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1、ξ-ζrelationM.Krishna∗InstituteofMathematicalSciencesTaramani,Chennai600113,IndiaE-mail:krishna@imsc.ernet.in7January1999AbstractInthisnoteweprovearelationbetweentheRiemannZetafunc-tion,ζandtheξfunction(Kreinspectralshift)associatedwiththeHarmonicOscillatorinonedimension.Thisgivesanewin
2、tegralrepre-sentationofthezetafunctionandalsoareformulationoftheRiemannhypothesisasaquestioninL1(R).∗PartoftalkpresentedattheConferenceonHarmonicAnalysis,13-15March1997,RamanujanInstitute,UniversityofMadras,Chennai11IntroductionInversespectraltheoryinonedimensioninvolvesrecoveringaSchr
3、¨odingeroperatorfromtheknowledgeofspectrumandaspectralfunctionasdonebyGelfand-Levitaninthefifties.Intherecentyearsthereisagreatdealofprogressachievedinparametrizingiso-spectralclassesofpotentials(seethereviewsofSimon[13],Gesztesy[4],andthepapersofLevitan[7],Kotani-Krishna[9],Craig[1]and
4、Sodin-Yuditskii[14]forSchr¨odingeroperators).OneoftheconsequencesofageneralformulationobtainedviausingtheKreinspectralshiftfunctionbyGesztesy-Simon[5]isgiveninthispaper.TheRiemannzetafunctionisawellstudiedobject,forexample,Titch-march[15]givesadetailedexpositionofthisfunction.Thereares
5、everalexpressionsforζ,andinthisnotewepresentanintegralrepresentationforζ,thatcomesfromtheKreinspectralshiftformulaofKrein[10,11].RecentlyGesztesy-Simon[5]generalizedthetraceformulaeforSchr¨odingeroperatorsusingtheKreinspectralshiftfunction,whichtheynamedtheξfunction,asitiscentraltoinve
6、rsespectraltheoriesinonedimensionandhadseveralim-portantapplicationsinspectraltheoriesofoperatorsinonedimension.ThisworkusedtheproofoftheKreinformula,giveninSimon[13],theoremI.10anditsgeneralizations.AproofoftheformulaforaslightlylargerclassisshowninMohapatra-Sinha[8].Werefertothesepap
7、ersforthehistoryandotherworkontheKreinspectralshiftfunction.FinallywenotethatthereformulationweobtainfortheRiemannhy-pothesisisaclosureprobleminthespaceL1(R).Though,viathepowerfulWiener’stheorem,theverificationonlyrequiresexhibitingasinglefunctiongσ(foreachσ∈(1/2,1))tolieinanexplicits
