# American Institute of Mathematical Sciences

January  2013, 7(1): 119-133. doi: 10.3934/jmd.2013.7.119

## Remarks on quantum ergodicity

 1 Laboratoire Paul Painlevé (UMR CNRS 8524), UFR de Mathématiques, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France

Received  October 2012 Revised  January 2013 Published  May 2013

We prove a generalized version of the Quantum Ergodicity Theorem on smooth compact Riemannian manifolds without boundary. We apply it to prove some asymptotic properties on the distribution of typical eigenfunctions of the Laplacian in geometric situations in which the Liouville measure is not (or not known to be) ergodic.
Citation: Gabriel Rivière. Remarks on quantum ergodicity. Journal of Modern Dynamics, 2013, 7 (1) : 119-133. doi: 10.3934/jmd.2013.7.119
##### References:
 [1] L. Barreira and Y. Pesin, Smooth ergodic theory and nonuniformly hyperbolic dynamics, in "Handbook of Dynamical Systems," Vol. 1B, Elsevier B. V., Amsterdam, (2006), 57-263. doi: 10.1016/S1874-575X(06)80027-5. [2] M. V. Berry, Regular and irregular semiclassical wavefunctions, J. Phys. A: Math. Gen., 10 (1977), 2083-2091. doi: 10.1088/0305-4470/10/12/016. [3] N. Burq, Mesures semi-classiques et mesures de défaut, Séminaire Bourbaki, Vol. 1996/97, Astérisque, 245 (1997), 167-195. [4] Y. Colin de Verdière, Ergodicité et fonctions propres du Laplacien, Comm. in Math. Phys., 102 (1985), 497-502. doi: 10.1007/BF01209296. [5] V. Donnay, Geodesic flow on the two-sphere. I. Positive measure entropy, Ergodic Theory Dynam. Systems, 8 (1988), 531-553. doi: 10.1017/S0143385700004685. [6] M. Einsiedler and T. Ward, "Ergodic Theory with a View Towards Number Theory," Graduate Texts in Mathematics, 259, Springer-Verlag London, Ltd., London, 2011. doi: 10.1007/978-0-85729-021-2. [7] J. Galkowski, Quantum ergodicity for a class of mixed systems, to appear in J. of Spectral Theory, arXiv:1209.2968, (2012). [8] P. Gérard, Microlocal defect measures, CPDE, 16 (1991), 1761-1794. doi: 10.1080/03605309108820822. [9] B. Gutkin, Note on converse quantum ergodicity, Proc. AMS, 137 (2009), 2795-2800. doi: 10.1090/S0002-9939-09-09849-9. [10] J. Marklof and S. O'Keefe, Weyl's law and quantum ergodicity for maps with divided phase space, With an appendix by S. Zelditch, Nonlinearity, 18 (2005), 277-304. doi: 10.1088/0951-7715/18/1/015. [11] J. Marklof and Z. Rudnick, Almost all eigenfunctions of a rational polygon are uniformly distributed, Journal of Spectral Theory, 2 (2012), 107-113. doi: 10.4171/JST/23. [12] I. C. Percival, Regular and irregular spectra, J. Phys. B: At. Mol. Opt. Phys., 6 (1973), L229-L232. [13] Y. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, (Russian) Uspehi Mat. Nauk, 32 (1977), 55-112, 287. [14] R. Schubert, "Semiclassical Localization in Phase Space," Ph.D Thesis, Ulm, (2001). [15] L. Schwartz, "Théorie des Distributions," (French) Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. IX-X, Nouvelle édition, entiérement corrigée, refondue et augmentée, Hermann, Paris, 1966. [16] A. Šnirel'man, Ergodic properties of eigenfunctions, Usp. Math. Nauk., 29 (1974), 181-182. [17] A. Šnirel'man, Asymptotic properties of eigenfunctions in the regions of chaotic motion, Addendum to "KAM Theory and Semiclassical Approximations of Eigenfunctions" (by V. F. Lazutkin), Springer-Verlag, Berlin, 1993. [18] J. Sjöstrand, Asymptotic distribution of eigenfrequencies for damped wave equations, Publ. RIMS, 36 (2000), 573-611. doi: 10.2977/prims/1195142811. [19] M. Taylor, "Pseudodifferential Operators," Princeton Mathematical Series, 34, Princeton University Press, Princeton, N.J., 1981. [20] F. Trèves, "Topological Vector Spaces, Distributions and Kernels," Academic Press, New York-London, 1967. [21] S. Zelditch, Uniform distribution of the eigenfunctions on compact hyperbolic surfaces, Duke Math. Jour., 55 (1987), 919-941. doi: 10.1215/S0012-7094-87-05546-3. [22] S. Zelditch, Quantum ergodicity on the sphere, Comm. in Mat. Phys., 146 (1992), 61-71. doi: 10.1007/BF02099207. [23] S. Zelditch, Recent developments in mathematical quantum chaos, in "Current Developments in Mathematics, 2009," Int. Press, Somerville, MA, (2010), 115-204. [24] S. Zelditch, Random orthonormal bases of spaces of high dimension, arXiv:1210.2069, (2012). [25] M. Zworski, "Semiclassical Analysis," Graduate Studies in Mathematics, 138, AMS, Providence, RI, 2012.

