178S.NonnenmacherSeminairePoincare - PDF document

178S.NonnenmacherSeminairePoincare Figure1:TwoeigenfunctionsoftheDirichletLaplacianinthestadiumbilliard,withwavevectorsk=60:196andk=60:220(see(9)).Largevaluesofj (x)j2correspondtodarkregions,whilenodallinesarewhite.Whilethelefteigenfunctionlooksrelatively\ergodic",therightoneisscarredbytwosymmetricperiodicorbits(seex4.2).Thesenotesareorganizedasfollows.Weintroduceinsection2theclassicaldynamicalsystemswewillfocuson(mostlygeodesic owsandmapsonthe2-dimensionaltorus),mentioningtheirdegreeof\chaos".Wealsosketchthequantiza-tionproceduresleadingtoquantumHamiltoniansorpropagators,whoseeigenstateswewanttounderstand.Wealsomentionsomepropertiesofthesemiclassical/high-frequencylimit.Insection3wedescribethemacroscopicpropertiesoftheeigen-statesinthesemiclassicallimit,embodiedbytheirsemiclassicalmeasures.Thesepropertiesincludethequantumergodicityproperty,whichforsomesystems(witharithmeticsymmetries)canbeimprovedtoquantumuniqueergodicity,namelythefactthatallhigh-frequencyeigenstatesare\ at"atthemacroscopicscale;ontheopposite,somespecicsystemsallowthepresenceofexceptionallylocalizedeigen-states.Insection4wefocusonmorerenedpropertiesoftheeigenstates,manyofstatisticalnature(valuedistribution,correlationfunctions).Verylittleisknownrigorously,soonehastoresorttomodelsofrandomwavefunctionstodescribethesestatisticalproperties.ThelargevaluesofthewavefunctionsorHusimidensitiesarediscussed,includingthescarphenomenon.Section5discussesthemost\quantum"ormicroscopicaspectoftheeigenstates,namelytheirnodalsets,bothinpositionandphasespace(Husimi)representations.Hereaswell,therandomstatemodelsarehelpful,andleadtointerestingquestionsinprobabilitytheory.2Whatisaquantumchaoticeigenstate?Inthissectionwerstpresentageneraldenitionofthenotionof\chaoticeigen-states".Wethenfocusourattentiontogeodesic owsonEuclideandomainsoroncompactriemannianmanifolds,whichformthesimplestsystemsprovedtobechaotic.Finallywepresentsomediscretetimedynamics(chaoticcanonicalmapsonthe2-dimensionaltorus).2.1AshortreviewofquantummechanicsLetusstartbyrecallingthatclassicalmechanicsonthephasespaceTRdcanbedened,intheHamiltonianformalism,byarealvaluedfunctionH(x;p)onthatphasespace,calledtheHamiltonian.Wewillalwaysassumethesystemtobeautonomous,namelythefunctionHtobeindependentoftime.Thisfunctionthen Vol.XIV,2010Anatomyofquantumchaoticeigenstates179generatesthe ow2(x(t);p(t))=tH(x(0);p(0));t2R;bysolvingHamilton'sequations:_xj(t)=@H @pj(x(t);p(t));_pj=�@H @xj(x(t);p(t)):(1)This owpreservesthesymplecticformPjdpj^dxj,andtheenergyshellsEE=H�1(E).Thecorrespondingquantummechanicalsystemisdenedbyanoperator^H~actingonthe(quantum)HilbertspaceH=L2(Rd;dx).Thisoperatorcanbefor-mallyobtainedbyreplacingcoordinatesx;pbyoperators:^H~=H(^x~;^p~);(2)where^x~istheoperatorofmultiplicationbyx,whilethemomentumoperator^p~=~ ir,isconjugateto^xthroughthe~-FouriertransformF~.Thenotation(2)assumesthatonehasselectedacertainorderingbetweentheoperators^x~and^p~;inphysicsoneusuallychoosesthefullysymmetricordering,alsocalledtheWeylquantization:ithastheadvantagetomake^H~aself-adjointoperatoronL2(Rd).QuantizationprocedurescanalsobedenedwhentheEuclideanspaceRdisreplacedbyacompactmanifoldM.Wewillnotdescribeitinanydetail.Thequantumdynamics,whichgovernstheevolutionofthewavefunction (t)2Hdescribingthesystem,isthengivenbytheSchrodingerequation:i~@ (x;t) @t=[^H~ ](x;t):(3)Solvingthislinearequationproducesthepropagator,thatisthefamilyofunitaryoperatorsonL2(Rd),Ut~=exp(�i^H~=~);t2R:Remark1Inphysicalsystems,Planck'sconstant~isaxednumber,whichisoforder10�34inSIunits.However,ifthesystem(atom,molecule,\quantumdot")isitselfmicroscopic,thevalueof~maybecomparablewiththetypicalactionofthesystem,inwhichcaseitismorenaturaltoselectunitsinwhich~=1.Ourpointofviewthroughoutthisworkwillbetheopposite:wewillassumethat~is(very)smallcomparedwiththetypicalactionofthesystem,andmanyresultswillbevalidasymptotically,inthesemiclassicallimit~!0.2.2Quantum-classicalcorrespondenceAtthispoint,letusintroducethecrucialsemiclassicalpropertyofthequantumevo-lution:itiscalled(inthephysicsliterature)thequantum-classicalcorrespondence,whileinmathematicsthisresultisknownasEgorov'stheorem.Thispropertystatesthattheevolutionofobservablesapproximatelycommuteswiththeirquantization.Forus,anobservableisasmooth,compactlysupportedfunctiononphasespace 2Wealwaysassumethatthe owiscomplete,thatisitdoesnotblowupinnitetime. Vol.XIV,2010Anatomyofquantumchaoticeigenstates181quantumsystem,isthatmosteigenstatesareeitherlocalizedintheregularregion,orinthechaoticsea[80].Tomyknowledgethisconjectureremainsfullyopenatpresent,inpartduetoourlackofunderstandingoftheclassicaldynamics.Forthisreason,Iwillrestrictmyself(asmostresearchesinquantumchaosdo)tothecaseofsystemsadmittingapurelychaoticdynamicsonEE.Iwillallowvariousdegreesofchaos,theminimalassumptionbeingtheergodicityofthe owtHonEE,withrespecttothenatural(Liouville)measureonEE.Thisassumptionmeansthat,foralmostanyinitialpositionx02EE,thetimeaveragesofanyobservablefconvergetoitsphasespaceaverage:limT!11 2TZT�Tf(t(x0))dt=ZEEf(x)dL(x)def=Zf(x)(H(x)�E)dx:(6)Astrongerchaoticpropertyisthemixingproperty,ordecayoftimecorrelationsbetweentwoobservablesf;g:Cf;g(t)def=ZEEg(ft)dL�ZfdLZgdLt!1���!0:(7)Therateofmixingdependsonboththe owtandtheregularityoftheobservablesf;g.Forverychaotic ows(Anosov ows,seex2.4.2)andsmoothobservables,thedecayisexponential.2.4GeometricquantumchaosInthissectionwegiveexplicitexamplesofchaotic ows,namelygeodesic owsinaEuclideanbilliard,oronacompactmanifold.Thedynamicsistheninducedbythegeometry,ratherthanapotential.Boththeclassicalandquantumpropertiesofthesesystemshavebeeninvestigatedalotinthepast30years.2.4.1BilliardsThesimplestformofergodicsystemoccurswhenthepotentialV(x)isaninnitebarrierdelimitingaboundeddomain Rd(say,withpiecewisesmoothboundary),sothattheparticlemovesfreelyinside andbouncesspecularlyattheboundaries.Forobviousreasons,suchasystemiscalledaEuclideanbilliard.Allpositiveenergyshellsareequivalenttooneanother,uptoarescalingofthevelocity,sowemayrestrictourattentiontotheshellE=f(x;p);x2 ;jpj=1g.Thelongtimedynamicalpropertiesonlydependontheshapeofthedomain.Forinstance,in2dimensions,arectangular,circularorellipticbilliardsleadtoanintegrabledynamics:the owadmitstwoindependentintegralsofmotion|inthecaseofthecircle,theenergyandtheangularmomentum.Aconvexbilliardwithasmoothboundarywillalwaysadmitsomestable\whisperinggallery"stableorbits.Ontheopposite,thefamousstadiumbilliard(seeFig.2)wasprovedtobeergodicbyBunimovich[32].Historically,therstEuclideanbilliardprovedtobeergodicwastheSinaibilliard,composedofoneorseveralcircularobstaclesinsideasquare(ortorus)[91].ThesebilliardsalsohavepositiveLyapunovexponents(meaningthatalmostalltrajectoriesareexponentiallyunstable,seetheleftpartofFig.2).Ithasbeenshownmorerecentlythatthesebilliardsaremixing,butwithcorrelationsdecayingatpolynomialorsubexponentialrates[34,14,73]. Vol.XIV,2010Anatomyofquantumchaoticeigenstates183E�x.Thestable(resp.unstable)subspaceisdenedbythepropertythatthe owcontractsvectorsexponentiallyinthefuture(resp.inthepast),seeFig.3:8v2Ex;8t�0;kdtx
