S. Zelditch. Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. Journal, 55:919--941, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... [2] 4] 5] 7] 10] 6] 8] 9] 12] 11] 13] 14] 15] 18] 16] 17] 19] 20] 21] 24] 25] 23] 27] 26] 28] 22] 29] 33] 30] 31] 32] 34] 35] 36] 37] 38] 39] 1] [40] quilibre instable en rgime semi classique I: Concentration microlocale. Yves Colin de Verdire ycolver Bernard Parisse parisse fourier.ujf grenoble.fr Institut Fourier URA 188 du CNRS BP 74 38402 St Martin d Hres Cedex June 11, 1997 Abstract On s intresse tout d abord aux fonctions ....

.... dans des situations plus gnriques, par exemple lorsque Y p admet des trajectoires priodiques stables gnriques, 3] 2] 6] 31] Dans le cas o le flot hamiltonien de p est ergodique sur le rsultat principal est le thorme de Schnirelmann [35] dont une dmonstration complte a t donne dans [40], 9] 23] les sous suites qui convergent vers une mesure # di#rente de la mesure de Liouville de (E 0 valeur rgulire de p) sont de densit 0 dans le spectre. Ce dernier rsultat laisse ouverte la possibilit de sous suites de densit 0 associes des fonctions propres se concentrant sur des ....

S. Zelditch. Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. Journal, 55:919--941, 1987.

....(i.e. microcanonical) measure corresponds to the property of quantum ergodicity. This concept goes back to Shnirelman [14] and denotes the semiclassical convergence of the phase space lifts of almost all quantum eigenfunctions to Liouville measure; it has been proven in several situations [16, 5, 10]. Quantum systems that possess spin degrees of freedom in addition to their translational ones require an extension of Egorov s theorem and of the concept of quantum ergodicity. In previous studies of non relativistic quantum systems with spin [1, 3] whose dynamics are generated by a Pauli type ....

S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55 (1987), 919--941. 10

....elements and obtain agreement with our conjecture. 1. Introduction A fundamental feature of quantum wave functions of classically chaotic systems is that the matrix elements of smooth observables tend to the phase space average of the observable, at least in the sense of convergence in the mean [14, 2, 16] or in the mean square [17] In many systems it is believed that in fact all matrix elements converge to the micro canonical average, however this has only been demonstrated for a couple of arithmetic systems: For quantum cat maps [11] and conditional on the Generalized Riemann Hypothesis ....

S. Zelditch. Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J., 55(4):919-941, 1987.

....jens.bolte physik.uni ulm.de E mail address: rainer.glaser physik.uni ulm.de Introduction The relation between dynamical properties of a quantum system and its classical limit is a central subject in the eld of quantum chaos. In this context quantum ergodicity is a wellestablished concept [Zel87, CdV85, HMR87, Zel96]. It states for quantisations of ergodic classical systems that the phase space lifts of almost all eigenfunctions of the quantum Hamiltonian converge in the semiclassical limit to an equidistribution on the level surfaces of the classical Hamiltonian. The principal goal of this paper is to ....

....over ;E O , so that this expression vanishes as T 1. We hence conclude that 2; E; 0: This, in turn, is equivalent to the existence of a subsequence f j ; g 2N f j; g j2N of density one, such that equation (6.2) holds. Finally, by a diagonal construction as in [Zel87, CdV85] one can extract a subsequence of f j ; g 2N that is still of density one in f j; g j2N , such that (6.2) holds independently of the operator B. The version of quantum ergodicity asserted in Theorem 6.1 means that in the semiclassical limit the lifts of almost all quasimodes j; to the ....

S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55 (1987), 919-941.

....classical limit, then most of its eigenfunctions equidistribute (on the energy surface) in phase space in the semi classical limit. This result has been proven in many di erent contexts: for the Laplace Beltrami operator on compact Riemannian manifolds with an ergodic geodesic ow in [Sc] [Z1] [CdV] for ergodic billiards in [GL] ZZ] for nonrelativistic quantum mechanics in the classical limit in [HMR] for quantum maps in [BDB] Z2] A precise statement in the latter context will be given below (Theorem 1.3) The theorem raises obvious questions: do there exist exceptional ....

Zelditch S., Uniform distribution of the eigenfunctions on compact hyperbolic surfaces, Duke Math. J 55 (1987), 919-941.

....the one obtained from the dwelling time of a typical billiard orbit, provided the billiard ow is ergodic. For billiards, the Schnirelman theorem was proven in [GL] and [ZZ] It was rst stated and proved in the context of geodesic ows and Laplace Beltrami operators on Riemannian manifolds [Sc][Z1][CdV] In addition to the Schnirelman theorem, there are several more detailed conjectures on the behaviour of the eigenfunctions j as j 1, but it is more di cult even to paraphrase them without additional mathematical tools so I won t go into this aspect of the problem any further. In the ....

