Absolutely continuous spectrum of perturbed Stark

AbsolutelycontinuousspectrumofperturbedStarkoperatorsAlexanderKiselevMathematicalSciencesResearchInstitute1000CentennialDriveBerkeleyCA94720AbstractWeprovenewresultsonthestabilityoftheabsolutelycontinuousspectrumforperturbedStarkoperatorswithdecayingorsatisfyingcertainsmoothnessassumptionperturbation.Weshowthattheab-solutelycontinuousspectrumoftheStarkoperatorisstableiftheperturbingpotentialdecaysattherate(1+x)13orifitiscontinu-ouslydierentiablewithderivativefromtheHolderspaceC(R);withany0:0.IntroductionInthispaper,westudythestabilityoftheabsolutelycontinuousspec-trumofone-dimensionalStarkoperatorsundervariousclassesofperturba-tions.StarkSchrodingeroperatorsdescribebehaviorofthechargedparticleintheconstantelectriceld.Theabsolutelycontinuousspectrumisaman-ifestationofthefactthattheparticledescribedbytheoperatorpropagatestoinnityataratherfastrate(see,e.g.[1],[11]).Itisthereforeinterestingtodescribetheclassesofperturbationswhichpreservetheabsolutelycon-tinuousspectrumoftheStarkoperators.Intherstpartofthiswork,westudyperturbationsofStarkoperatorsbydecayingpotetnials.ThispartisinspiredbytherecentworkofNabokoandPushnitski[13].ThegeneralpicturethatweproveisverysimilartothecaseofperturbationsoffreeSchrodingeroperators[8].Inaccordancewithphysicalintuition,however,theabsolutelycontinuousspectrumisstableunderstrongerperturbationsthaninthefreecase.Ifinthefreecasetheshortrangepotentialspreserv-ingpurelyabsolutelycontinuousspectrumofthefreeoperatoraregivenbycondition(onthepowerscale)jq(x)jC(1+jxj)1;intheStarkopera-torcasethecorrespondingconditionreadsjq(x)jC(1+jxj)12:Ifisallowedtobezerointheabovebounds,imbeddedeigenvaluesmayoccurinbothcases(see,e.g.[13],[14]).Moreover,inbothcasesifweallowpotential1todecayslowerbyanarbitraryfunctiongrowingtoinnity,veryrichsingu-larspectrum,suchasadensesetofeigenvalues,mayoccur(see[12]forthefreecaseandeNaPufortheStarkcaseforpreciseformulationandproofsoftheseresults).Therstpartofthisworkdrawsthepaprallelfurther,show-ingthattheabsolutelycontinuousspectrumofStarkoperatorsispreservedunderperturbationssatisfyingjq(x)jC(1+jxj)13;inparticularevenintheregimeswhereadensesetofeignevaluesoccurs;henceinsuchcasestheseeigenvaluesaregenuinelyimbedded.Similarresultsforthefreecasewereprovenin[8],[9].Ourmainstrategyoftheproofhereissimilartothatin[8]and[9]:westudytheasymptoticsofthegeneralizedeigenfunctionsandthenapplyGilbert-Pearsontheory[6]toderivespectralconsequences.WhilethemainnewtoolweintroduceinourtreatmentofStarkoperatorsisthesameasinthefreecase,namelythea.e.convergenceoftheFourier-typeintegraloperators,therearesomemajordierences.Firstofall,thespectralparameterentersthenalequationsthatwestudyinadierentwayandthismakesanalysismorecomplicated.Secondly,weemployadierentmethodtoanalyzetheasymptotics.InsteadofHarris-LutzasymptoticmethodwestudyappropriatePrufertransformvariables,simplifyingtheoverallconsideration.Inthesecondpartoftheworkwediscussperturbationsbypotentialshavingsomeadditionalsmoothnessproperties,butwithoutdecay.ItturnsoutthatforStarkoperatorstheeectsofdecayorofadditionalsmoothnessofpotentialonthespectralpropertiesaresomewhatsimilar.ItwasknownforalongtimethatifapotentialperturbingStarkoperatorhastwoboundedderivativesthespectrumremainspurelyabsolutelycontinuous(actually,cer-taingrowthofderivativesisalsoallowed,seeSection3fordetailsorWalter[19]fortheoriginalresult).WenotethattheresultssimilartoWalter’sonthepreservationonabsolutelycontinuousspectrumwerealsoobtainedin[3]byapplyingdierenttypeoftechnique(Mourremethodinsteadofstudyingasymptoticsofsolutions).Ontheotherhand,iftheperturbingpotentialisasequenceofderivativesoffunctionsinintegerpointsonRwithcertaincouplings,thespectrummayturnpurepoint[2],[4].Insomesense,the0interactionisthemostsingularandleast\dierentiableamongallavailablenaturalperturbationsofone-dimensionalSchrodingeroperators[10].Hencewehaveverydierentspectralpropertiesontheveryoppositesidesofthesmoothnessscale.Thisworkclosesapartofthegap.Weimprovethewell-knownresultsofWalter[19]concerningtheminimalsmoothnessrequiredforthepreservationoftheabsolutelycontinuousspectrumandshowthatinfact2existenceandminimalsmoothnessoftherstderivativeissucienttoimplyabsolutecontinuityofthespectrum.1.Decayingperturbations.Consideraself-adjointoperatorHqdenedbythedierentialexpressionHqu=u00xu+q(x)uontheL2(1;1):Letusintroducesomenotation.Forthefunctionf2L2wedenotebyfitsFouriertransform:f(k)=L2limN!1NZNexp(ikx)f(x)dx:ForlocallyintegrablefunctiongwedenotebyM+gthefunctionM+g(x)=sup1h012hhZ0jg(x+t)+g(xt)jdt:WedenotebyM+fthesetwhereM+fisnite.Bythegeneralresultsonthemaximalfunctions(see,e.g.[15]),itiseasytoconcludethatthecomple-mentofthesetM+hasLebesguemeasurezero.Wewillprovethefollowingtheorem:Theorem1.1Supposethatthepotentialq(x)satisesjq(x)jC(1+x)13;0:Thentheabsolutelycontinuousspectrumofmultiplicityonellsthewholerealaxis.Imbeddedsingularspectrummayoccur,butonlyonthecom-plementofthesetS=M+expix3iZx30q(ct23)ct23dt!q(cx2)x16!!;wherec=(32)23:Wewillprov