941 y(Casilla)31 b(306,)f(Correo)i(22)f(Santia)n(go,)g(Chile.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF 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