Zelditch S., Uniform distribution of the eigenfunctions on compact hyperbolic surfaces, Duke Math. J 55 (1987), 919-941.

....corresponding quantum observable. The quantization of the cat map is a unitary operator UN (A) on HN , the quantum propagator, unique up to a phase factor, characterized by an exact 2 version of Egorov s theorem (1. 2) UN (A) 1 Op N (f)UN (A) Op N (f A) 8f 2 C 1 (T 2 ) 1 see Zelditch [24] and Colin de Verdiere [5] for proofs. 2 This exact version of Egorov s theorem is very special and is a consequence of the map being linear. HECKE THEORY AND EQUIDISTRIBUTION FOR QUANTIZED CAT MAPS 3 The eigenvectors of the quantum propagator UN (A) are the analogues of the eigenmodes of ....

S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55 (1987), 919-941.

....by a grant from the Israel Science Foundation. 1 2 P AR KURLBERG AND ZE EV RUDNICK In this paper we will use the quantized cat map to illuminate one of the few rigorous results available on the semi classical limit of eigenstates of classically chaotic systems, namely Quantum Ergodicity [18, 3, 21]. To formulate this notion, recall that if the classical dynamics are ergodic, then almost all trajectories of a particle cover the energy shell uniformly. The intuition a orded by the Correspondence Principle leads one to look for an analogous statement about the semi classical limit of ....

S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55 (1987), 919-941.

....Quantum ergodicity was first established for the free motion of a particle on a compact Riemannian manifold, where the quantum dynamics is generated by minus the LaplaceBeltrami operator on that manifold. This goes back to Shnirelman [Shn74] and the first complete proofs are due to Zelditch [Zel87] and Colin de Verdi ere [CdV85] In the systems considered by these authors the semiclassical limit is actually realised as a high energy limit. However, for Schrodinger operators involving a potential and possibly also a magnetic field the semiclassical limit can in general only be performed in ....

....an extended criterion in order that quantum ergodicity holds. It is the primary goal of the present work to elaborate on this question in some detail for the case of a non relativistic quantum particle with spin 1 2. Our proof of quantum ergodicity in this context generalises the methods of [Shn74, Zel87, CdV85, HMR87] to the situation of Weyl operators with 2 Theta 2 matrix valued symbols. This requires two essential ingredients. The first one is an Egorov Theorem, which relates the semiclassical limit of the quantum mechanical time evolution of an observable to the classical time evolution of the ....

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S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55 (1987), 919--941.

.... diffeomorphisms and flows, since there exist some very good books describing this material (notably [19] 28] and [31] However, we have listed a few original papers specific to dynamical zeta functions, although the contents of some of them have been presented in [28] References [19] [21] contain the basic theory of Axiom A diffeomorphisms and flows, showing how many of their ergodic properties can be studied (via Markov partitions and symbolic dynamics) by understanding subshifts of finite type (and their suspensions under Lipschitz or Holder return times) with Lipschitz or ....

....and Axiom A [19] R. Bowen. Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Springer (Lecture Notes in Math. Vol. 470) Berlin, 1975. 20] R. Bowen and O.E. Lanford III. Zeta functions of restrictions of the shift transformation. Proc. Sympos. Pure Math. 14:43 50, 1970. [21] R. Bowen and D. Ruelle. The ergodic theory of Axiom A flows. Invent. Math. 29:181 202, 1975. 22] D. Dolgopyat. On decay of correlations in Anosov flows. Preprint, 1996. 23] D. Dolgopyat. Prevalence of rapid mixing for hyperbolic flows. Preprint, 1996. 24] G. Gallavotti. Funzioni zeta ed ....

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Zelditch, S.: Uniform Distribution of Eigenfunctions on Compact Hyperbolic Surfaces. Duke Math. J. 55, 919--941 (1987)

....flow, Delta the corresponding Laplace Beltrami operator, j an orthonormal basis of eigenfunction with eigenvalues 0 = 0 1 2 : Let d j denote the probability measures j j j 2 dv where dv is the volume form on X. Shnirelman, Zelditch and Colin de Verdiere ( Sn74] Sn93] [Ze87], CdV] have proved that there exists a subsequence j k of j of the full density (i.e. #fk : j k ng n as n 1) such that for every f 2 C 1 (X) R f d j k R f dv. Zelditch in [Ze92] extended that result to non compact X = PSL 2 (Z)nH. He also proved that for every f 2 C 1 0 (X) ....