vkCe�tkvk:(10)Anosovsystemspresentthestrongestformofchaos,buttheirergodicpropertiesare(paradoxically)betterunderstoodthanforthebilliardsoftheprevioussection.The owhasapositivecomplexity,re ectedintheexponentialproliferationoflongperiodicgeodesics.Forthisreason,thisgeometricmodelhasbeenatthecenterofthemathematicalinvestigationsofquantumchaos,inspiteofitsminorphysicalrelevance.Generalizing Figure3:Topleft:geodesic owonasurfaceofnegativecurvature(suchasurfacehaveagenus2).Right:fundamentaldomainforan\octagon"surface�nHofconstantnegativecurvature(thegureisduetoC.McMullen)Bottonleft:aphasespacetrajectoryandtwonearbytrajectoriesapproachingitinthefutureorpast.Thestable/unstabledirectionsatxandt(x)areshown.Theredlinesfeaturetheexpansionalongtheunstabledirection,measuredbytheunstableJacobianJut(x)=det(dtE+x).thecaseoftheEuclideanbilliards,thequantizationofthegeodesic owon(M�g)isgivenbythe(semiclassical)Laplace-Beltramioperator,^H~=�~2
g 2;(11)actingontheHilbertspaceisL2(M;dx)associatedwiththeLebesguemeasure.Theeigenstatesof^H~witheigenvalues1=2(equivalently,thehigh-frequencyeigen-statesof�
g)constituteaclassofquantumchaoticeigenstates,whosestudyisnotimpededbyboundaryproblemspresentinbilliards.ThespectralpropertiesofthisLaplacianhaveinterestedmathematicianswork-inginriemanniangeometry,PDEs,analyticnumbertheory,representationtheory,foratleastacentury,whilethespecic\quantumchaotic"aspectshaveemergedonlyinthelast30years.TherstexampleofamanifoldwithnegativecurvatureisthePoincarehalf-space(ordisk)Hwithitshyperbolicmetricdx2+dy2 y2,onwhichthegroupSL2(R) 184S.NonnenmacherSeminairePoincareactsisometricallybyMoebiustransformations.Forcertaindiscretesubgroups�ofPSL2(R)(calledco-compactlattices),thequotientM=�nHisasmoothcompactsurface.ThisgroupstructureprovidesdetailedinformationonthespectrumoftheLaplacian(forinstance,theSelbergtraceformulaexplicitlyconnectsthespectrumwiththeperiodicgeodesicsofthegeodesic ow).Furthermore,forsomeofthesediscretesubgroups�,(calledarithmetic),onecanconstructacommutativealgebraofHeckeoperatorsonL2(M),whichalsocommutewiththeLaplacian;itthenmakesensetostudyinprioritythejointeigenstatesof
andoftheseHeckeoperators,whichwewillcalltheHeckeeigenstates.Thisarith-meticstructureprovidesnontrivialinformationontheseeigenstates(seex3.3.3),sotheseeigenstateswillappearseveraltimesalongthesenotes.Theirstudycomposesapartofarithmeticquantumchaos,alivelyeldofresearch.2.5ClassicalandquantumchaoticmapsBesidetheHamiltonianorgeodesic ows,anothermodelsystemhasattractedmuchattentioninthedynamicalsystemscommunity:chaoticmapsonsomecompactphasespaceP.Insteadofa ow,thedynamicsisgivenbyadiscretetimetransfor-mation:P!P.Becausewewanttoquantizethesemaps,werequirethephasespacePtohaveasymplecticstructure,andthemaptopreservethisstructure(inotherwords,isaninvertiblecanonicaltransformationonP.Theadvantageofstudyingmapsinsteadof owsismultifold.Firstly,amapcanbeeasilyconstructedfroma owbyconsideringaPoincaresectiontransversaltothe ow;theinducedreturnmap:!,togetherwiththereturntime,containallthedynamicalinformationonthe ow.Ergodicpropertiesofchaoticmapsareusuallyeasiertostudythantheir owcounterpart.Forbilliards,thenaturalPoincaremaptoconsideristheboundarymapdenedonthephasespaceassociatedwiththeboundary,T@ .Theergodicofthisboundarymapwereunderstood,andusedtoaddressthecaseofthebilliard owitself[14].Secondly,simplechaoticmapscanbedenedonlow-dimensionalphasespaces,themostfamousonesbeingthehyperbolicsymplectomorphismsonthe2-dimensionaltorus.ThesearedenedbytheactionofamatrixS=abcdwithintegerentries,determinantunityandtracea+d�2(equivalently,Sisunimodularandhyperbolic).Suchamatrixobviouslyactsonx=(x;p)2T2linearly,throughS(x)=(ax+bp;cx+dp)mod1:(12)AschematicviewofSforthefamousArnold'scatmapScat=1112isshowsinFig.5.ThehyperbolicityconditionimpliesthattheeigenvaluesofSareoftheformfegforsome�0.Asaresult,ShastheAnosovproperty:ateachpointx,thetangentspaceTxT2splitsintostableandunstablesubspaces,identiedwiththeeigenspacesofS,andaretheLyapunovexponents.ManydynamicalpropertiesofScanbeexplicitlycomputed.Forinstance,everyrationalpointx2T2isperiodic,andthenumberofperiodicorbitsofperiodngrowslikeen(thusalsomeasuresthecomplexityofthemap).Thislinearityalsoresultsinthefactthatthedecayofcorrelations(forsmoothobservables)issuperexponentialinsteadofexponentialforagenericAnosovdieomorphism.MoregenericAnosovdieomorphismsofthe 186S.NonnenmacherSeminairePoincare2-toruscanbeobtainedbysmoothlyperturbingthelinearmapS.Namely,foragivenHamiltonianH2C1(T2),thecomposedmapHSremainsAnosovifissmallenough,duetothestructuralstabilityofAnosovdieomorphisms. Figure6:Schematicviewofthebaker'smap(13).Thearrowsshowthecontraction/expansiondirections.Anotherfamilyofcanonicalmapsonthetoruswasalsomuchinvestigated,namelytheso-calledbaker'smaps,whicharepiecewiselinear.Thesimplest(sym-metric)baker'smapisdenedbyB(x;p)=((2xmod1;p 2);0x1=2;(2xmod1;p+1 2);1=2x1:(13)Thismapisconjugatedtoaverysimplesymbolicdynamics,namelytheshiftontwosymbols.Indeed,ifoneconsidersthebinaryexpansionsofthecoordinatesx=0;12


,p=12


,thenthemap(x;p)7!B(x;p)equivalentwiththeshifttotheleftonthebi-innitesequence


21
12


