Upper bounds for the energy expectation in time de

UpperBoundsfortheEnergyExpectationinTime-DependentQuantumMechanics.AlainJoyeyDepartmentofMathematicsandCenterforTransportTheoryandMathematicalPhysicsVirginiaPolytechnicInstituteandStateUniversityBlacksburg,Virginia24061-0123,U.S.A.April26,1997AbstractWeconsiderquantumsystemsdrivenbyhamiltoniansoftheformH+W(t),wherethespectrumofHconsistsinaninnitesetofbandsandW(t)dependsarbitrarilyontime.LethHi’(t)denotetheexpectationvalueofHwithrespecttotheevolutionattimetofaninitialstate’.WeproveupperboundsofthetypehHi’(t)=O(t),0,underconditionsonthestrengthofW(t)withrespecttoH.NeithergrowthofthegapsbetweenthebandsnorsmoothnessofW(t)arerequired.SimilarestimatesareshownfortheexpectationvalueoffunctionsofH.Sucientconditionstohaveuniformlyboundedexpectationvaluesareexplicitedandtheconsequencesonotherapproachesofquantumstabilityarediscussed.CPT-95/PE.3255Keywords:Quantumstability,energyexpectations,quantumdiusion.SupportedbyFondsNationalSuissedelaRechercheScientique,Grant8220-037200yPermanentaddress:CentredePhysiqueTheorique,CNRSMarseille,LuminyCase907,F-13288Mar-seilleCedex9,FranceandPHYMAT,UniversitedeToulonetduVar,B.P.132,F-83957LaGardeCedex,France11IntroductionConsideratimedependentsystemcharacterizedbyahamiltonianoftheformH+W(t)(1.1)whereHisapositiveselfadjointoperatorwhosespectrumconsistsofseparatedbandsfjg1j=1suchthatj[j;j](1.2)andW(t)isatimedependentsymmetricperturbation.LetU(t)bethecorrespondingevolutionoperatorsatisfyingtheSchrodingerequationiU0(t)’=(H+W(t))U(t)’;U(0)=I;(1.3)wherethe0denotestimederivative.Ourmainconcernisthetimebehaviorast!1oftheexpectationvalueoftheenergyoperatorHhHi’(t)=hU(t)’jHU(t)’i(1.4)andofsimilaroperators.Theseareamongthequantitiesofinterestinthestudyofquantumstabilityforgeneraltimedependentsystems,seee.g.[EV,BJLPN,Ja,JL,dO]andreferencestherein.Suchquantitieshavebeenstudiedanalyticallyfordrivenquantumoscillators[EV,HLS,BJLPN]forvarioustimedependences.Thesolubilityofthequantumproblemanditsstronglinkswiththeclassicaldynamicsofthesystemmakeitpossibletogetaratherprecisedescriptionoftheexpectationvalueofthekineticenergy.Forexample,itcanbededucedfromtheanalysisprovidedin[HLS]thatperiodicallyforcedharmonicoscillatorscanleadtoabehaviorofthetypehHi’(t)’et;0(1.5)forsomeparametersandsomeinitialcondition’a.Forgeneralhamiltoniansoftheform(1.1)witharbitrarytimedependence,theonlyanalyticalresultsweareawareofarethoseofNenciu[N2],whotacklesthisproblembymeansoftoolscomingfromtheadiabatictheory.Theadiabaticmachineryalreadyprovedtobeusefulinthedeterminationofthespectralpropertiesofthemonodromyoperatorincaseofperiodictimedependenceofthehamiltonian,seee.g.[H1,H2,N1,Jo,N2,DS1,DS2].Nenciuconsiderssystemswithincreasinggapsin[N2]andthemainresultregardinghHi’(t)isessentiallythatifthegapsjj1betweenthebandsgrowlikej,with0,andifW(t)isstronglyCnwithnh1+2i+1,thenhHi’(t)=O(t1+n);(1.6)ast!1,providedsupt2R+k(d=dt)kW(t)k1,k=0;1;;n.ThisestimateholdsforarbitrarytimedependenceofW(t)andregardlessofthenatureofthespectruminthebandsj.Thelengthofthebandsmustnotgrowtofasterthanj.NotethatthenecessaryaG.Hagedorn,privatecommunication.2growthofthegapsinthespectrumofHpreventstheapplicationofthisresulttothedrivenharmonicoscillator.InthispaperwealsodealwithH’swhosespectrumconsistsinaninnitesetofbandsandweobtainresultswhichcanbeconsideredascomplementarytothoseofNenciu[N2]inthefollowingsense.Weproveestimatessimilarto(1.6),withoutrestrictiononthesizeofgapsandwithoutsmoothnessassumptiononW(t).Ofcourse,thereisapricetopayfordroppingthesehypotheses:thestrengthofperturbationW(t)withrespecttoHmustbesmallinsomesense.Typically,iftheoperatorW(t)issuchthatsups2R+1Xj=12qjkPjW(s)k21(1.7)forsomeq1=2,then,ast!1,hHi’(t)=O(t1=q);(1.8)seesection2.Hereagain,(1.8)holdsforarbitrarytimedependenceofW(t)andregardlessofthenatureofthespectruminthebandsjprovidedthebandsarenottoolong.SuchalgebraicboundsonthegrowthofhHi’(t)alreadygivesomeinformationonthesystem,see(1.5).However,asnoticedin[N2],forC1perturbationssuchthatbothkW(t)kandkW0(t)kareuniformlyboundedintimewehavethetrivialboundforanyHhHi’(t)=O(t):(1.9)Ifweworkunderstrongerhypothesesmorepreciseestimatesaregiven,someofwhichleadingtouniformlyboundedexpectationvalues.Thisisthecaseforinstanceifweassumeinsteadof(1.7)thatthefunctionofsdenedby1Xj=11=2jkPjW(s)k(1.10)isintegrableonR+.Remarkingthat(1.8)isindependentofthecharacteristicsofthebandsinthespectrum,wecangeneralizeourresultstoestimatetheexpectationvaluesofreasonablepositivefunc-tionsofH,f(H)inthefollowingway,seesection3.Assumeforsimplicitythatf:R+!R+isstrictlyincreasingandthat(1.7)issatisedwithf(j)inplaceofj.Thenhf(H)i’(t)=O(t1=q):(1.11)Also,if(1.10)issatisedwithf(j)inplaceofj,supt2Rhf(H)i’(t)1.Thusitmaybepossibletohavesupt2Rhf(H)i’(t)1foracarefulchoiceoff,whereashHi’(t)cannotbeuniformlyboundedbyourmethods.Aspecicexampledisplayingthesefeaturesisdealtwithinsection4.Suchcasesareofinterestwhenweconsiderotherrelevantquantitiesinthestudyofquantumstability,seesection6.Insection5weconsiderhamiltoniansHwithdiscretespectrums