....f d j j 2 ;f 1=2 (8) Wigner distribution d j is not a positive distribution. In some problems it is necessary to consider its positive counterpart which we shall call d j and which is defined by hoe; d j i = hoe F ; d j i (9) where oe F is a Friedrichs symmetrization of oe (cf. [Ze87], Ze91] One can show ( Ze91, Prop. 3.8] that oe Gamma oe F is a symbol of order Gamma1 for any 0, so jhf; d j i Gamma hf; d j ij j j j Gamma1=2 This implies that the estimate (8) also holds with d j s replaced by d j s. Theorem 2 is proved by an approximation ....

S. Zelditch. Uniform distribution of Eigenfunctions on compact hyperbolic surfaces. Duke Math. Jour, 55:919--941, 1987.

....Z S X oe A d ; where d = dxdyd 2 y 2 is a Liouville measure 1 on GammanG and ( is a scalar product on S X. This is a generalization to finite area surfaces of a well known result for compact surfaces due to Shnirelman, Zelditch, and Colin de Verdiere (cf. CdV] Sn1] Sn2] and [Z1]) Actually, Zelditch showed that Maass cusp forms j and Eisenstein series E(z; 1 2 it) become on average equidistributed in S X (cf. Z2] for the precise definitions) While results in [Z2] are valid not only for Gamma = SL 2 (Z) but also for an arbitrary finite area surface GammanG, ....

....transform satisfies the defining property of d t ( Fo] Proposition 2.5) As remarked by Zelditch, d t , though useful for asymptotic computations, is not a positive distribution. To make it positive, one uses the technique of Friedrichs symmetrization. A new distribution d F t is defined (cf. [Z1], Z2] by (0.5) oe; d F t ) oe F ; d t ) for oe 2 C c ( GammanG) where oe F is a Friedrichs symmetrization of oe (cf. Z1] Z2] The expression d F t is now a positive distribution and is asymptotically equivalent (cf. x4) to d t ( Z2] Proposition 3.8) While first defined for ....

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S. Zelditch, Uniform distribution of Eigenfunctions on compact hyperbolic surfaces, Duke Math. Jour. 55 (1987), 919--941.

....given by specifying a unitary operator UN on the state space HN , which satisfies a version of the correspondence principle (Egorov s theorem) kU Gamma1 N Op N (f) UN Gamma Op N (f ffi A)k N 1 Gamma 0; 8f 2 C 1 (T 2 ) 1. 2) 1 announced in [16] with full proofs given by Zelditch [17] for hyperbolic surfaces and Colin de Verdiere [1] in general, see also [9] 2 There are other notions of ergodicity in quantum mechanics, such as von Neumann s [14, 12] which are not related to the one used here. QUANTUM UNIQUE ERGODICITY FOR PARABOLIC MAPS 3 where f ffi A( p q ) f(A( ....

....evolution becomes classical evolution. The analogue of eigenmodes are then the eigenfunctions of the propagator UN . The main focus in the literature has so far been on hyperbolic transformations of the torus, the so called cat maps [7, 11, 4, 5] to which the proof of Schnirelman s theorem [16, 17, 1] can be adapted to prove quantum ergodicity, but not QUE [2, 18] Assuming the Generalized Riemann Hypothesis, Degli Esposti, Graffi and Isola [5] found an explicit infinite (though sparse) subsequence of values of N , for which they show that the expectation values for all eigenfunctions 2 HN ....

S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55 (1987) 919-941.

....the quantum system become equidistributed with respect to the Liouville measure . Made precise (see below) this is a statement about the diagonal matrix elements of quantized observables between eigenstates and is commonly referred to as the Schnirelman Theorem. It has been proven in many cases [Z1, CdV, HMR, GL, Z3, Z4, DIG, Sa, BD, ZZ]. If the system is in addition mixing, more can be inferred: in that case (most) off diagonal matrix elements tend to zero [Z2, CR] We will be interested here in the classical limit of quantized, discontinuous, ergodic or mixing symplectic transformations of the two torus. Combining ideas of [Z4] ....

.... 2I j e k ; j 2 : Here the e k , k = 0; N Gamma 1, are the usual position eigenvectors (see Section 2 for the precise definition) Theorem 2 has now the following main corollary, which is the usual statement of the equidistribution of the eigenfunctions in the position variables [CdV, S, Z1]. Corollary 3 Under the assumptions of Theorem 2: j N (I) X k N 2I j e k ; j N j 2 Gamma jIj; when N 1 The paper is organized as follows. In section 2 we briefly recall the basic elements of quantization on the torus. In section 3 we prove an Egorov estimate for quantized ....