.Thisconjugacyallowstoeasilyprovethatthemapisergodicandmixing,identifyallperiodicorbits,andprovidealargesetofnontrivialinvariantprobabilitymeasures.Alltrajectoriesnotmeetingthediscontinuitylinesareuniformlyhyperbolic.Simplecanonicalmapshavealsobedenedonthe2-spherephasespace(likethekickedtop),buttheirchaoticpropertieshave,tomyknowledge,notbeenrigorouslyproven.Theirquantizationhasbeenintensivelyinvestigated.2.5.1Quantummapsonthe2-dimensionaltorusAsopposedtothecaseofHamiltonian ows,thereisnonaturalruletoquan-tizeacanonicalmaponacompactphasespaceP.Already,associatingaquantumHilbertspacetothisphasespaceisnotobvious.Therefore,fromtheverybeginning,quantummapshavebeendenedthroughsomewhatarbitrary(orrather,adhoc)procedures,oftenspecictotheconsideredmap:P!P.Stilltheserecipesarealwaysrequiredtosatisfyacertainnumberofproperties:oneneedsasequenceofHilbertspaces(HN)N2NofdimensionsN.HereNisinterpretedastheinverseofPlanck'sconstant,inagreementwiththeheuristicsthateachquantumstateoccupiesavolume~dinphasespace.Wealsowanttoquantizeobservablesf2C(P)intohermitianoperators^fNonHN. Vol.XIV,2010Anatomyofquantumchaoticeigenstates187ForeachN1,thequantizationofisgivenbyaunitarypropagatorUN()actingonHN.Thewholefamily(UN())N1iscalledthequantummapasso-ciatedwith.inthesemiclassicallimitN~�1!1,thispropagatorsatisessomeformofquantum-classicalcorrespondence.Namely,forsome(largeenough)familyofobservablesfonP,weshouldhave8n2Z;U�nN^fNUnN=\(fn)N+On(N�1)asN!1.(14)Thecondition(14)istheanalogueoftheEgorovproperty(4)satisedbythepropa-gatorU~associatedwithaquantumHamiltonian,whichquantizesthestroboscopicmapx7!1H(x).Letusbrie ysummarizetheexplicitconstructionofthequantizationsUN(),forthemaps:T2!T2presentedintheprevioussection.Letusstartbycon-structingthequantumHilbertspace.OnecanseeT2asthequotientofthephasespaceTR=R2bythediscretetranslationsx7!x+n;n2Z2.Hence,itisnaturaltoconstructquantumstatesonT2bystartingfromstates 2L2(R),andrequiringthefollowingperiodicityproperties (x+n1)= (x);(F~ )(p+n2)=(F~ )(p);n1;n22Z:Itturnsoutthatthesetwoconditionscanbesatisedonlyif~=(2N)�1,N2N,andthecorrespondingstates(whichareactuallydistributions)thenformavectorspaceHNofdimensionN.AbasisofthisspaceisgivenbytheDiraccombsej(x)=1 p NX2Z(x�j N�);j=0;:::;N�1:(15)ItisnaturaltoequipHNwiththehermitianstructureforwhichthebasisfej;j=0;:::;N�1gisorthonormal.LetusnowexplainhowthelinearsymplectomorpismsSareconstructed[48].GivenaunimodularmatrixS,itsactiononR2canbegeneratedbyaquadraticpolynomialHS(x;p);thisactioncanthusbequantizedintotheunitaryoperatorU~(S)=exp(�i^HS;~=~)onL2(R).ThisoperatoralsoactsondistributionsS0(R),andinparticularonthenitesubspaceHN.ProvidedthematrixSsatisessome\checkerboardcondition",onecanshow(usinggrouptheory)thattheactionofU~(S)onHNpreservesthatspace,andactsonitthroughaunitarymatrixUN(S).Thefamilyofmatrices(UN(S))N1denesthequantizationofthemapSonT2.Grouptheoryalsoimpliesthatanexactquantum-classicalcorrespondenceholds(thatis,theremaindertermin(14)vanishes),andhasotherimportantconsequencesregardingtheoperatorsUN=UN(S)(foreachNthematrixUNisperiodic,ofpe-riodTN2N).ExplicitexpressionsforthecoecientsmatricesUN(S)canbeworkedout,theydependsquitesensitivelyonthearithmeticpropertiesofthedi-mensionN.Theconstructionofthequantizedbaker'smap(13)proceedsverydierently.AnAnsatzwasproposedbyBalasz-Voros[13],withthefollowingform(weassumethatNisaneveninteger):UN(B)=FNFN=2FN=2;(16) Vol.XIV,2010Anatomyofquantumchaoticeigenstates189Thesequestionsleadustothenotionofphasespacelocalization,ormicrolo-calization4.Wewillsaythatthefamilyofstates( ~)ismicrolocalizedinsideasetBT if,foranysmoothobservablef(x;p)vanishingnearB,thematrixelementsh ~;^f~ ~i.decreasefasterthananypowerof~when~!0.Microlocalpropertiesarenoteasytoguessfromplotsofthespatialdensityj j(x)j2likeFig.1,buttheyaremorenaturaltostudyifwewanttoconnectquantumtoclassicalmechanics,sincethelattertakesplaceinphasespaceratherthanpositionspace.Indeed,themajortoolwewilluseisthequantum-classicalcorrespondence(4);forallthe owswewillconsider,anyinitialspatialtestfunctionf(x)evolvesintoagenuinephasespacefunctionft(x;p),soapurelyspatialformalismisnotveryhelpful.Thesemicrolocalpropertiesareeasiertovisualizeon2-dimensionalphasespaces(seebelowtheguresonthe2-torus).3.1ThecaseofcompletelyintegrablesystemsInordertomotivateourfurtherdiscussionofchaoticeigenstates,letusrstrecallafewfactsabouttheoppositesystems,namelycompletelyintegrableHamiltonian ows.Forsuchsystems,theenergyshellEEisfoliatedbyd-dimensionalinvariantLagrangiantori.EachsuchtorusischaracterizedbythevaluesofdindependentinvariantactionsI1;:::;Id,soletuscallsuchatorusT~I.TheWKBtheoryallowsonetoexplicitlyconstruct,inthesemiclassicallimit,precisequasimodesof^H~associatedwithsomeofthesetori5,thatisnormalizedstates ~I= ~;~Isatisfying^H~ ~I=E~I ~I+O(~1);(19)withenergiesE~IE.SuchaLagrangian(orWKB)state ~Itakesthefollowingform(awayfromcaustics): ~I(x)=LX`=1A`(x;~)exp(iS`(x)=~):(20)HerethefunctionsS`(x)are(local)generatingfunctions6forT~I,andeachA`(x;~)=A0`(x)+~A1`(x)+


isasmoothamplitude.Fromthisveryexplicitexpression,onecaneasilycheckthatthestate ~IismicrolocalizedonT~I.Ontheotherhand,ourknowledgeof ~ismuchmoreprecisethanthelatterfact.OnecaneasilyconstructotherstatesmicrolocalizedonT~I,whichareverydierentfromtheLagrangianstates ~I.Forinstance,foranypointx0=(x0;p0)2T~ItheGaussianwavepacket(orcoherentstate)'x0(x)=(~)�1=4e�jx�x0j2=2~eip0
x=~(21)ismicrolocalizedonthesinglepointx0,andthereforealsoonT~I.Thisexamplejustre ectsthefactthatastatementaboutmicrolocalizationofasequenceofstatesprovidesmuchlessinformationonthanaformulalike(20). 