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PITAEVSKII HIERARCHY FOR WEAKLY COUPLED TWO-DIMENSIONAL BOSONS

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PITAEVSKII HIERARCHY FOR WEAKLY COUPLED TWO-DIMENSIONAL BOSONSFOR,TWO,for

    PITAEVSKII HIERARCHY FOR

    WEAKLY COUPLED

    TWO-DIMENSIONAL BOSONS

    Availableonlineatwww.sciencedirect.com

    帮毋

    

    爨萨ScienceDirect

    ActaMathematicaScientia2010,30B(3):841856

    数学物理

    http://actams.wipm.ac.ca

    ARIGoRoUSDERIVAT10N0FTHE

    GR0SSPITAEVSKIIHIERARCHYFoRWEAKI

    C0UPLEDTW0DIMENS10NALBoSoNS

    LiuChuangye(刘创业)

    WuhanInstituteofPhysicsandMathematics,ChineseAcademyofSciences P.0.Box71010,Wuhan430071,China

    E-mail:Chuangyeliu1130@126.corn

    AbstractInthisarticle,weconsiderthedynamicsofNtwo-dimensionalbosonsystems interactingthroughapairpotentialN_.Va(Xixj)where()=a-2V(x/a).Itiswell

    knownthattheGrossPitaevskii(GP)equationisanonlinearSchr6dingerequationand theGPhierarchyisaninfiniteBBGKYhierarchyofequationsSOthatif'Lttsolvesthe GPequation,thenthefaznilyofk-particledensitymatrices{kUt,k1)solvestheGP hierarchy.Denoteby砂?.tthesolutiontotheNparticleSchr6dingerequation.Underthe assumptionthata=N.for0<?<3/4.weprovethatasN}?thelimitpointsof

    thekparticledensitymatricesofCN,taresolutionsoftheGPhierarchywiththecoupling constantinthenonlineartermoftheGPequationgivenby.Jry(x)dx.

    KeywordsGrossPitaevskiiequation;Bosonsystem;densitymatrix;BBGKYhierarchy 2000MRSubjectClassification35Qa0;35Q55

    1Introduction

    MotivatedbyrecentexperimentalrealizationsofBoseEinsteincondensationthetheory

    ofdilute,inhomogeneousBosesystemswascurrentlyasubjectofintensivestudies1].The

    groundstateofbosonicatomsinatrapWaSshownexperimentallytodisplayBoseEinstein

    condensation(BEC).ThisfactwasprovedtheoreticallybyLiebf9,10,111forbosonswithtwo

    bodyrepulsiveinteractionpotentialsinthedilutelimit,startingfromthebasicSchr6dinger equation.Ontheotherhand.itiSwellknownthatthedynamicsofBose.Einsteincondensates arewelldescribedbythetheGross

    Pitaevskiiequation.Arigorousderivationofthisequation

    fromthebasicmany

    bodySchr5dingerequationinanappropriatelimitisnotasimplematter, however,andhasonlybeenachievedrecentlyinthreespatialdimension[2,3,4,5.Thisarticle

    iSconcernedwiththejustificationoftheGross

    Pitaevskiiequationintwospatia1dimensions.

    Inthiscase,severalnewissuesariseandwereferto11]fordetails.

    ReceivedSepetember11,2007;revisedApril7,2008.ThisworkispartiallysupportedbyNSFC(10571176)

    ACTAMATHEMATICASCIENTIAVl01.30Ser.B

    ConsiderNbosonsinthetwodimensionspace.Thebosonsinteractviaatwobody

    potentia1

    ():a-2Y(x/a)(1.1)

    WeassumethatthepotentialVissmooth,symmetric,andpositive,withcompactsupport. Theparameteradeterminestherangeandthestrengthofthepotential:aandNwillbe coupledSOthata__+0asN__?o..ThusthepotentialconvergestoDiracfunction.The

    NbodyHamiltonianfortheNweaklycoupledbosonsisgivenby

    HN

    N

??J+1?()

    j=li<j

    (1.2)

    whereAj=?,:(92+forxj=(j,)?.ThedynamicsoftheBosesystemis

    governedbytheNbodySchr6dingerequation

    i0tCN,t=HNCN,t

    Here,thewavefunction?,

    tliesinL2(X?)whichisthesubspaceofL(?)consisting

    offunctionssymmetricwithrespecttopermutationoftheNparticles.Moregenerally,we

    candescribetheNbodysystembyitsdensitymatrix3'N,

    t.Thedensitymatrixisapositive

    sel~adjointoperatoractingonL2(R?)with=1.Thedensitymatrixcorrespondingto thewavefunctionCN.tisgivenbytheonedimensionalorthogonalprojectiononto?,t,that

    is,?,

    t=l?,t)(?,t1.QuantummeChanicalstatesdescribedbyone-dimensionalorthogonal projectionsarecalledpurestates.Ingeneral,adensitymatrix(mixedstates)isaweighted

    averageoftheone

    dimensionalorthogonalprojections.Thetimeevolutionofthedensitymatrix ?,tisthengivenby

    ",t:HN,3'g,t],(1.4)

    whichisequivalenttotheSchr5dingerequation(1.3). Asfollows,wedenotebyageneralvariableinandbyX=(Xl,,XN)apointin'N.

