ABSOLUTELY CONTINUOUS SPECTRUM OF STARK OPERATORS

ABSOLUTELYCONTINUOUSSPECTRUMOFSTARKOPERATORSMICHAELCHRISTANDALEXANDERKISELEVAbstrat.Weproveseveralnewresultsontheabsolutelyontinuousspetraofperturbedone-dimensionalStarkoperators.First,wendnewlassesofperturba-tions,haraterizedmainlybysmoothnessonditions,whihpreservepurelyabso-lutelyontinuousspetrum.Thenweestablishstabilityoftheabsolutelyontinuousspetruminmoregeneralsituations,whereimbeddedsingularspetrummayour.Wepresenttwokindsofoptimalonditionsforthestabilityofabsolutelyontinuousspetrum:deayandsmoothness.Inthedeaydiretion,weshowthatasuÆient(inthepowersale)onditionisjq(x)jC(1+jxj)14;inthesmoothnessdire-tion,asuÆientonditioninHolderlassesisq2C12+(R).Ontheotherhand,weshowthatthereexistpotentialswhihbothsatisfyjq(x)jC(1+jxj)14andbelongtoC12(R)forwhihthespetrumbeomespurelysingularonthewholerealaxis,sothattheaboveresultsareoptimalwithinthesalesonsidered.1.IntrodutionInthispaperweonsidertheStarkoperator(1.1)Hq=d2dx2x+q(x)denedonthewholereallineR:Thisoperatordesribesahargedquantumpartileinaonstanteletrieldsubjettoanadditionaleletripotentialq(x):ThereexistsanextensivephysialandmathematialliteratureonStarkoperators;forareview,seee.g.[12℄.Whenq(x)=0;theoperatorhaspurelyabsolutelyontinuousspetrum.Thequestionwewishtoaddressiswhihlassesofperturbationsqpreservethisproperty.Wewillonsidertwolassesofonditionsthatensurepreservationoftheabsolutelyontinuousspetrum:smoothnessanddeay.TherstresultonthesmoothnessonditionwasprovenbyWalter[39℄,whoshowedthatifthepotentialisboundedandhastwoboundedderivatives,thespetrumremainspurelyabsolutelyontinuous.SimilarresultswereobtainedbyBentosela,Carmona,Dulos,Simon,SouillardandWederin[4℄usingMourre’smethod.Aorollarynotedin[4℄isadrastihangeinthespetralpropertiesofShrodingeroperatorsofAndersonmodelDate:January29,2001.TherstauthorwassupportedinpartbyNSFgrantDMS-9970660andompletedthisresearhwhileonappointmentasaMillerResearhProfessorintheMillerInstituteforBasiResearhinSiene.TheseondauthorwassupportedinpartbyNSFgrantDMS-9801530.12MICHAELCHRISTANDALEXANDERKISELEVtype,sayH!=d2dx2+Xnan(!)V(xn)whereanareindependentidentiallydistributedrandomvariablesandV2C20((0;1));whenaonstanteletrieldisswithedon.Thespetrumhangesfromalmostsurelypurepointtopurelyabsolutelyontinuous.Reently,Sahbani[31,32℄relaxedsmoothnessonditionsof[39℄and[4℄(seetheremarkafterTheorem1.6).Ontheoppositesideofthesmoothnesssale,Delyon,SimonandSouillard[15℄showedthatforaperiodiarrayofÆfuntionpotentialswithrandomouplingsinaonstanteletrield,thespetrumispurelysingular.Avron,ExnerandLast[3℄realizedthatthespetrummaybepurelysingularevenforadeterministiperiodiarrayofverysingularinterations,suhasÆ0:Generalizationsoftheseresults,aswellasothermodelswithsingularpotentials,wereonsideredin[25,16,1,24,5,2℄.Thereremainedagap,however,betweenthelassesofpotentialsforwhihloalizationwasknowntoour,andthoseforwhihthespetrumwasknowntoremainabsolutelyontinuous.Asfarasdeayonditionsareonerned,itiswellknownthatifq(x)satisesjq(x)jC(1+jxj);1=2;thenthespetrumremainspurelyabsolutelyontin-uous[37℄.Moreover,thereareexampleswherejq(x)x1=2jCandisolatedimbeddedeigenvaluesappear.Ifjq(x)jx1=2!1;itwasshownbyNabokoandPushnitski[26℄thatdense(imbedded)pointspetrummayappearonallofR.Weremarkthatfortheoperatorwithouteletrield,thedeaythresholdwhereimbeddedeigenvaluesmayappearisthepower1;ofourse,itisphysiallynaturalthatitismorediÆulttogetanimbeddedeigenvalueinthepreseneoftheonstanteletrield.However,ifwedonotwishtoruleoutimbeddedsingularspetrum,ithasbeenshownin[19℄thattheabsolutelyontinuousspetrumofaperturbedStarkoperatorstillllsthewholerealaxiswhenjq(x)jC(1+jxj);1=3:Thequestionwhatistheritialrateofdeayforwhihthespetrummaybeomepurelysingularremainedopen.OurmaingoalinthispaperistoprovetwosharpresultsonthepreservationoftheabsolutelyontinuousspetrumofStarkoperators.Reallthatf(x)isalledHolderontinuouswithexponent(f2C(R))ifkfkC=supxjf(x)j+supx;yjf(x)f(y)jjxyj1:Wewillanalyzesolutionsofthegeneralizedeigenfuntionequation(1.2)u00xu+q(x)u=Eu:Herexrepresentsabakgroundpotentialduetoaonstanteletrialeld,whileqissomeperturbation.Theorem1.1.Assumethatthepotentialq(x)isHolderontinuouswithexponent1=2:Thenanessentialsupportoftheabsolutelyontinuouspartofthespetralmeasureoinideswiththewholerealaxis.Moreover,fora.e.E;allsolutionsu(x;E)ofequation(1.2)satisfyu(x;E)=O(x1=4);u0(x;E)=O(x1=4)asx!+1:ABSOLUTELYCONTINUOUSSPECTRUMOFSTARKOPERATORS3Remark.1.AnessentialsupportofisasetSsuhthat(RnS)=0and(S1)0foranyS1SofpositiveLebesguemeasure.2.Inthisandsubsequenttheorems,onlybehaviorofq(x)forjxjlargematters.Wewillalwaysimpliitlyassumeqtobeloallyintegrable,andwillstateonlyadditionalhypotheseswhihonernitsbehaviorforlargex.Onthenegativepartoftherealaxis,itissuÆientforallouronlusionstorequirethatq(x)x!+1asx!1.Weprefertostatetheresultsinaslightlyweakerformtoavoidmakingstatementstooumbersome.Theorem1.2.Assumethatthepotentialq(x)isloallyintegrable,andthatq(x2)2Lpforsome1p2:Thenanessentialsupportoftheabsolutelyontinuouspartofthespetralmeasureoinideswiththewholerealaxis.M