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##### References:
 [1] L. Barreira and Y. Pesin, Smooth ergodic theory and nonuniformly hyperbolic dynamics, in "Handbook of Dynamical Systems," Vol. 1B, Elsevier B. V., Amsterdam, (2006), 57-263. doi: 10.1016/S1874-575X(06)80027-5. [2] M. V. Berry, Regular and irregular semiclassical wavefunctions, J. Phys. A: Math. Gen., 10 (1977), 2083-2091. doi: 10.1088/0305-4470/10/12/016. [3] N. Burq, Mesures semi-classiques et mesures de défaut, Séminaire Bourbaki, Vol. 1996/97, Astérisque, 245 (1997), 167-195. [4] Y. Colin de Verdière, Ergodicité et fonctions propres du Laplacien, Comm. in Math. Phys., 102 (1985), 497-502. doi: 10.1007/BF01209296. [5] V. Donnay, Geodesic flow on the two-sphere. I. Positive measure entropy, Ergodic Theory Dynam. Systems, 8 (1988), 531-553. doi: 10.1017/S0143385700004685. [6] M. Einsiedler and T. Ward, "Ergodic Theory with a View Towards Number Theory," Graduate Texts in Mathematics, 259, Springer-Verlag London, Ltd., London, 2011. doi: 10.1007/978-0-85729-021-2. [7] J. Galkowski, Quantum ergodicity for a class of mixed systems, to appear in J. of Spectral Theory, arXiv:1209.2968, (2012). [8] P. Gérard, Microlocal defect measures, CPDE, 16 (1991), 1761-1794. doi: 10.1080/03605309108820822. [9] B. Gutkin, Note on converse quantum ergodicity, Proc. AMS, 137 (2009), 2795-2800. doi: 10.1090/S0002-9939-09-09849-9. [10] J. Marklof and S. O'Keefe, Weyl's law and quantum ergodicity for maps with divided phase space, With an appendix by S. Zelditch, Nonlinearity, 18 (2005), 277-304. doi: 10.1088/0951-7715/18/1/015. [11] J. Marklof and Z. Rudnick, Almost all eigenfunctions of a rational polygon are uniformly distributed, Journal of Spectral Theory, 2 (2012), 107-113. doi: 10.4171/JST/23. [12] I. C. Percival, Regular and irregular spectra, J. Phys. B: At. Mol. Opt. Phys., 6 (1973), L229-L232. [13] Y. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, (Russian) Uspehi Mat. Nauk, 32 (1977), 55-112, 287. [14] R. Schubert, "Semiclassical Localization in Phase Space," Ph.D Thesis, Ulm, (2001). [15] L. Schwartz, "Théorie des Distributions," (French) Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. IX-X, Nouvelle édition, entiérement corrigée, refondue et augmentée, Hermann, Paris, 1966. [16] A. Šnirel'man, Ergodic properties of eigenfunctions, Usp. Math. Nauk., 29 (1974), 181-182. [17] A. Šnirel'man, Asymptotic properties of eigenfunctions in the regions of chaotic motion, Addendum to "KAM Theory and Semiclassical Approximations of Eigenfunctions" (by V. F. Lazutkin), Springer-Verlag, Berlin, 1993. [18] J. Sjöstrand, Asymptotic distribution of eigenfrequencies for damped wave equations, Publ. RIMS, 36 (2000), 573-611. doi: 10.2977/prims/1195142811. [19] M. Taylor, "Pseudodifferential Operators," Princeton Mathematical Series, 34, Princeton University Press, Princeton, N.J., 1981. [20] F. Trèves, "Topological Vector Spaces, Distributions and Kernels," Academic Press, New York-London, 1967. [21] S. Zelditch, Uniform distribution of the eigenfunctions on compact hyperbolic surfaces, Duke Math. Jour., 55 (1987), 919-941. doi: 10.1215/S0012-7094-87-05546-3. [22] S. Zelditch, Quantum ergodicity on the sphere, Comm. in Mat. Phys., 146 (1992), 61-71. doi: 10.1007/BF02099207. [23] S. Zelditch, Recent developments in mathematical quantum chaos, in "Current Developments in Mathematics, 2009," Int. Press, Somerville, MA, (2010), 115-204. [24] S. Zelditch, Random orthonormal bases of spaces of high dimension, arXiv:1210.2069, (2012). [25] M. Zworski, "Semiclassical Analysis," Graduate Studies in Mathematics, 138, AMS, Providence, RI, 2012.
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