4Theprexmicromustn'tmisleadus:wearestilldealingwithmacroscopiclocalizationpropertiesof ~!5The\quantizable"toriT~IsatisfyBohr-SommerfeldconditionsIi=2~(ni+i),withni2Zandi2[0;1]xedindices.6AbovesomeneighbourhoodU2Rd,thetorusT~IistheunionofLlagrangianleavesf(x;rS`(x));x2Ug,`=1;:::;L Vol.XIV,2010Anatomyofquantumchaoticeigenstates191quantizedergodicdieomorphismsonthetorus[28]oronmoregeneralcompactphasespaces[99]ageneralframeworkofCdynamicalsystems[98]afamilyofquantizedergodicmapswithdiscontinuities[72]andthebaker'smap[39]certainquantumgraphs[17]Letusgivetheideasusedtoprovetheabovetheorem.Wewanttostudythestatisti-caldistributionofthematrixelementsWj(f)=h j;^f~j jiintherangefkjKg,withKlarge.Therststepistoestimatetheaverageofthisdistribution.ItisestimatedbythegeneralizedWeyllaw:XkjKj(f)Vol(M)d (2)dKdZEfdL;K!1;(22)wheredisthevolumeoftheunitballinRd[53].Inparticular,thisasymptoticsallowstocountthenumberofeigenvalueskjK:#fkjKgVol(M)d (2)dKd;K!1;(23)andshowsthattheaverageofthedistributionfj(f);kjKgconvergestothephasespaceaverageL(f)whenK!1.Now,wewanttoshowthatthedistributionisconcentratedarounditsaverage.ThiswillbedonebyestimatingitsvarianceVarK(f)def=1 #fkjKgXkjKjh j;(^f~j�L(f)) jij2;whichhasbeencalledthequantumvariance.Becausethe jareeigenstatesofU~,wemayreplace^f~jbyitsquantumtimeaverageuptosomelargetimeT,^fT;~j=1 2TZT�TU�t~j^f~jU�t~jdt;withoutmodifyingthematrixelements.Atthisstep,weusethesimpleinequalityjh j;A jj2h j;AA i;foranyboundedoperatorA;togetthefollowingupperboundforthevariance:VarK(f)1 #fkjKgXkjKh j;(^fT;~j�L(f))(^fT;~j�L(f)) ji:TheEgorovtheorem(4)showsthattheproductoperatorontherighthandsideisapproximatelyequaltothequantizationofthefunctionjfT�L(f)j2,wherefTistheclassicaltimeaverageoff.ApplyingthegeneralizedWeyllaw(22)tothisfunction,wegettheboundVarK(f)L(jfT�L(f)j2)+OT(K�1): Vol.XIV,2010Anatomyofquantumchaoticeigenstates193 cannotbeasemiclassicalmeasure.SuchanunlikelysubsequencewaslatercalledastrongscarbyRudnickandSarnak[84],inreferencetothescarsdiscoveredbyHelleronthestadiumbilliard(seex4.2).Inthesamepapertheauthorsformulatedastrongerconjecture:Conjecture1[Quantumuniqueergodicity]Let(M;g)beacompactriemannianmanifoldwithnegativesectionalcurvature.Thenallhigh-frequencyeigenstatesbe-comeequidistributedwithrespecttotheLiouvillemeasure.Thetermquantumuniqueergodicityreferstothenotionofuniqueergodicityinergodictheory:asystemisuniquelyergodicsystemifitadmitsuniqueinvariantprobabilitymeasure.Thegeodesic owsweareconsideringadmitmanyinvariantmeasures,buttheconjecturestatesthatthecorrespondingquantumsystemselectsonlyoneofthem.3.3.3ArithmeticquantumuniqueergodicityThisconjecturewasmotivatedbythefollowingresultprovedinthecitedpaper.Theauthorsspecicallyconsideredarithmeticsurfaces,obtainedbyquotientingthePoincarediskHbycertaincongruentco-compactgroups�.Asexplainedinx2.4.2,onsuchasurfaceitisnaturaltoconsiderHeckeeigenstates,whicharejointeigen-statesoftheLaplacianandthe(countablymany)Heckeoperators7.Itwasshownin[84]thatanysemiclassicalmeasurescassociatedwithHeckeeigenstatesdoesnotcontainanyperiodicorbitcomponent .Themethodsof[84]wererenedbyBourgainandLindenstrauss[27],whoshowedthatthemeasurescofan-thintubealonganygeodesicsegmentisboundedfromabovebyC2=9.ThisboundimpliesthattheKolmogorov-Sinaientropyofscisboundedfrombelowby2=98.Finally,usingadvancedergodictheorymethods,LindenstrausscompletedtheproofoftheQUEconjectureinthearithmeticcontext.Theorem2[ArithmeticQUE][69]LetM=�nHbeanarithmetic9surfaceofcon-stantnegativecurvature.Consideraneigenbasis( j)j2NofHeckeeigenstatesof�
M.Then,theonlysemiclassicalmeasureassociatedwiththissequenceistheLiouvillemeasure.LindenstraussandBrooksrecentlyannouncedanimprovementofthistheorem:theresultholdstrue,assumingthe( j)arejointeigenstatesof
MandofasingleHeckeoperatorTn0.Theirproofusesanewdelocalizationestimateforregulargraphs[30],whichreplacestheentropyboundsof[27].ThispositiveQUEresultwasprecededbyasimilarstatementforthehyperbolicsymplectomorphismsonT2introducedinx2.5.1.ThesequantummapsUN(S0)havethenongenericpropertytobeperiodic,soonehasanexplicitexpressionfortheireigenstates.Itwasshowsin[38]thatforacertainfamilyhyperbolicmatricesS0andcertainsequencesofprimevaluesofN,theeigenstatesofUN(S0)becomeequidis-tributedinthesemiclassicallimit,withanexplicitboundontheremainder.Some 7ThespectrumoftheLaplacianonsuchasurfaceisbelievedtobesimple;ifthisisthecase,thenaneigenstateof
isautomaticallyaHeckeeigenstate.8ThenotionofKSentropywillbefurtherexplainedinx3.4.9Fortheprecisedenitionofthesesurfaces,see[69]. Vol.XIV,2010Anatomyofquantumchaoticeigenstates195Conjecture2[GeneralizedQUE]LettHbeanergodicHamiltonian owonsomeenergyshellEE.Then,alleigenstates ~;jof^H~ofenergiesE~;jEbecomeequidis-tributedwhen~!0.LetbeacanonicalergodicmaponT2.Then,theeigenstates N;jofUN()becomeequidistributedwhenN!1.AnintensivenumericalstudyforeigenstatesofaSinai-likebilliardwascarriedonbyBarnett[15].ItseemstoconrmQUEforthissystem.Inthenextsubsectionwewillexhibitparticularsystemsforwhichthisconjec-turefails.3.3.4CounterexamplestoQUE:half-scarredeigenstatesInthissectionwewillexhibitsequencesofeigenstatesconvergingtosemiclassicalmeasuresdierentfromL,thusdisprovingtheaboveconjecture.LetuscontinueourdiscussionofsymplectomorphismsonT2.ForanyN1,thequantumsymplectomorphismUN(S0)areperiodic(uptoaglobalphase)ofperiodTN3N,sothatitseigenvaluesareessentiallyTN-rootsofunity.ForvaluesofNsuchthatTNN,thespectrumofUN(S0)isverydegenerate,inwhichcaseimposingtheeigenstatestobeofHecketypebecomesastrongrequirement.In[62]itisshownthat,providedtheperiodisnottoosmall(namely,TNN1=2�,whichisthecaseforalmostallvaluesofN),thenQUEholdsforanyeigenbasis.Ontheopposite,thereexist(sparse)valuesofN,forwhichtheperiodcanbeassmallasTNClogN,sothattheeigenspacesofhugedimensionsC�1N=logN.ThisfreedomallowedFaure,DeBievreandtheauthortoexplicitlyconstructeigen-stateswithdierentlocalizationproperties[42,43].Theorem4TakeS02SL2(Z)a(quantizable)hyperbolicmatrix.Then,thereexistsaninnite(sparse)sequenceSNsuchthat,foranyperiodicorbit ofS0,onecanconstructasequenceofeigenstates( N)N2SofUN(S0)associatedwiththesemiclassicalmeasuresc=1 2 +1 2L:(24)Moregenerally,foranyS0-invariantmeasureinv,onecanconstructsequencesofeigenstatesassociatedwiththesemiclassicalmeasuresc=1 2inv+1 2L:ThisresultprovidedtherstcounterexampletothegeneralizedQUEconjecture.Theeigenstatesassociatedwith1 2 +1 2Lcanbecalledhalf-localized.Thecoe-cient1=2infrontofthesingularcomponentof wasshowntobeoptimal[43],aphenomenonwhichwasthengeneralizedusingentropymethods(seex3.4).Letusbrie yexplaintheconstructionofeigenstateshalf-localizedonaxedpointx0.TheyareobtainedbyprojectingonanyeigenspacetheGaussianwavepacket'x0(see(21)).EachspectralprojectioncanbeexpressedasalinearcombinationoftheevolvedstatesUN(S0)n'x0,forn2[�TN=2;TN=2�1].Now,weusethefactthat,forNinaninnitesubsequenceSN,theperiodTNoftheoperatorUN(S0)isclosetotwicetheEhrenfesttimeTE=log~�1 ;(25) Vol.XIV,2010Anatomyofquantumchaoticeigenstates197indeedfailtoequidistribute.Tostatehisresult,letusparametrizetheshapeofastadiumbilliardisbytheratiobetweenthelengthandtheheightoftherectangle.Theorem5[49]Forany�0,thereexistsasubsetB[1;2]ofmeasure1�4andanumberm()�0suchthat,forany2B,the-stadiumadmitsasemiclassicalmeasurewithaweightm()onthebouncing-ballorbits.Althoughthetheoremonlyguaranteesthatafractionm()ofthesemiclassicalmeasureislocalizedalongthebouncing-ballorbits,numericalstudiessuggestthatthemodesareasymptoticallyfullyconcentratedontheseorbits.Besides,suchmodesareexpectedtoexistforallratios�0. Figure10:Twoeigenstatesofthestadiumbilliard(=2).Left(k=39:045):themodehasascaralongtheunstablehorizontalorbit.Right(k=39:292):themodeislocalizedinthebouncing-ballregion.3.4EntropyofthesemiclassicalmeasuresToendthissectiononthemacroscopicproperties,letusmentionarecentapproachtocontrainthepossiblesemiclassicalmeasuresoccurringinachaoticsystem.Thisapproach,initiatedbyAnantharaman[1],consistsinprovingnontriviallowerboundsfortheKolmogorov-Sinaientropyofthesemiclassicalmeasures.TheKS(ormetric)entropyisacommontoolinclassicaldynamicalsystems, owsormaps[57].Tobebrief,theentropyHKS()ofaninvariantprobabilitymeasureisanonnegativenumber,whichquantiestheinformation-theoreticcomplexityof-typicaltrajec-tories.Itdoesnotdirectlymeasurethelocalizationof,butgivessomeinformationaboutit.Herearesomerelevantproperties:thedeltameasure onaperiodicorbithasentropyzero.foranAnosovsystem( owordieomorphism),theentropyisconnectedtotheunstableJacobianJu(x)(seeFig.3)throughtheRuelle-Pesinformula:invariant,HKS()ZlogJu()d;withequalityiistheLiouvillemeasure.theentropyisananefunctiononthesetofprobabilitymeasures:HKS(1+(1�)2)=HKS(1)+(1�)HKS(2). 198S.NonnenmacherSeminairePoincareInparticular,theinvariantmeasure +(1�)Lofahyperbolicsymplectomor-phismShasentropy(1�),whereisthepositiveLyapunovexponent.Anantharamanconsideredthecaseofgeodesic owsonmanifoldsMofnegativecurvature,seex2.4.2.SheprovedthefollowingconstraintsonsemiclassicalmeasuresofM:Theorem6[1]Let(M;g)beasmoothcompactriemannianmanifoldofnegativesectionalcurvature.Thenthereexistsc�0suchthatanysemiclassicalmeasurescof(M;g)satisesHKS(sc)c.Inparticular,thisresultforbidssemiclassicalmeasuresfrombeingsupportedonunionsofperiodicgeodesics.Amorequantitativelowerboundwasobtainedin[2,4],relatedwiththeinstabilityofthe ow.Theorem7[4]Underthesameassumptionsasabove,anysemiclassicalmeasuremustsatisfyHKS(sc)ZlogJudsc�(d�1)max 2;(26)whered=dimMandmaxisthemaximalexpansionrateofthe ow.ThislowerboundwasgeneralizedtothecaseoftheWalsh-quantizedbaker'smap[2],andthehyperbolicsymplectomorphismsonT2[29,78],whereittakestheformHKS(sc) 2.Forthesemaps,theboundissaturatedbythehalf-localizedsemi-classicalmeasures1 2( +L).Thelowerbound(26)iscertainlynotoptimalincasesofvariablecurvature.Indeed,therighthandsidemaybecomenegativewhenthecurvaturevariestoomuch.AmorenaturallowerboundhasbeenobtainedbyRiviereintwodimensions:Theorem8[82,83]Let(M;g)beacompactriemanniansurfaceofnonpositivesec-tionalcurvature.ThenanysemiclassicalmeasuresatisesHKS(sc)1 2Z+dsc;(27)where+isthepositiveLyapunovexponent.ThesamelowerboundwasobtainedbyGutkinforafamilyofnonsymmetricbaker'smap[46];healsoshowedthattheboundisoptimalforthatsystem.Thelowerbound(27)isalsoexpectedtoholdforergodicbilliards,likethestadium;inparticular,itwouldnotcontradicttheexistenceofsemiclassicalmeasuressupportedonthebouncingballorbits.Inhigherdimension,oneexpectsthelowerboundHKS(sc)1 2RlogJudsctoholdforAnosovsystems.Kelmer'scounterexamples[60]showthatthisboundmaybesaturatedforcertainAnosovdieomorphismsonT2d.Toclosethissection,wenoticethattheQUEconjecture(whichremainsopen)amountstoimprovingtheentropiclowerbound(26)toHKS(sc)RlogJudsc.4StatisticaldescriptionThemacroscopicdistributionpropertiesdescribedintheprevioussectiongiveapoordescriptionoftheeigenstates,comparedwithourknowledgeofeigenmodesof 200S.NonnenmacherSeminairePoincareforthisfunctionintherange0jrj1:C ;R(x;r)J0(kjrj) Vol( ):(28)Suchahomogeneousandisotropicexpressioncouldbeexpectedfromourapprox-imation.ReplacingtheWignerdistributionbyLsuggeststhat,neareachpointx2 ,theeigenstate isanequalmixtureofparticlesofenergyk2travellinginallpossibledirections.4.1.2ArandomstateAnsatzYet,theapproximationLfortheWignerdistributionsW isNOTtheWignerdistributionofanyquantumstate10.Thenextquestionisthus[95]:canoneexhibitafamilyofquantumstateswhoseWignermeasuresresembleL?Orequivalently,whosemicroscopiccorrelationsbehavelike(28)?BerryproposedarandomsuperpositionofplanewavesAnsatztoaccountfortheseisotropiccorrelations.OneformofthisAnsatzreads rand;k(x)=2 NVol( )1=2NXj=1ajexp(k^nj
x)
;(29)where(^nj)j=1;:::;Nareunitvectorsdistributedontheunitcircle,andthecoecients(aj)j=1;:::;Nareindependentidenticallydistributed(i.i.d.)complexnormalGaussianrandomvariables.Inordertospanallpossiblevelocitydirections(withintheuncer-taintyprinciple),oneshouldincludeNkdirections^njThenormalizationensuresthatk rand;kkL2( )1withhighprobabilitywhenk1.Alternatively,onecanreplacetheplanewavesin(29)bycircular-symmetricwaves,namelyBesselfunctions.Incircularcoordinates,therandomstatereads rand;k(r;)=(Vol( ))�1=2MXm=�MbmJjmj(kr)eim;(30)wherethecoecientsi.i.d.complexGaussiansatisfyingthesymmetrybm=b�m,andMk.Bothrandomensemblesasymptoticallyproducethesamestatisticalresults.Therandomstate rand;ksatisestheequation(
+k2) =0intheinte-riorof .Furthermore, rand;ksatisesa\localquantumergodicity"property:foranyobservablef(x;p)supportedintheinteriorofT ,thematrixelementsh rand;k;^fk�1 rand;kiL(f)withhighprobability(moreisknownabouttheseelements,seex4.1.3).Thestrongerclaimisthat,intheinteriorof ,thelocalstatisticalpropertiesof rand;k,includingitsmicroscopicones,shouldbesimilarwiththoseoftheeigenstates jwithwavevectorskjk.Thecorrelationfunctionofeigenstatesofchaoticplanarbilliardshasbeennu-mericallystudied,andcomparedwiththisrandommodels,seee.g.[71,11].The 10CharacterizingthefunctiononTRdwhichareWignerfunctionsofindividualquantumstatesisanontrivialquestion. 