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1、PROPERTIESOFTHEGREEN'SFUNCTIONSOFSOMESCHRODINGEROPERATORSE.B.DAVIESWereviewandre-derivesomepropertiesoftheeigenfunctionsandGreen'snfunctionforafairlygeneralclassofSchrodingeroperatorsonR.TheSchrodingeroperatoristakentobeoftheformH=-A/2+V,whereVisanon-negativelocallyIfpot
2、entialforsomep>n/2.Weprovethattheeigenfunctionsarecontinuousandobtainauniformboundonanyeigenfunctionwhichdependsonlyontheeigenvalue,andwhichwebelievetobebestpossible.Weprove,foralargeclassofdiscretespectrumSchrodingeroperators,thattheGreen'sfunctioniscontinuousandistheun
3、iformlimitofitseigenfunctionexpansion.Bywayofcomparisonwith[9],ourproofsmakenouseofWienerspaceintegrationandarenomorethanelementaryfunctionalanalysis.Wehavemadeanefforttogiveaself-containedanalysis,subject,ofcourse,toanassumptionoffamiliaritywiththebookofT.Kato[17].Theau
4、thorshouldliketoexpresshisthankstoProfessorsBazleyandSimonforvaluablereferencestotheliterature.1.Somefunctionalanalysis2LetJP=L(ft,dco),where(Cl,dco)isaa-finitemeasurespace.Wesaythatfel3(Cl)ispositive,/5»0,if/(cw)^0almosteverywhere.ThesetofpositiveelementsinJfisaconeclos
5、edinthenormandweaktopologies.WesaythatanoperatorAe$£(^f)ispositive,A>0,ifAf^0forall/^0.Then{Ae<£{&):A>0}isacone,closedundermultiplication,adjointsandweakoperatorlimits.MoreoverA>0ifandonlyif{Af,g)^0forall/,g^0.IfAhasanintegralkernelitiseasytoshowthatA^>0ifandonlyiftheker
6、nelisnon-negativealmosteverywhereonftxfl;thisappliesinparticulartoHilbert-Schmidtoperators.WecontinuethegeneralconventionofusingA^0tomean(Af,f)>0forallfe3>A,thedomainofA.WeletA~denotetheclosureofasymmetricoperatorA.LEMMA1.1.SupposethatA,B>0areself-adjointandA+Bisessentia
7、llyself-tAadjointonSiAnQ)B.Ife~>0forallt^0andBisamultiplicationoperatoron2L(Q)thenforallt>0Proof.ThisisanimmediateconsequenceoftheTrotterproductformula[5,27]e-t(A+B)~=s.]im(e-tAlne-tB/ny^^n-»ooReceived11July,1972.[J.LONDONMATH.SOC.(2),7(1973),483-491]484E.B.DAVIEStogethe
8、rwiththeobservationthatforallt^0(1.3)LEMMA1.2.IfA,BareboundedoperatorsandA>B>0thenMil£11*11;IMIIH.S.>\
