Convergence of an exact quantization scheme

arXiv:math/0306218v1 [math.DS] 13 Jun 2003CONVERGENCEOFANEXACTQUANTIZATIONSCHEMEARTURAVILAAbstract.IthasbeenshownbyVoros[V1]thatthespectrumoftheone-dimensionalhomo-geneousanharmonicoscillator(Schr¨odingeroperatorwithpotentialq2M,M>1)isafixedpointofanexplicitnon-lineartransformation.Weshowthatthisfixedpointisgloballyandexponentiallyattractiveinspacesofproperlynormalizedsequences.

1.IntroductionLet00,define(1.1)θ(E′,E)=tan−1sinθ

π󰀅kθ(Xk,Yj).

LetQ=(Qi)∞i=1beaconstantvector,andconsidertheoperatorT≡Tθ,Qgivenimplicitlybyφ(X,T(X))=Q.OfcourseT(X)isonlydefinedforcertainsequencesX.WeremarkthatTisdilatationequivariant(T(λX)=λT(X)forλ>0)andpositiveinthesensethatif0′kforallk>0andifT(X)=YandT(X′)=Y′aredefinedthenYk≤Y′kforallk>0.

InthispaperwewillbeinterestedinthedescriptionofthedynamicsofTactingoncertainspacesofnormalizedsequences,underappropriateconditionsonQ.

1.1.Relationtoexactanharmonicquantization.Wenowdescribethephysicalmotivationoftheproblem(forfutherdetailsandreferences,see[V1],andformorerecentrelatedwork,see[V2]).Letusconsidertheone-dimensionalanharmonicoscillatorwithevenhomogeneouspolynomialpotential,thatis,theSchr¨odingeroperator

(1.3)(Hu)(q)=−d2u

M+1π,(1.6)αθ=π+θM+1.2ARTURAVILAItisknownthatEkhaspolynomialgrowth,moreprecisely:Proposition1.1(see[V1],§2.1).Thespectrum(1.4)oftheoperator(1.3)satisfies(1.7)ν=limk→∞k−αθEk,

whereαθisgivenby(1.6)andνispositiveandfinite.Thesemiclassicalanalysisprovidemuchmoreinformationthenwhatiscontainedintheaboveproposition,forinstance,νcanbeexplicitelycomputed

(1.8)ν=󰀈2π1/2MΓ󰀂32M󰀃Γ󰀂1

4+M−14−M−1CONVERGENCEOFANEXACTQUANTIZATIONSCHEME3TheoperatorTactuallycomesaboutasa(non-linear)quantizationofasemiclassicalBohr-Sommerfeldlinearoperator.ThemainstepsofouranalysisinvolvesshowingthatTbehavesasaperturbationofthelinearoperator.Theasymptoticlimitk→∞isgiventocertainaccuracybythesemiclassicallinearoperator,whichcanbeshowntohavetherequiredproperties.WeusetwoobviousfeaturesofTtoshowthatthequantizationdoesnotdestroythoseproperties.Thefirstoneispositivity,andthesecondoneisequivariancebydilatation.Thosepropertiesarepresentbothattheinfinitesimalanalysis(theyareusedinperturbativeestimatesoftheoperatornormofthederivativeDT)asintheglobalanalysis(wheretheyareusedinakeyprecompactnessargument).

2.ProofofTheorem1.32.1.Settingandnotations.Wewillactuallyproveaslightlymoregeneralresult,Theorem2.1,abouttheoperatorsTθ,Q.ThisresultimpliesTheorem1.3immediately,usingPropositions1.1and1.2.Theremaininganalysisiscompletelyself-contained.Wewillneedtomakenorestrictionon0sequenceQk:

(2.1)Qk=k+O(1)

(2.2)Qk>󰀂k−1π.Thefirstconditioncomesfromthephysicalproblem,andcanberelaxedtoQk=eo(1)kwithoutanychangesinouranalysis.NoticethatforanyX,

(2.3)k󰀅j=1φj(X,X)>θk24ARTURAVILA(2.10)u0(α,ǫ)=u0((αlnk)∞k=1,ǫ),α>0,ǫ≥0.Noticethatifx∈u(α,ǫ)thenu(x,ǫ′)=u(α,ǫ′)providedǫ′≤ǫ.TheseveralaffinespacesuparametrizebyexponentiationspacesU,forinstance

(2.11)U(α,ǫ)={(Xk)∞k=1,Xk>0,Xk=kα+O(kα−ǫ)},

(2.12)U0(α,0)={(Xk)∞k=1,Xk>0,Xk=kα+o(kα)}.Wecannowstateourmainresult:Theorem2.1.Thereexistsauniqueαθ>0forwhichthereexistsafixedpointX∈U(αθ,0)forT.Moreover,(1)ThespaceU0(αθ,0)isinvariantforT,(2)ThereexistsafixedpointP∈U(αθ,1),(3)PisaglobalattractorinU0(αθ,0),thatis,foranyX∈U0(αθ,0),

(2.13)limn→∞󰀑Tn(x)−p󰀑0=0,

(4)ThespacesU(P,ǫ)areinvariantfor0≤ǫ(5)PisaglobalexponentialattractorinU(P,ǫ),0

(2.14)󰀑Tn(x)−p󰀑ǫ≤Cλn,whereC=C(ǫ,󰀑x−p󰀑ǫ)>0andλ=λ(ǫ)<1.Theproofofthisresultwilltaketheremainingofthissection.2.3.LipschitzcontinuityinU(X,0).LetuswriteX≤X′ifXk≤X′kforallk.ThenX≤X′andY≥Y′impliesφ(X,Y)≥φ(X′,Y′),whichimpliesthepositivityofTwestatedbefore:X≤X′impliesT(X)≤T(X′).Inparticular,T(X)≤Xifφ(X,X)≥QandT(X)≥Xifφ(X,X)≤Q.ThisalsogivesusawaytoshowthatTisdefinedatsomeX:ifφ(X,YY)thenT(X)=YisdefinedandYY.

Lemma2.2.AssumethatT(X)=Yisdefined.ThenTisdefinedonU(X,0)andT(U(X,0))=U(Y,0).Moreover,T:U(X,0)→U(Y,0)is1-Lipschitz.

Proof.IfC−1X≤X′≤CXthenφ(X′,C−1Y)≤φ(C−1X,C−1Y)=Q=φ(CX,CY)≤φ(X′,CY).󰀁

2.4.Thederivative.Let(2.15)P(E,E′)=EE′

dxk

(X,Y)=−sinθ

dyj

(X,Y)=

󰀅

ksinθ

dyk

(X,Y)=0,j=k.