202S.NonnenmacherSeminairePoincarevariance,aswellastheGaussiandistributionforthematrixelementsathighfre-quency.Still,theconvergencetothislawcanbesloweddownforbilliardsadmittingbouncing-balleigenmodes,likethestadiumbilliard[10].Foragenericchaoticsystem,rigoroussemiclassicalmethodscouldonlyprovelogarithmicupperboundsforthequantumvariance[88],Var~(f)C=jlog~j.Schu-bertshowedthatthisslowdecaycanbesharpforcertaineigenbasesofthequantumcatmap,inthecaseoflargespectraldegeneracies[89](aswehaveseeninx3.3.4,suchdegeneraciesarealsoresponsiblefortheexistenceexceptionallylocalizedeigen-states,soalargevarianceisnotsurprising).Theonlysystemsforwhichanalgebraicdecayisknownareofarithmeticnature.LuoandSarnak[70]provedthat,inthecaseofonthemodulardomainM=SL2(Z)nH(anoncompact,nitevolumearithmeticsurface),thequantumvariancecorrespondingtohigh-frequencyHeckeeigenfunctions11isoftheformVarK(f)=B(f) K:thepolynomialdecayisthesameasin(32),butthecoecientB(f)isequaltheclassicalvariance,\decorated"byanextrafactorofarithmeticnature.Morepreciseresultswereobtainedforquantumsymplectomorphismsonthe2-torus.KurlbergandRudnick[64]studiedthedistributionofmatrixelementsfp Nh N;j;^fN N;ji;j=1;:::;Ng,wherethe( N;j)formaHeckeeigenbasisofUN(S)(seex3.3.3).Theycomputedthevariance,whichisasymptoticallyoftheformB(f) N,withB(f)a\decorated"classicalvariance.Theyalsocomputedthefourthmo-mentofthedistribution,whichsuggeststhatthelatterisnotGaussian,butgivenbyacombinationofseveralsemi-circlelawson[�2;2](orSato-Tatedistributions).Thesamesemi-circlelawhadbeenshownin[63]tocorrespondtotheasymptoticvaluedistributionoftheHeckeeigenstates,atleastforNalongasubsequenceof\splitprimes".4.1.4MaximaofeigenfunctionsAnotherinterestingquantityisthestatisticsofthemaximalvaluesofeigenfunctions,thatistheirL1norms,ormoregenerallytheirLpnormsforp2(2;1](wealwaysassumetheeigenfunctionstobeL2-normalized).Themaximabelongtothefartailofthevaluedistribution,sotheirbehaviourisaprioriuncorrelatedwiththeGaussiannatureofthelatter.Therandomwavemodelgivesthefollowingestimate[8]:forC�0largeenough,k rand;kk1 k rand;kk2Cp logkwithhighprobabilitywhenk!1(33)NumericaltestsonsomeEuclideanchaoticbilliardsandasurfaceofnegativecur-vatureshowthatthisorderofmagnitudeiscorrectforchaoticeigenstates[8].Smallvariationswereobservedbetweenarithmetic/non-arithmeticsurfacesofconstantnegativecurvature,thesup-normsappearingslightlylargerinthearithmeticcase,butstillcompatiblewith(33).Fortheplanarbilliards,thelargestmaximaoccuredforstatesscarredalongaperiodicorbit(seex4.2).Mathematicalresultsconcerningthemaximaofeigenstatesofgenericmanifoldsofnegativecurvaturearescarce.Ageneralupperboundk jk1Ck(d�1)=2j(34) 11Theproofisfullywrittenfortheholomorphiccuspforms,buttheauthorsclaimthatitadaptseasilytotheHeckeeigenfunctions. 204S.NonnenmacherSeminairePoincare Figure11:L1(left)andL2(right)normsoftheHusimidensitiesforeigenstatesofthequantumsymplectomorphismUN(SDEGI)(crosses;thedotsindicatethestatesmaximallyscarredon(0;0)).Thedataarecomparedwiththevaluesformaximallylocalizedstates,lagrangianstates,randomstatesandthemaximallydelocalizedstate.(reprintedfrom[79]).In[63]theHeckeeigenstatesofUN(S),expressedasN-vectorsinthepositionbasis(e`),wereshowntosatisfynontrivialL1bounds12:k N;jk1CN3=8+:Besides,forNalongasubsequenceof\splitprimes",thedescriptionoftheHeckeeigenstatesismuchmoreprecise.Theirpositionvectorsareuniformlybounded,k N;jk12,andthevaluedistributionofindividualeigenstatesfj N;j(`=N);`=0;:::;N�1jgisasymptoticallygivenbythesemicirclelawon[0;2],showingthattheseeigenstatesareverydierentfromGaussianrandomstates.Inspiteofthisfact,thevaluedistributionoftheirHusimifunctionx!hj'x; N;jij2,whichcombinesp Npositioncoecients,seemstobeexponential.4.2ScarsofperiodicorbitsAroundthetimethequantumergodicitytheoremwasproved,aninterestingphe-nomenonwasobservedbyHellerinnumericalstudiesonthestadiumbilliard[51].Hellernoticedthatforcertaineigenfunctions,thespatialdensityj j(x)j2isabnor-mallyenhancedalongoneorseveralunstableperiodicgeodesics.Hecalledsuchanenhancementascaroftheperiodicorbitontheeigenstate j.SeeFig.10(left)andFig.12forscarsatlowandrelativelyhighfrequencies.ThisphenomenonwasobservedtopersistathigherandhigherfrequenciesforvariousEuclideanbilliards[94,8],butwasnotdetectedonmanifoldsofnegative 12Inthechosennormalization,thetrivialboundreadsk k1ke`k1=N1=2. Vol.XIV,2010Anatomyofquantumchaoticeigenstates205curvature[6].Couldscarredstatesrepresentcounterexamplestoquantumergodic- Figure12:Ahigh-energyeigenmodeofthestadiumbilliard(k130).Doyouseeanyscar?ity?Moreprecisenumerics[15]showedthatthattheprobabilityweightsoftheseenhancementsneartheperiodicorbitsdecayinthehigh-frequencylimit,becausetheareascoveredbytheseenhancementsdecayfasterthantheirintensities.Asaresult,asequenceofscarredstatesmaystillbecomeequidistributedintheclassicallimit[55,15].Inordertoquantitativelycharacterizethescarringphenomenon,itturnedmoreconvenienttoswitchtophasespacerepresentations,inparticulartheHusimidensityoftheboundaryfunction (q)=@ (q),whichlivesonthephasespaceT@ ofthebilliardmap[37,93].Periodicorbitsarethenrepresentedbydiscretepoints.ScarswerethendetectedasenhancementsoftheHusimidensityH j(x)onperiodicphasespacepointsx.Similarstudieswereperformedinthecaseofquantumchaoticmapsonthetorus,likethebaker'smap[86]orhyperbolicsymplectomorphisms[79].ThescarredstateshowedinFig.8(left)hasthelargestvalueoftheHusimidensityamongalleigenstatesofUN(SDEGI),butitisneverthelessaHeckeeigenstate,sothatitsHusimimeasureshouldbe(macroscopically)closetoL.Thisexampleunambigu-ouslyshowsthatthescarringphenomenonisamicroscopicphenomenon,compatiblewithquantumuniqueergodicity.4.2.1AstatisticaltheoryofscarsHellerrsttriedtoexplainthescarringphenomenonusingthesmoothedlocaldensityofstatesS;x0(E)=Xj(E�Ej)jh'x0; jij2=h'x0;(E�^H~)'x0i;(36)where'x0isaGaussianwavepacket(21)sittingonapointoftheperiodicorbit;theenergycutoisconstrainedbythefactthatthisexpressionisestimatedthrough 206S.NonnenmacherSeminairePoincareitsFouriertransform,thatisthetimeautocorrelationfunctiont7!h'x0;Ut~'x0i~(t);(37)where~isthe~-Fouriertransformof.Becausewecancontroltheevolutionof'x0onlyuptotheEhrenfesttime(25),wemusttake~supportedontheinterval[�TE=2;TE=2],sothathaswidth&~=jlog~j13.SinceUt~'x0comesbacktothepointx0ateachperiodT,S;x(E)haspeaksattheBohr-Sommerfeldenergiesoftheorbit,separatedby2~=Tfromoneanother;however,duetothehyperbolicspreadingofthewavepacket,thesepeakshavewidths~=T,whereistheLyapunovexponentoftheorbit.Hence,thepeakscanonlybesignicantforsmallenough,thatisweaklyunstableorbits.Evenifissmall,thewidth~=Tbecomesmuchlargerthanthemeanlevelspacing1=C~2inthesemiclassicallimit,sothatS;x(E)isamixtureofmanyeigenstates.Inparticular,thismechanismcannotpredictwhichindividualeigenstatewillshowanenhancementatx0,norcanitpredictthevalueoftheenhancements.FollowingHeller'swork,Bogomolny[23]andBerry[19]showedthatcertainlinearcombinationsofeigenstatesshowsome\extradensity"inthespatialdensity(resp.