ESTIMATES OF GLOBAL BOUNDS FOR SOME

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1、ILLINOISJOURNALOFMATHEMATICSVolume44,Number3,2000ESTIMATESOFGLOBALBOUNDSFORSOMESCHRODINGERHEATKERNELSONMANIFOLDSQIS.ZHANGANDZ.ZHAOABSTRACT.WeestablishglobalboundsfortheheatkernelofSchr6dingeroperators--A/VwhereVisacertainlongrangepotential.Asaconsequencewefin

2、dsomeconditionsfortheheatkerneltohaveglobalGaussianlowerandupperbound.Someoftheconditionsaresharpifthepotentialdoesnotchangesign.WealsoprovideageneralizedLiouvilletheoremforSchr6dingeroperatorsandarefinedversionofthetraceformulaofSaBarretoandZworski[SZ].1.Int

3、roductionAfundamentalresultprovedin[A]statesthatthefundamentalsolutionofasecondorderuniformlyparabolicequationindivergenceformhasGaussianlowerandupperbounds.However,ingeneral,theseboundsarenotglobalintimesincetheparametersintheseboundsdepend,inanimplicitmanne

4、r,onthelowerordertermsoftheequation.Thefollowingexampleillustratestheneedforabetterunderstandingofthebounds.Bystandardestimates,thefundamentalsolutionofA+VtwithVLsatisfies-Ilvllttx-rcnellVlltCnee4,

5、ande-IIVIltmasksawealthofinformationandmakestheboundslessusefulwhenAnimportantquestionarises:Doesthereexistaglobalestimateontheheatkernelof-A+V,whichrevealsanexplicitdependenceonthepotentialV?Manyauthorshavestudiedtheabovetheproblem.Wereferthereaderto[Sil],[M

6、],[DS],INS],[LY],[N],[SZ],[Se],[Zg2],[Zo2]andthepapersquotedthere.Letussketchtwointerestingrecentresultsin[SZ]and[Se].InthemainLemma3.2in[SZ],SaBarretoandZworskiprovedthattheheatkernelof-A+VhasglobalGaussianupperboundprovidedthatVhassuperexponentialdecayand-A

7、+Vhasnonegativeeigenvalueand0isnotapoleoftheresolvent.Ontheotherhand,Semenov[Se]provedthat-A+VwithV>0hasaglobalGaussianlowerboundifandonlyifA-VLC(Rn).AsindicatedbelowinRemark1.1,thisclassoffunctions,alsocalledGreenboundfunctions,belongtoshortrangepotentialssi

8、ncethesefunctionsessentiallydecayfasterthan1/Ixlnearinfinity.ReceivedFebruary19,1999;receivedinfinalformAugust12,1999.1991MathematicsSubjectClassification.Primary35K10,35J10;Secondary58G1