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1、Commun.Math.Phys.258,741Ð750(2005)CommunicationsinDigitalObjectIdentiÞer(DOI)10.1007/s00220-005-1366-xMathematicalPhysicsSchrödingerOperatorswithFewBoundStatesDavidDamanik1,,RowanKillip2,,BarrySimon1,1Mathematics253Ð37,CaliforniaInstituteofTechnology,Pasadena,CA
2、91125,USA.E-mail:damanik@caltech.edu;bsimon@caltech.edu2DepartmentofMathematics,UniversityofCalifornia,LosAngeles,CA90055,USA.E-mail:killip@math.ucla.eduReceived:16August2004/Accepted:28January2005Publishedonline:2June2005ЩSpringer-Verlag2005Abstract:Weshowthatwhole-
3、lineSchrdingeroperatorswithÞnitelymanyboundstateshavenoembeddedsingularspectrum.Incontradistinction,weshowthatembed-dedsingularspectrumispossibleevenwhentheboundstatesapproachtheessentialspectrumexponentiallyfast.Wealsoprovethefollowingresultforone-andtwo-dimensionalS
4、chrdingeroper-ators,H,withboundedpositivegroundstates:GivenapotentialV,ifbothH±Vareboundedfrombelowbytheground-stateenergyofH,thenV≡0.1.IntroductionThispaperhasitsrootsinthefollowingresultofKillipandSimon[14]:Adiscretewhole-lineSchrdingeroperatorhasspectrum[−2,2]ifand
5、onlyifthepotentialvan-ishesidentically.Tobemoreprecise,givenapotentialV:Z→R,wedeÞnetheSchrdingeroperator[hVφ](n)=φ(n+1)+φ(n−1)+V(n)φ(n)(1)on2(Z).Thetheoremmentionedabove,[14,Theorem8],saysthatσ(hV)⊆[−2,2]impliesV≡0.Asimplevariationalproofofthistheoremwasgivenin[4],wh
6、eretheresultwasalsoextendedtotwodimensions;itdoesnotholdinthreeormoredimensions,noronthehalf-line.Itwasalsoshownin[4]that,forboundedpotentials,theessentialspectrumofhViscontainedin[−2,2]ifandonlyifV→0.Thisshowsthatσess(hV)=[−2,2]inthiscase.D.D.wassupportedinpartbyNSF
7、grantDMSÐ0227289.R.K.wassupportedinpartbyNSFgrantDMSÐ0401277.B.S.wassupportedinpartbyNSFgrantDMSÐ0140592.742D.Damanik,R.Killip,B.SimonDamanikandKillip,[6],investigatedhalf-lineSchrdingeroperatorswithspec-trumcontainedin[−2,2].Byahalf-lineSchrdingeroperatorwemeana
8、noperatoroftheform(1),actingon2(Z+),withφ(−1)=0.Wewilldenotethisoperatorbyh+.Itwasshownthatifσ(h+)=[−2,2],thenh+haspurelyab
