The absolutely continuous spectrum of Jacobi matri

THEABSOLUTELYCONTINUOUSSPECTRUMOFJACOBIMATRICESCHRISTIANREMLINGAbstract.IexploresomeconsequencesofagroundbreakingresultofBrei-messerandPearsonontheabsolutelycontinuousspectrumofone-dimensionalSchr¨odingeroperators.TheseincludeanOracleTheoremthatpredictsthepotentialandrathergeneralresultsontheapproachtocertainlimitpotentials.Inparticular,weproveaDenisov-Rakhmanovtypetheoremforthegeneralfinitegapcase.Themainthemeisthefollowing:Itisextremelydifficulttoproduceabso-lutelycontinuousspectruminonespacedimensionandthusitsexistencehasstrongimplications.1.Introductionandstatementofmainresults1.1.Introduction.Thispaperdealswithone-dimensionaldiscreteSchr¨odingeroperatorsonℓ2,(1.1)(Hu)(n)=u(n+1)+u(n−1)+V(n)u(n),withsomeabsolutelycontinuousspectrum.WewillalsoconsiderJacobimatrices,(Ju)(n)=a(n)u(n+1)+a(n−1)u(n−1)+b(n)u(n);theseofcourseinclude(1.1)asthespecialcasea(n)=1.ThepurposeofthispaperistoexploreastunningresultofBreimesserandPear-son[3,4](whichseemstohavegonealmostunnoticed).IwillgiveareformulationinTheorem1.4below,which(Ibelieve)shouldhelptoclarifythesignificanceofthebrilliantworkofBreimesser-Pearson.Infact,itseemstomethat[3,4]revealnewfundamentalpropertiesoftheabsolutelycontinuousspectrum.Thesituationisperhapsreminiscentofthereevaluationofthesingularcontinuousspectrumsometenyearsago(showntobeubiquitous,contrarytothencommonpopularbelief)[7,15,16,17,31,41].Asisverywellknown,theabsolutelycontinuousspectrumisthatpartofthespectrumthathasthebeststabilitypropertiesundersmallperturbations.Oncethisisadmitted,itturnsoutthatitisextremelydifficulttoproduceabsolutelycontinuousspectruminonespacedimensioninanyotherway(otherthanasmallperturbationofoneofthefewknownexamples).Thisisthemainmessageofthispaper.(Iusedtobelievetheexactopposite:absolutelycontinuousspectrumiswhatyounormallygetunlesssomethingspecialhappens,butthisnowturnsouttobeagrossmisinterpretation.)Date:June7,2007.Keywordsandphrases.Absolutelycontinuousspectrum,Jacobimatrix,reflectionlesspotential.12CHRISTIANREMLING1.2.Commentonnotation.Wewilldiscusstheseissuesinmoredetailinamo-ment,butletmefirstintroduceanotationalconventionthatwillbeusedthroughoutthispaper.Everythingwedohereworksinthegeneral(Jacobi)setting,buttheneedtodealwithtwosequencesofcoefficientsa(n),b(n)oftenmakesthenotationawkward.SoitmightseemwisetoonlydealwiththeSchr¨odingercase,butthisisnotanidealsolutioneitherbecausesometimesthegreatergeneralityoftheJacobisettingisessential.Ihavedecidedonaperhapssomewhatunusualremedyagainstthispredicament:SinceusuallytheextensiontotheJacobicaseisobvious,IwillsimplyworkintheSchr¨odingeroperatorsettingmostofthetime.Occasionally,IwillhavetoswitchtotheJacobicase,though.Forexample,Sections5,6,7don’tmakemuchsensewithoutthisgenerality,andinSect.3,it’snottotallyclearhowtoincorporatethea(n)’s.However,IwillusuallyquicklyswitchbacktotheSchr¨odingernotationwhenfeasible.Ihopethatthisleadstoamoreeasilyreadablepresentationwithoutconfusingthereadertoomuch.Ifnecessary,IwillalsouseallpreviousresultsasiftheyhadbeenprovedforJacobioperators.Inotherwords,everythinginthispaperis(atleastimplicitly)assertedforthegeneralJacobicase.Sotherearetwoextremewaysofreadingthispaper:(1)Schr¨odingerreader:specializetoa(n)=1andidentifyb(n)=V(n)when-everyouseecoefficientsa(n),b(n);(2)Jacobireader:replaceV(n)with(a(n),b(n))throughoutandmakeotherad-justmentsasnecessary(frequently,nosuchadditionaladjustmentsarenecessary).SomewhatmoredetailedinstructionsfortheJacobireaderwillbegivenaswego.1.3.TheOracleTheorem.ThebasicresultofthispaperisTheorem1.4below,butletmebeginthediscussionbymentioningtwoconsequencesthatareparticu-larlyaccessible:Theorem1.1.Supposethatthe(halfline)potentialV(n)takesonlyfinitelymanyvaluesandσac6=∅.ThenViseventuallyperiodic:Thereexistn0,p∈NsothatV(n+p)=V(n)foralln≥n0.Forergodicpotentials,thisisawellknownTheoremofKotani[20].Thatitholdsforarbitraryoperatorscameasamildsurprise,atleasttome.(Recallalsothatbyourgeneralconvention,thesamestatementholdsforJacobioperators:ifσac(J)6=∅,theneventuallya(n+p)=a(n),b(n+p)=b(n)forsomep∈N.)Theorem1.1isaconsequenceofthefollowingmoregeneralresult,whichsaysthatthereareuniversaloraclesthatwillpredictfuturevaluesofpotentialswithsomeabsolutelycontinuousspectrumwithanydesiredaccuracy,basedon(partial)informationonpastvalues.Theorem1.2(TheOracleTheorem).LetC0,ǫ0,andletA⊂RbeaBorelsetofpositiveLebesguemeasure.ThenthereexistL∈NandasmoothfunctionΔ:[−C,C]L+1→[−C,C](theoracle),suchthatthefollowingholds:Forany(halfline)potentialVwithkVk∞≤CandΣac(V)⊃A,thereexistsn0∈Nsothatforalln≥n0,|V(n+1)−Δ(V(n−L),V(n−L+1),...,V(n))|ǫ.Here,weusethesymbolΣactodenoteanessentialsupportoftheabsolutelycon-tinuouspartofthespectralmeasure.Inotherwords,themeasuresdρacandχΣacdtABSOLUTELYCONTINUOUSSPECTRUM3havethesamenullsets.ThisconditiondeterminesΣacuptosetsof(Lebesgue)measurezero.Theabsolutelycontinuousspectrum,σac,istheessentialclosureofΣac.Jacobireader:InterprettheassumptionthatkVk∞≤CasV∈VC+;thiswillbeexplainedinmoredetailbelow.Theoraclewillnowpredict(a(n+1),b(n+1)),asafunctionof(a(j),b(j))forn−L≤j≤n.Notethaton