Proof of the Variational Principle for a Pair Hami

arXiv:math-ph/0603061v124Mar2006ProofoftheVariationalPrincipleforaPairHamiltonianBosonModelJosephV.Pul´eaSchoolofMathematicalSciencesUniversityCollegeDublinBelfield,Dublin4,IrelandEmail:[e-mail]´edelaM´editerran´eeandCentredePhysiqueTh´eoriqueLuminy-Case907,13288Marseille,Cedex09,FranceEmail:[e-mail]metervariationalformulaforthegrand-canonicalpressureofthePairBosonHamiltonianmodel.ByusingtheApproximatingHamiltonianMethodweprovidearigorousproofofthisvariationalprinciple.Keywords:PairBosonHamiltonian,ApproximatingHamiltonianmethod,general-izedBose-EinsteincondensationPACS:05.30.Jp,03.75.Hh,03.75.Gg,67.40.-wAMS:82B10,82B26,82B21,81V70Contents1.Introduction2.Superstability3.TheFirstApproximation4.TheSecondApproximation5.DiscussionAcknowledgementsAppendixA:CommutatorsAppendixB:BoundsReferencesaResearchAssociate,SchoolofTheoreticalPhysics,DublinInstituteforAdvancedStudies.1IntroductionThefirstversionofthePairBosonHamiltonian(PBH)modelwasproposedbyZubarevandTserkovnikovin1958[1].TheirintentionwastogeneralizetheBogoliubovmodeloftheWeaklyImperfectBoseGas[2]byincludingmoretermsfromthetotalinteraction,withoutlosingthepossibilityofhavinganexactsolution.Wereferthereaderto[3]andto[4]foramorerecentdiscussionofthisquestion.ThesuggestionofZubarevandTserkovnikov[1]wastoconsideratruncatedHamiltonianwhichincludesadiagonaltermrepresentingforward-scatteringandexchange-scatteringaswellasanon-diagonalBCS-typeinteractionterm.Themodelcontainingonlytheforward-scatteringpartoftheinteractioncorrespondstotheMean-Field(ortheImperfect)Bosegas,see[4]and[5]fordetails.UsingthesamemethodastheyhadusedearlierforthefermionBCSmodel[6],theauthorsgivein[1]a“solution”ofthePBHmodel.LaterthisHamil-tonianbecamethesubjectofveryintensiveanalysis[7]-[9],leadingessentiallytothesameconclusionasin[1],namely,thatthePBHhasthesamethermodynamicpropertiesasacer-tainapproximatingHamiltonianquadraticinthecreationandannihilationoperators.UsingthisHamiltonianwhichcanbediagonalizedbythecanonicalBogoliubovtransformation,itsthermodynamicpropertieswereinvestigatedanditwasshowntohavesomeintriguingproperties.Oneoftheseispossibilityoftheoccurrenceoftwokindsofcondensation,thestandardone-particleBose-EinsteincondensationaswellasaBCS-typepaircondensationwhichmayappearintwostages,seee.g.[10],[11].Anotheroneconcernsthegapinthespectrumof“elementaryexcitations”[7]-[9].InspiteoffairlyconvincingargumentsthesepapersdidnotproverigorouslythattheabovementionedsolutionofthePBHmodelisexact.AmathematicaltreatmentofthePBHmodel,relatedtorepresentationsoftheCanonicalCommutationRelations(CCR)appearedin[12].InthepresentpaperwegiveavariationalformulaforthepressureforthePBHmodelandprovidearigorousderivationoftheformula.Thelatteryieldsthesameexpressionforthepressureaswasobtainedin[1],thecorrespondingEuler-Lagrangeequationscoincidingwithself-consistencyequationsstudiedin[1]and[7]-[12].Inanearlierpaper[13]weconjecturedthatthepressurecanbeexpressedasthesupremumofavariationalfunctionaldependingontwomeasures:apositivemeasuredescribingtheparticledensityandacomplexmeasuredescribingthepairdensity,similartotheCooperpairsdensityintheBCSmodel.Thiscon-firmedtheconclusionof[10],[11]aboutthecoexistenceofone-particleandpaircondensates.Thestudyin[13]wasinspiredbytheLargeDeviationPrinciple(LDP)developedfortheanalysisofbosonsystemsin[14]-[17].Thismethodgivesrigorousresultsforthepressureinthecaseofmodelswithdiagonal(commutative)bosoninteractions.Asimilartechniquewasdevelopedin[18]-[23]basedonthework[22],extendingtheLDPtononcommutativeMean-Fieldmodels(includingtheBCSone)withonlyboundedoperatorsinvolvedinHamil-tonians.SinceneitherofthesemethodsapplytothePBHwithoutextensivemodifications,hereweoptedfortheApproximatingHamiltonianMethod(AHM)[24],whichhasbeenal-readysuccessfullyappliedtomanymodels,includingsomeinteractingbosonmodels(seeforexample[4],[5],[25]).ThereisrenewedinterestinthepropertiesofthePBHinteractioninthecontextoffinitebosonsystemsconfinedinamagneto-optictrap,seee.g.[26]-[28].Wedonotdiscussthisaspectintheframeworkofourapproachleavingitforfuturepublications.NowweturntotheexactformulationofthePBHmodelinitssimplestform,thatis,withconstantpairandmean-fieldbosoncouplings[13].LetΛ⊂RνbeacubeofvolumeV=|Λ|centeredattheorigin.Thenthekineticenergyoper-atorforaparticleofmassmconfinedtothecubicboxΛ,thatistheoperator−Δ/2mwithpe-riodicboundaryconditions,haseigenvaluesǫ(k)=kkk2/2m,k∈Λ∗:={2πs/V1/ν|s∈Zν}.ConsiderasystemofidenticalbosonsofmassmenclosedinΛ.Fork∈Λ∗leta∗kandakbetheusualbosoncreationandannihilationoperatorssatisfyingtheCCR[ak,a∗k′]=δk,k′andletNk:=a∗kakbethek-modeparticlenumberoperator.Thekinetic-energyoperatorTΛforthePerfectBose-gas,canbeexpressedintheformTΛ:=Pk∈Λ∗ǫ(k)Nk.TointroduceapairingtermintheHamiltonianweshallneedtheoperatorsAk=A−k:=aka−k,k∈Λ∗.(1.1)LetNΛ:=Xk∈Λ∗Nkand˜QΛ:=Xk∈Λ∗˜λ(k)Ak,(1.2)wherethefunction˜λ:Rν7→Csatisfiesthefollowingconditions:|˜λ(k)|≤|˜λ(0)|=1,˜λ(k)=˜λ(−k)forallk∈Rν,thereexistsC∞andδ0suchthat|˜λ(k)|≤C1+kkkmax(ν,ν/2+1)+δ(1