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S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. Jour., 55 (1987), 919-941.

....scalar products hu n ; Au n i turn out to be independent of n. This property cannot obviously hold in any sensible quantum system: however it may be valid at the classical limit, at which hu n ; Au n i may tend to the phase average of the r.h.s. of (1. 1) as verified in many instances (see e.g.[Sc, CdV, HMR, Ze, DEGI]) provided H is the quantization of a Hamiltonian generating an ergodic flow. This fact has even led some authors to assume the validity of this limiting property as the very definition of quantum ergodicity. The persisting dependence on the initial datum after time average can be looked at as a ....

S.Zelditch, Uniform Distribution of Eigenfunctions on Compact Hyperbolic Surfaces. Duke Math.J. 55 (1987), 919-941

....classical notions as h 0. As a final remark let us mention that if A is a pseudodifferential operator also the Von Neumann definition (1. 6) reproduces the classical one at the classical limit if hu n ; Au n i tends to the phase average of the symbol of A, as verified in many instances (see e.g.[Sc, CdV, HMR, Ze1, Ze2, DEGI]) in which H is the quantization of a Hamiltonian generating an ergodic flow. Some authors ( Sa, Ze2] assume this limiting property as the very definition of quantum ergodicity. Ergodic Properties of Quantum Harmonic Crystals 29 ....

S.Zelditch, Uniform Distribution of Eigenfunctions on Compact Hyperbolic Surfaces. Duke Math.J. 55 (1987), 919-941

....for the diagonal elements we have: lim h 0 A jj ( h) a E (the average of a for the Liouville measure on Sigma E ) 3) and for the non diagonal elements: lim [ h 0; j 6=k] A jk ( h) 0 (4) Untill now theses claims are not completly proved. Following the works by Shnirelmann [1] Zelditch [3], Colin de Verdi ere [2] Helffer Martinez Robert [5] it can be proved that (3) is true almost everywhere . One of the main goals of this paper is to discuss the claim (4) and in particular to extend and improve some results obtained by Zelditch [4] We will also discuss the variance of the ....

Zelditch. S: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math.J. 55, 919-941, 1987

....of a Hamiltonian dynamical system one expects to find in the corresponding quantum system is the equipartition of its eigenfunctions in the classical limit. Such a phenomenon has been proved to occur in several cases. For the geodesic flow on compact Riemannian manifolds it is proved in [Sc] [Z1] [CdV] for Hamiltonian flows on 2n in [HMR] and for smooth convex two dimensional ergodic billiards in [GL] In this paper we study the quantization and the classical limit of certain area preserving ergodic maps on the two torus T (2) viewed as phase space, with canonical coordinates (q; ....

....Mar echal de Lattre de Tassigny, 75775 Paris Cedex 16. E mail: bouzoui ccr.jussieu.fr y UFR de Math ematiques, Universit e Paris VII. E mail: debievre mathp7.jussieu.fr large class of models for which the desired equipartition result can be proved very easily. We will use the original idea of [Z1] [CdV] which can be applied here with considerably less technical complications. Before doing so, we nevertheless first need to decide how to quantize an area preserving map on the torus. In section 2 we describe the quantum Hilbert spaces associated to the torus. This problem has been addressed ....

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S.Zelditch, Uniform distribution of the eigenfunctions on compact hyperbolic surfaces, Duke.Math.J. 55 (1987), 919-941.

....at the classical limit. x4: The proof of the results rests on the convergence of the diagonal matrix elements of the quantum observables to the ergodic mean of the corresponding classical symbols at the classical limit. Results of this type, at least on subsequences, have been proved in [S,CdV,HMR,Z] in the context of geodesic flows on compact manifolds with negative curvature and of ergodic flows on constant energy surfaces, respectively. Here we give two independent convergence proofs. The first is based on a direct computation of the matrix elements in terms of exponential sums fulfilling ....

.... . By the normalization condition, the dependence on the initial state disappears if for instance he s ; b f e s i is independent of s. This property does not hold in general, but only at the classical limit, provided one deals with the quantization of a classically ergodic system as verified in [CdV,HMR,S,Z]. Moreover under the assumption oe ess = the quantum system is obviously quasi periodic, which prevents the validity of the mixing property. This definition becomes interesting only at the classical limit. To state the results of the present paper, we assume from now on: 1) Classical ....

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S.Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. Jour. 55 (1987), 919-941.

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S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math J. 55 (1987), 919-941.

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S. Zelditch. Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. Jour. 55 (1987), 919-941.

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