\oscillatorycorrections"intheWignerdensity)aroundacertainnumberofclosedgeodesics.Inthesemiclassicallimit,thesecombinationsalsoinvolvemanyeigenstatesinsomeenergywindow.Adecadelater,HellerandKaplandevelopeda\nonlinear"theoryofscarring,whichproposesastatisticaldenitionofthescarringphenomenon[55].Theyno-ticedthat,givenanenergyintervalIofwidth~2jIj~,thedistributionoftheoverlapsfjh'x; jij2;Ejj2Igdependsonthephasepointx:ifxliesona(mildlyunstable)periodicorbit,thedistributionisspreadbetweensomelargeval-ues(scarredstates)andsomelowvalues(antiscarredstates).Ontheopposite,ifxisa\generic"point,thedistributionoftheoverlapsisnarrower.Thisremarkwasmadequantitativebydeningastochasticmodelfortheun-smoothedlocaldensityofstatesSx(E)(thatis,takingin(36)tobeadeltafunction),asaneectivewaytotakeintoaccountthe(uncontrolled)longtimere-currencesintheautocorrelationfunction(37).Accordingtothismodel,theoverlaps'x; jiinanenergywindowI3EshouldbehavelikerandomGaussianvariables,ofvariancegivenbythesmoothedlocaldensityS;x(E).Hence,ifxliesonashortperiodicorbit,thestatesinenergywindowsclosetotheBohr-Sommerfeldenergies(whereS;x(E)ismaximal)statisticallyhavelargeroverlapswith'x,whilestateswithenergiesEjclosetotheanti-Bohr-Sommerfeldenergiesstatisticallyhavesmalleroverlaps.TheconcatenationoftheseGaussianrandomvariableswithsmoothly-varyingvariancesproducesanon-Gaussiandistribution,withataillargerthantheonepredictedbyBerry'srandommodel.Ontheopposite,ifxisa\generic"point,thevarianceshouldnotdependontheenergyEjandthefulldistributionofthe'x; jiremainsGaussian.Althoughnotrigorouslyjustied,thisstatisticaldenitionofscarringgivesquantitativepredictions,andcanbeviewedasaninterestingdynamicalcorrectionoftherandomstatemodel(29). 13Westickhereto2-dimensionalbilliards,ormapsonT2,sothattheunstablesubspacesare1-dimensional. 208S.NonnenmacherSeminairePoincare Figure13:Left:nodaldomainsofaneigenfunctionofthequarter-stadiumwithk100:5(doyouseetheboundaryofthestadium?).Right:nodaldomainsofarandomstate(30)withk=100(reprintedfrom[26]).modelstartsfromaseparableeigenfunction,oftheformcos(kx=p 2)cos(ky=p 2),forwhichnodallinesformagrid;theyperturbthisfunctionneareachintersection,sothateachcrossingbecomesanavoidedcrossing(seeFig.14).Althoughthelengthofthenodalsetisalmostunchanged,thestructureofthenodaldomainsisdrasticallymodiedbythisperturbation.Assumingtheselocalperturbationsareuncorrelated,theyobtainarepresentationofthenodaldomainsasclustersofacriticalbondpercolationmodel,awell-knownmodelin2-dimensionalstatisticalmechanics.The(somewhatamazing)claimmadein[25]isthatsuchaperturbationoftheseparablewavefunctionhasthesamenodalcountstatisticsasarandomfunction,eventhoughthelatterisverydierentfromtheformerinmayways.Thisfactcanbeattributedtotheinstabilityofthenodaldomainsoftheseparablewavefunction(duetothelargenumberofcrossings),asopposedtotherelativestabilityofthedomainsofrandomstates(withgenericallynocrossing).Theuncorrelatedlocalperturbationsinstantaneouslytransformmicroscopicsquaredomainsintomesoscopic(sometimesmacroscopic)\fractal"domains.Thehigh-frequencylimit(k!1)fortherandomstate(29,30)correspondstothethermodynamicslimitofthepercolationmodel,alimitinwhichthestatisticalpropertiesofpercolationclustershavebeenmuchinvestigated.Thenodalcountratio(39)countsthenumberofclustersonalatticeofNtot=2 N(k)sites.Thedistributionofthenumberofdomains/clusters(k)wascomputedinthislimit[25]:itisaGaussianwithpropertiesh(k)i N(k)!0:0624;2((k)) N(k)!0:0502:(39)NazarovandSodin[76]haveconsideredrandomsphericalharmonicsonthe2-sphere Vol.XIV,2010Anatomyofquantumchaoticeigenstates209 Figure14:Constructionoftherandom-bondpercolationmodel.Left:startingfromasquaregridwhere (x)alternativelytakespositive(+)andnegative(�)values,asmallperturbation (x)neareachcrossingcreatesanavoidedcrossing.Right:theresultingpositiveandnegativenodaldo-mainscanbedescribedbysettingupbonds(thick/dashedlines)betweenadjacentsites(reprintedfrom[25]).(namely,Gaussianrandomstateswithineach2n+1-dimensionaleigenspace,n0),andprovedthatthenumberofnodaldomainsontheeigenspaceassociatedwiththeeigenvaluen(n+1)statisticallybehaveslike(a+o(1))n2,forsomeconstanta�0.Althoughtheycomputeneithertheconstanta,northevarianceofthedistribution,thisresultindicatesthatthepercolationmodelmayindeedcorrectlypredictthenodalcountstatisticsforGaussianrandomstates.Remark2Wenowhavetwolevelsofmodelization.First,thechaoticeigenstatesarestatisticallymodelledbytherandomstates(29,30).Second,thenodalstructureofrandomstatesismodelledbycriticalpercolation.Thesetwoconjecturesappealtodierentmethods:thesecondoneisapurelystatisticalproblem,whiletherstonebelongstothe\chaotic=random"meta-conjecture.5.2OthernodalobservablesThepercolationmodelalsopredictsthestatisticaldistributionoftheareasofnodaldomains:thisdistributionhastheformP(s)s�187=91,wheresisthearea.Ofcourse,thisscalingcanonlyholdinthemesoscopicrangek�2s1,sinceanydomainhasanareaC=k2.Thenumber~jofnodallinesof jtouchingtheboundary@ isalsoaninterstingnodalobservable.Therandomstatemodel(30)wasusedin[22]topredictthefollowingdistribution:h~(k)i kk!1���!Vol(@ ) 2;2(~(k)) kk!1���!0:0769Vol(@ ):ThesameexpectationvaluewasrigorouslyobtainedbyTothandWigman[92],whenconsideringadierentrandommodel,namelyrandomsuperpositionsofeigenstates joftheLaplacianinfrequencyintervalskj2[K;K+1]: rand;K=Xkj2[K;K+1]aj j;(40) 