Seminaire BOURBAKI Juin 59eme annee no

Stéphane Nonnenmacher

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Seminaire BOURBAKI Juin 2007 59eme annee, 2006-2007, no 978 SEMI-CLASSICAL MEASURES AND ENTROPY [after Nalini Anantharaman and Stephane Nonnenmacher] by Yves COLIN de VERDIERE INTRODUCTION This report is about recent progress on semi-classical localization of eigenfunctions for quantum systems whose classical limit is hyperbolic (Anosov systems); the main example is the Laplace operator on a compact Riemannian manifold with strictly nega- tive curvature whose classical limit is the geodesic flow; the quantizations of hyperbolic cat maps, called “quantum cat maps”, are other nice examples. All this is part of the field called “quantum chaos”. The new results are: – Examples of eigenfunctions for the cat maps with a strong localization (“scarring”) effect due to S. de Bievre, F. Faure and S. Nonnenmacher [16, 17] – Uniform distribution of Hecke eigenfunctions in the case of arithmetic Riemann surfaces by E. Lindenstrauss [26] – General lower bounds on the entropy of semi-classical measures due to N. Anan- tharaman [1] and improved by N. Anantharaman–S. Nonnenmacher [2] and N. Anantharaman–H. Koch–S. Nonnenmacher [3]. This lower bound is sharp with respect to the cat maps examples. We will mainly focus on this last result. 1. THE 2 BASIC EXAMPLES 1.1.

• riemannian manifold


• measure dl


• maps examples


• sinaı start


• semi-classical measures


• points z ?


• lim t?∞


• compact connected

Sujets

• Riemannian manifold

Informations

• Publié par: pefav
• Publié le: 01 juin 2007
• Nombre de lectures: 9
• Langue: English

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S´eminaireBOURBAKI 59`emeann´ee,2006-2007,n o 978 SEMI-CLASSICALMEASURESANDENTROPY [afterNaliniAnantharamanandSte´phaneNonnenmacher] byYvesCOLINdeVERDI ` ERE INTRODUCTION Juin2007 Thisreportisaboutrecentprogressonsemi-classicallocalizationofeigenfunctions forquantumsystemswhoseclassicallimitishyperbolic(Anosovsystems);themain exampleistheLaplaceoperatoronacompactRiemannianmanifoldwithstrictlynega- tivecurvaturewhoseclassicallimitisthegeodesicflow;thequantizationsofhyperbolic catmaps,called“quantumcatmaps”,areotherniceexamples.Allthisispartofthe fieldcalled“quantumchaos”.Thenewresultsare: –Examplesofeigenfunctionsforthecatmapswithastronglocalization(“scarring”) effectduetoS.deBi`evre,F.FaureandS.Nonnenmacher[16,17] –UniformdistributionofHeckeeigenfunctionsinthecaseofarithmeticRiemann surfacesbyE.Lindenstrauss[26] –Generallowerboundsontheentropyofsemi-classicalmeasuresduetoN.Anan- tharaman[1]andimprovedbyN.Anantharaman–S.Nonnenmacher[2]andN. Anantharaman–H.Koch–S.Nonnenmacher[3].Thislowerboundissharpwith respecttothecatmapsexamples. Wewillmainlyfocusonthislastresult. 1.THE2BASICEXAMPLES 1.1.Catmaps WestartwithamatrixA ∈ SL 2 ( Z )whichisassumedtobehyperbolic:theeigen- valuesλ ± ofAsatisfy0< | λ − | <1< | λ + | .TheactionofAonto R 2 definesasymplectic actionUofAonthetorus R 2 / Z 2 byconsideringactiononpointsmod Z 2 .Suchamap isasimpleexampleofachaoticmap.Ithasbeenobservedsincealongtimethat suchamapcanbequantized:foreachintegerN,weconsidertheHilbertspace H N of dimensionNofSchwartzdistributionsfwhichareperiodicofperiodoneandofwhich P FouriercoefficientsareperiodicofperiodN:iff(x)= k ∈ Z a k e 2πikx ,wehave,forall k ∈ Z ,a k+N =a k .UsingthemetaplecticrepresentationappliedtoA,wegetanatural unitaryactionU ˆ N ontothespace H N .Wearemainlyinterestedintheeigenfunctions

978–02 ofU ˆ N .Thesemi-classicalparameteris ~ =1/Nandtheclassicallimitcorrespondsto largevaluesofN.Agoodreferenceis[8]. 1.2.TheLaplaceoperators OnasmoothcompactconnectedRiemannianmanifold(X,g)withoutboundary,we considertheLaplaceoperatorΔgiveninlocalcoordinatesby Δ= −| g | − 1 ∂ i g ij | g | ∂ j with | g | =det(g ij ).TheLaplaceoperatorΔisessentiallyself-adjointonL 2 (X)with domainthesmoothfunctionsandhasacompactresolvent.Thespectrumisdiscrete anddenotedby 0=λ 1 <λ 2 ≤∙∙∙≤ λ k ≤∙∙∙ withanorthonormalbasisofeigenfunctionsϕ k satisfyingΔϕ k =λ k ϕ k .Itisusefult 1 o − introduceaneffectivePlanckconstant(thesemi-classicalsmallparameter) ~ :=λ k 2. Wewillrewritetheeigenfunctionequation ~ 2 Δϕ=ϕ.Thesemi-classicallimi √ t ~ → 0 correspondstothehighfrequencylimitfortheperiodicsolutionsu(x,t)=exp(iλ k t)ϕ k ofthewaveequationu tt +Δu=0.Insteadofthewaveevolution,wewillusethe Schr¨odingerevolutionwhichisgivenby 2 ~ u t = − ~ Δu, i2 andintroducetheunitarydynamicsdefinedbythe1-parametergroup U ˆ t =exp( − it ~ Δ/2),t ∈ R . Forthebasicdefinitions,onecanread[5]. 1.3.Thegeodesicflow If(X,g)isaRiemannianmanifoldandv ∈ T x Xatangentvectoratthepointx ∈ X, wedefine,fort ∈ R ,G t (x,v)=(y,w)asfollows:ifγ(t)isthegeodesicwhichsatisfies γ(0)=x,˙γ(0)=v,weputy:=γ(t)andw:=γ˙(t).Byusingtheidentificationofthe tangentbundlewiththecotangentbundleinducedbythemetricg(whichisalsothe LegendretransformoftheLagrangian 1 2 g ij (x)v i v j ),wegetaflow(G t ) ? onT ? Xwhich preservestheunitcotangentbundledenotedbyZ.WedenotebyU t therestrictionof (G t ) ? toZ.TheLiouvillemeasuredLonZistheRiemannianmeasurenormalizedas aprobabilitymeasure.TheLiouvillemeasuredLisinvariantbythegeodesicflow. 2.CLASSICALCHAOS Goodtextbooksontheclassicalchaosare[21,30,10].