    Wewilla1sousethenotationX=(1,,X)??匙2andX?一=(+l,,XN)??2(N-k).

    For=1,,?一1,thek-particlemarginaldistribution7of?,

    tisdefinedthrough

    itskernelby

    r

    ,y(xk,x)=/dxNkTN,t(xk,XNx,XNk),(1.5)J

    wherex=(,?,)and3'Y,t(x;X)denotesthekernelofthedensitymatrix"INFrom

Try,?,

    t=1,itisconcludedthatTJr-r,y?(k

    .

    )

    t=1forevery=1,,?一1.Inthesequel,weset

    7=tand=0,>Nforconvenience.

    From(1.4)andthesymmetryof3'N,twithrespecttopermutationsoftheNparticles,we

    concludethattheevolutionofthemarginaldistributionsof1N.

    tisdeterminedbythefollowing

    hierarchyofNequations,commonlycalledtheBBGKYhierarchy,

    7j(x;x)=?(一?,+?;){(x;x)

    +_[G(zj-x1)Ya(7;

    NO.3Liu:DERIVAT10N0FGRoSSPETAEVSKIIHIERARCHY843 fork=1,?,?

    

    +[Yo(x5-Xk+1)Ya(t-Xk+1)]

    ×+?x,+l;x,+1)

    Rewritingthishierarchyinintegralform,wehave (xtd

    s

    (;)()

    

    it

    ds[f)_()()

    1

    )dsdXk+l[Ya(Xj--Xk+1)--Ya(x;-Xk+1)] ×7'(x%,%+1;x,Xk+1)

    Settinga=NandlettingN}?.onehasthattheBBGKYhierarchyconvergesformally

    tothefollowinginfinitehierarchyofequations

(xxtds(z;)(x

    dsdXk+l[~(Xj--Xk+1--Xk+1)]

    ×+(x,Xk+1;x,X+1)

    fork=1,2,,whereb:2V(x)tdx.Eq.(1.8)issaidtobetheinfiniteBBGKYhierarchy,or theGross

    Pitaevskii(GP)hierarchy.Itturnsoutthat(1.8)hasafactorizedsolution.Specially, thefamily.fmargina1distributi.n((x,x):k

    j=l

    t(J)t()

    onlyifthefunctiontsatisfiesthenonlinearSchr5dingerequation t=一?t+hietlt

    ThisistheGross-Pitaevskii(GP)equation

    ofthenonlinearinteractionisgivenbyb.

    isasolutionof(1.8)ifand

    [7,8,12],exceptthatthecouplingconstantinfront

    Theaimofthisarticleistoprovetheconvergenceofsolutionsof(1.7)toonesof(1.8).More precisely,wewillprovethatforevery0<?<3/4andn=N,thesequenceFN,{)

    hasatleastasN_-+?onelimitpointF={,yk>1withrespecttosomeweaktopology, andthatanyweaklimitpointro.,tsatisfiestheinfinitehierarchy(1.8). TheremainderofthisarticleiSdividedintofoursections.InSection2weprovesoree Sobolev-typeinequalitiesandenergyestimates.Somenotationsandthemainresultarepre

    sentedinSection3.Section4isdevotedtoproofsoftwolemmas,whichplayacrucialrolein theproofofthemainresult.Finally,inSection5.themainresultiSproved. ?

    ,?/

    七一?

    ACTAMATHEMATICASCIENTIAVo1.3OSer.B

    2EnergyEstimates

    WebeginwiththefollowingSobolevtypeinequalities,

Lemma1(i)SupposethatV?LP(IR),1<P?,and?H().Then,

    .

    f)f.IVa(x)ldxc).;.fffILP(N2)fJcfl~(R.)

    ?豫0,

    (ii)LetV?L(.)beanonnegativefunction.Then,thereisanabsoluteconstantC>0

    suchthat

    ()J(,y)ldxdycIIVIIL.)((1一?)(1一?),)R0×R2

    (2.2)

    forall?Hf1t(2×?).

    Thefollowingpropositionplaysakeyroleintheproofofthemainresult,whichpresents

    boundsfortheL2normofthederivativesofawavefunctionintermsofthemeanvalueof

    powersoftheHamiltonian日?inthestatedescribedby.