210S.NonnenmacherSeminairePoincarewiththeajarei.i.d.normalGaussians.5.3MacroscopicdistributionofthenodalsetThevolumeofthenodalsetofeigenfunctionsisanotherinterestingquantity.Apriori,thisvolumeshouldbelesssensitivetoperturbationsthanthenodalcountj.Severalrigorousresultshavebeenobtainedonthismatter.DonnellyandFeerman[40]showedthat,foranyd-dimensionalcompactreal-analyticmanifold,the(d�1)-dimensionalvolumeofthenodalsetoftheLaplacianeigenstatessatisestheboundsC�1kjVold�1N( j)Ckj;forsomeC�0dependingonthemanifold.Thestatisticsofthisvolumehasbeeninvestigatedforvariousensemblesofrandomstates[16,20,85].TheaveragelengthgrowslikecMkwithaconstantcM�0dependingonthemanifold;estimatesforthevariancearemorediculttoobtain.Berry[20]arguedthatforthe2-dimensionalrandommodel(29),thevarianceshouldbeoforderlog(k),showinganunusuallystrongconcentrationpropertyforthisrandomvariable.SuchalogarithmicvariancewasrecentlyprovedbyWigmaninthecaseofrandomsphericalharmonicsofthe2-sphere[96].Countingorvolumeestimatesdonotprovideanyinformationonthespatiallocalizationofthenodalset.Atthemicroscopiclevel,Bruning[31]showedthatforanycompactriemannianmanifold,thenodalsetN( j)is\dense"atthescaleofthewavelength:forsomeconstantC�0,anyballB(x;C=kj)intersectsN( j).Onecanalsoconsiderthe\macroscopicdistribution"ofthezeroset,byinte-gratingweightfunctionsoverthe(d�1)-dimensionalriemannianmeasureonN( j):8f2C0(M);~Z j(f)def=ZN( j)f(x)dVold�1(x):(41)Similarlywiththecaseofthedensityj j(x)j2oritsphasespacecousins,thespatialdistributionofthenodalsetcanthenbedescribedbytheweak-limitsoftherenormalizedmeasuresZ j=~Z j ~Z j(M)inthehigh-frequencylimit.Inthecaseofchaoticeigenstates,thefollowingconjecture16seemsareasonable\dual"totheQUEconjecture:Conjecture3Let(M;g)beacompactsmoothriemannianmanifold,withanergodicgeodesic ow.The,foranyorthonormalbasis( j)j1,theprobabilitymeasuresZ jweak-convergetotheLebesguemeasureonM,inthelimitj!1.Thisconjectureiscompletelyopen.Oneslightweakeningwouldbetorequestthattheconvergenceholdsonadensity1subsequence,asinThm.1.Indeed,asimilarpropertycanbeprovedinthecomplexanalyticsetting(seex5.5).ForMarealanalyticmanifold,eigenfunctionscanbeanalyticallycontinuedinsomecomplexneighbourhoodofM,intoholomorphicfunctions Cj.For( j)j2Sasequenceofergodiceigenfunctions,Zelditchhasobtainedtheasymptoticdistributionofthe(complex)nodalsetof Cj[100];however,hisresultsaysnothingabouttherealzeros(thatis,N( Cj)\M). 16probablyrstmentionedbyZelditch 212S.NonnenmacherSeminairePoincare Figure15:Stellarrepresentationfor3eigenstatesofthequantumcatmapUN(SDEGI),theHusimidensitiesofwhichwereshowninFig.8,top,inadierentorder.Canyouguessthecorrespondence?pointsinC,whichwewilldenote17byZ .Moreinterestingly,throughHadamard'sfactorizationonecanessentiallyrecovertheBargmannfunctionB fromitsnodalset,andthereforethequantumstate .Assuming062Z ,wehaveB (z)=ez2+z+ Y06=zi2Z (1�z=zi)ez zi+1 2z2 z2i;leavingonly3undeterminedparameters.LeboeufandVoros[67]calledthesetZ thestellarrepresentationof ,andproposedtocharacterizethechaoticeigenstatesusingthisrepresentation.Thisideaisespeciallyappealinginthecaseofacompactphasespacelikethe2-torus:inthatcasetheBargmannfunctionB ofastate N2HNisanentirefunctiononCsatisfyingquasiperiodicityconditions,sothatitsnodalsetisZ2-periodic,andcontainsexactlyNzerosineachfundamentalcell.Onecanthenreconstructthestate NfromthissetofNpointsonT2(whichwedenotebyZT2 N):B N(z)=e Yzi2ZT2 N(z�zi);where(z)isaxedJacobithetafunctionvanishingonZ+iZ.Thisstellarrepre-sentationisexact,minimal(Ncomplexpointsrepresent 2HNCN)andlivesinphasespace.Theconjugationofthesethreepropertiesmakesitinterestingfromasemiclassicalpointofview.In[67]theauthorsnoticedastarkdierencebetweenthenodalpatternsofintegrablevs.chaoticeigenstates.Intheintegrablecase,zerosareregularlyalignedalongcertaincurves,whichwereidentiedasanti-StokeslinesinacomplexWKBformalism.Namely,theBargmannfunctioncanbeapproximatedbyaWKBAnsatzsimilarwith(20)withphasefunctionsSj(z),andanti-Stokeslinesaredenedbyequations=(Sj(z)�Sk(z))=0inregionswhereeiSj(z)=~andeiSk(z)=~dominatetheotherterms.Thesecurvesofzerosaresittingatthe\antipodes"ofthelagrangiantoruswheretheHusimidensityisconcentrated(seeFig.7(center)). 17Noconfusionwillappearbetweenthissetandtherealnodalsetsoftheprevioussection. Vol.XIV,2010Anatomyofquantumchaoticeigenstates213Ontheopposite,thezerosofchaoticeigenstatesappearmoreorlessequidis-tributedacrossthewholetorus(seeFig.15),liketheHusimidensityitself.Thisfactwascheckedonothersystems,e.g.planarbilliards,forwhichthestellarrep-resentationsoftheboundaryfunctions@ j(x)wereinvestigatedin[93],leadingtosimilarconclusions.Thisobservationwasfollowedbyarigorousstatement,whichweexpressbydening(usingthesamenotationasintheprevioussection)the\stellarmeasure"ofastate N2HN.Z Ndef=N�1Xzi2ZT2 Nzi:Theorem9[79]Assumethatasequenceofnormalizedstates( N2HN)N1be-comesequidistributedonT2inthelimitN!1(thatis,theirHusimimeasuresH Nweak-convergetotheLiouvillemeasureL).Then,thecorrespondingstellarmeasuresZ Nalsoweak-convergetoL.Usingthequantumergodicitytheorem(orquantumuniqueergodicitywhenavail-able),onededucesthat(almost)allsequencesofchaoticeigenstateshaveasymptot-icallyequidistributedHusiminodalsets.Thisresultwasprovedindependently(andingreatergenerality)byShimanandZelditch[90].Thestrategyistorstshowthattheelectrostaticpotentialu N(x)=N�1logH N(x)=2N�1logjB N(z)j�jzj2decays(inL1)inthesemiclassicallimit,andthenusethefactthatZ N=4
u N.Thisuseofpotentialtheory(specictotheholomorphicsetting)explainswhysuchacorrespondingstatementhasnotbeenprovedyetforthenodalsetofrealeigen-functions(seethediscussionattheendofx5.3).Tomyknowledge,thisresultistheonlyrigorousoneconcerningthestellarrepresentationofchaoticeigenstates.Inparallel,manystudieshavebeendevotedtothestatisticalpropertiesofstellarrepresentationofrandomstates(35),whichcanthenbecomparedwiththoseofchaoticeigenstates.Zerosofrandomholomorphicfunctions(e.g.polynomials)havealonghistoryinprobabilitytheory,seee.g.therecentreview[77],whichmentionstheworksofKac,Littlewood,Oord,Rice.Thetopichasbeenrevivedintheyears1990throughquestionsappearinginquantumchaos[66,24,47].ForaGaussianensemblelike(35),onecanexplicitlycomputethen-pointcor-relationfunctionsofthezeros:besidesbeingequidistributed,thezerosstatisticallyrepeleachotherquadraticallyonthemicroscopicscaleN�1=2(thetypicaldistancebetweennearbyzeros),butareuncorrelatedatlargerdistances.Suchalocalrepul-sion(whichimposesacertainrigidityofrandomnodalsets)holdsingreatgenerality,showingaformofuniversalityatthemicroscopicscale[21,77].Thestudyof[79]suggestedthatthelocalizationpropertiesoftheHusimimea-sure(e.g.ascaronaperiodicorbit)couldnotbedirectlyvisualizedinthedistribu-tionofthefewzerosnearthescarringorbit,butratherinthecollectivedistributionofallzeros.WethusstudiedindetailtheFouriercoecientsofthestellarmeasures,Z (e2ik
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