Eigenvalue asymptotics for Sturm--Liouville operat

arXiv:math/0407252v2[math.SP]3Oct2006EIGENVALUEASYMPTOTICSFORSTURM–LIOUVILLEOPERATORSWITHSINGULARPOTENTIALS†ROSTYSLAVO.HRYNIVANDYAROSLAVV.MYKYTYUKAbstract.WederiveeigenvalueasymptoticsforSturm–Liouvilleoperatorswithsin-gularcomplex-valuedpotentialsfromthespaceWα−12(0,1),α∈[0,1],andDirichletorNeumann–Dirichletboundaryconditions.Wealsogiveapplicationoftheobtainedresultstotheinversespectralproblemofrecoveringthepotentialfromthesetwospectra.1.IntroductionInthispaperweshallstudyeigenvalueasymptoticsforSturm–Liouvilleoperatorsontheinterval[0,1]withdistributionalpotentials.Namely,weassumethatqisacomplex-valueddistributionfromtheSobolevspaceWα−12(0,1),α∈[0,1],andconsideranoperatorTthat(formally)correspondstothedifferentialexpression(1.1)l(f):=−f′′+qfand,say,Dirichletboundaryconditions.Explicitly,theoperatorTisdefinedbytheregularisationmethodthatwassuggestedin[4]fortheparticularpotentialq(x)=1/xandwasdevelopedbySavchukandShkalikovin[30]fortheclassofdistributionalpotentialsfromW−12(0,1).(Incidentally,inthissituationtheform-sum[1,10]andthegeneralizedsum[20]methodsyieldthesameoperator).WeobservethattheconsideredclassofsingularpotentialsincludeDiracδ-typeandCoulomb1/x-typeinteractionsthatarewidelyusedinquantummechanicsandmathematicalphysics;seealso[18]forotherphysicalmodelsleadingtopotentialsfromnegativeSobolevspaces.Theregularisationmethodconsistsinrewriting(1.1)as(1.2)l(f)=lσ(f):=−(f′−σf)′−σf′,whereσisanydistributionalprimitiveofq.Wefixonesuchprimitiveσ∈L2(0,1)inwhatfollowsandcalltheexpressionf′−σf=:f[1]thequasi-derivativeofthefunctionf.ThenaturalL2-domainoflσisdomlσ={f∈W11(0,1)|f[1]∈W11(0,1),lσ(f)∈L2(0,1)},andweobservethatforf∈domlσthederivativef′=σf+f[1]belongstoL2(0,1)(butneednotbecontinuous),sothatdomlσ⊂W12(0,1).Inthepresentpaper,weshallonlyfocusonSturm–LiouvilleoperatorsTD=Tσ,DandTN=Tσ,NthataregeneratedbylσandtheDirichletandtheNeumann–Dirichletboundaryconditionsrespectively,althoughotherboundaryconditionscanalsobetreatedinasimilarmanner(see,e.g.,[18]forperiodicand[31]forgeneralregularDate:February1,2008.2000MathematicsSubjectClassification.Primary34L20,Secondary34B24,34A55.Keywordsandphrases.Eigenvalueasymptotics,Sturm–Liouvilleoperators,singularpotentials.†TheworkwaspartiallysupportedbyUkrainianFoundationforBasicResearchDFFDundergrantNo.01.07/00172.12R.O.HRYNIV,YA.V.MYKYTYUKboundaryconditions).Inotherwords,TDandTNaretherestrictionsoflσontothedomains(1.3)domTD={f∈domlσ|f(0)=f(1)=0},domTN={f∈domlσ|f[1](0)=f(1)=0}.Itisknown[30]thattheoperatorsTDandTNareclosed,denselydefinedandhavediscretespectratendingto+∞.Wedenotebyλ2n(resp.,μ2n)theeigenvaluesofTD(resp.,TN)countedwithmultiplicitiesandarrangedbyincreasingofthereal—andthen,ifequal,imaginary—partsofλn(resp.,μn).Fordefiniteness,weshallalwaystakeλnandμnfromtheset(1.4)Ω:={z∈C|−π/2argz≤π/2}∪{z=0}.Ifα=0,i.e.,ifσ∈L2,thenthenumbersλnandμnobeytheasymptotics[15,29–31](1.5)λn=πn+eλn,μn=π(n−12)+eμn,where(eλn)n∈Nand(eμn)n∈Naresomeℓ2-sequences.Itisreasonabletoexpectthatifσbecomessmoother,thentheremainderseλnandeμndecayfaster;forinstance,ifα=1,i.e.,ifq∈L2(0,1),thentheclassicalresult(see,e.g.,[26,Theorem3.4.1]or[28,Theorem2.4])statesthateλn,eμn=O(n−1).Thustheproblemarisestocharacterizethedecayofeλn,eμndependingonα∈[0,1].OurinterestintheaboveproblemhasstemmedfromtheinversespectraltheoryforSturm–Liouvilleoperatorswithsingularpotentials.Namely,weprovedin[16]that,assoonasthenumbersλ2nandμ2narereal,strictlyincreasewithn,interlace,andobey(1.5)withℓ2-sequences(eλn)and(eμn),thenthereexistsauniquereal-valuedσ∈L2(0,1)suchthat{λ2n}and{μ2n}arespectraoftheSturm–LiouvilleoperatorsTσ,DandTσ,Nrespectivelywithdistributionalpotentialq=σ′∈W−12(0,1).Itisreasonabletobelievethatifeλnandeμndecayfaster,thenσwillbesmoother.Forexample,theclassicalresultofMarchenko[26,Theorem3.4.1]claimsthatif,undertheaboveassumptions,wehaveinaddition(1.6)eλn=An+eλ′nn,eμn=An+eμ′nnwithrealAandℓ2-sequences(eλ′n)and(eμ′n),thenthecorrespondingpotentialqisinL2(0,1)(thusα=1)andA=12πR10q,cf.also[24,Theorem3.3.1].Itwouldbedesirableto“interpolate”betweenα=1andα=0andsolvetheinversespectralprob-lemforallintermediateα∈(0,1).TheessentialsteptowardssuchprojectistoderiveeigenvalueasymptoticsforSturm–LiouvilleoperatorswithpotentialsinWα−12(0,1)—i.e.,totreatthedirectspectralproblem.Andindeed,basedontheresultsobtainedhere,wecompletelysolvetheinversespectralproblemforSturm–LiouvilleoperatorswithpotentialsinthescaleWα−12(0,1),α∈[0,1],inourpaper[17].Anothermotivationforthisworkistherecentpapers[18]and[31],wheresimilarquestionsareconsidered.Inparticular,KappelerandM¨ohrin[18]foundeigenvalueasymptoticsfortheDirichletandperiodicSturm–Liouvilleoperatorswithcomplex-valuedpotentialsthatareperiodicdistributionsfromthespaceWα−12(0,1),α∈(0,1].TheDirichleteigenvaluesλ2nwereprovedtheretoobeytheasymptotics(1.7)λ2n=π2n2+bq(0)−bq(−2n)+bq(2n)2+νn,EIGENVALUEASYMPTOTICS3wherebq(n)isthen-thFouriercoefficientofqandthesequence(νn)belongstoℓ2α−1−ε2withε0arbitrary(theweightedℓspspacesaredefinedattheendofIntroduction);see[18]formorepreciseformulations.TheauthorsperformedtheF