    Proposition1SupposethatthepotentialV(x)ispositive,smooth,compactlysup

    ported,andsymmetric,thatis,y(x)=().set(z):(/0)andassumethat

    a=N.with0<?<3/4.Put

    HN?ya(xl

    1<f<m<N

    )=HN+N

    Hereandinthesequel,=(1一?j)/forJ=l,,N.Fixk?Nand0<C<1.Then, thereisNo=No(,C)suchthat

    (,(_?))CN(,2)(2.3)

    forallN>Noandall?D((日?))(isassumedtobesymmetricwithrespecttoany permutationofallitsvariables).

    ProofTheproofofthepropositionusesatwostepinductionover.For=0and

    k=1.theclaimistrivialbecauseofthepositivityofthepotentialandthesymmetryof.

    Now,weassumethepropositionistrueforallk?n,andweproveitfork=n+2.Tothis end,weapplytheinductionassumptionandwefind,forN>?0(n,),

    (,())

    (n)

    (,l?(厅?)_?)CN"(,HNS....?) +

    1

    J=1

    withvjm=Va(xjm).Then,wehave N…叠N>

    ?=

    1J<mN

    ) ?(,

    N>_

    jl,J2n1

    +?22??2Hn)+c.c.]?Jn+1

    =:1+

    (2.4)

    (2.5)

    一?

    ??

    ??一

    N0.3Liu:DERIVAT10N0FGR0SSPETAEVSKIIHIERARCHY

    becauseof()…日(0.

    Hereandafterwards,c_c_denotesthecomplexconjugateof

    thetermfollowed.Forand/2,usingthesymmetrywithrespecttopermutations,wehave

    and

    ^=(N一几)(?一佗一1)(,S--?++2)+(N一佗)(,s?-?+1),(2.6)

    /22n(N

    +

    +

    )(,s4

    n(n+1)(?一n)

2?

    (n+1)(?一n

    

    +)

    s2.._22)+c.c.]

    1)(?一n)

    N

    Here,wehaveusedthefactthat

    (,vjS}..

    ,n+2

    2

    .2)+c].(2.7)

    .

    S2+1)0

    ifJ,m>n+l,becauseofthepositivityofthepotentia1.Then,combiningwith(2.5),(2.6)

    and(2.7),weget

    (,SFINe)

    (?一几)(?一n1)(,S++2)+(2n+1)(?一他)(,s.+)

    +

    +

    n(n+1)(?一佗)

    2?

    (n+1)(?一n

    v12}2.2)+cJ

    1)(?一n)

    N

    (,?,n+...?+)+c-c-J(2.8)

    Next,weconsiderthelasttwotermsontherighthandof(2.8).Setting=+1,we

    have

(,V12S+1)+c.c.2(,?2p22)+(,2p2122)+c.c.

    wherePl=1,P2=2,andhence (,v12S}2+1)+c.c.

    2(,VV12V2~)+(2,VV12V1V2~)+(,2p2)+C.C where2:a-3()((1x2)/a).ApplyingtheSchwarzinequalityandLemma1,wefind

    that

    (,V12S+1)+c.c.

    

    C{ala-l(~,)+.吾一.(,2))

    

    {2n;.(V2,s2)+n2.(12,s12))

    C{a3a-l(~,})+la-3(V~V2,;2)).(2.9)

    OptimizingthechoicesofO11,0:2,and3,wecanconcludethat

    +)+?(,S+))

    +2)+?(,S+1))

    2121c,)

    ,I,,\

    22??

    r?Lr,,L

    8

    一一??

    一一

    >>

    C

    C

    +

    

    l

    +

21(,)

    2

    ,f

    846ACrrAMATHEMATICASCIENTIAVol_30Set.B Incontrast,letting=S2???+1,weconcludefromLemma1that (,?,+2s}...s+1)+c.c.=(西,?,+2s)+c.c.c.1(,s???S,2+2)

    Inserting(2.10)and(2.11)intotherighthandsideof(2.8),weobtain (,s…百_?)(?一几)(?一n1)f1N0

    2;1

    )()

    +(2n+1)(?一佗)(1)(,s42??+(2_12)

    F.r0<?<3/4and1<p<,.nehas>0,1一?(2)>0.Consequently,?口一

    吉》1

    and?》1.ForanyfixedC<1andn?N,by(2.12)wecanfindNoSOthat 西_?}…豆_?砂)C.N.(,S+2)

    foreveryNNo.This,togetherwith(2.4),completestheproofoftheproposition

    Wehavetheenergyestimatesasfollows.

    Corollary1SupposethattheinitialdensitymatrixVN, 0satisfies

    Tr(HN)7N,0cN

    (2.13)

    forall1.Let,y?,tbethes.luti.n.f(1.4)and{,y)?_.c.rrespondingmargina1distri

    butions.Then,foranyC>2(C0+1)andany?N,thereisNo=NO(k,Co,c),such that

    TrS1s7s...S1C,(2.14)

    forallt?andal1NNo

    3TheMainRsuit

    SinceOurmainresultstatespropertiesoflimitpointsofthesequenceN,t

    f0r_+O0'

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