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

Remarks on quantum ergodicity

Gabriel Rivière 1,

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.
Keywords: Eigenfunctions of the Laplacian, quantum ergodicity, nonuniformly hyperbolic dynamical systems..
Mathematics Subject Classification: Primary: 58J51, 35P20; Secondary: 37D2.
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.  Google Scholar

[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.  Google Scholar

[3]

N. Burq, Mesures semi-classiques et mesures de défaut, Séminaire Bourbaki, Vol. 1996/97, Astérisque, 245 (1997), 167-195.  Google Scholar

[4]

Y. Colin de Verdière, Ergodicité et fonctions propres du Laplacien, Comm. in Math. Phys., 102 (1985), 497-502. doi: 10.1007/BF01209296.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. Galkowski, Quantum ergodicity for a class of mixed systems, to appear in J. of Spectral Theory, arXiv:1209.2968, (2012). Google Scholar

[8]

P. Gérard, Microlocal defect measures, CPDE, 16 (1991), 1761-1794. doi: 10.1080/03605309108820822.  Google Scholar

[9]

B. Gutkin, Note on converse quantum ergodicity, Proc. AMS, 137 (2009), 2795-2800. doi: 10.1090/S0002-9939-09-09849-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

I. C. Percival, Regular and irregular spectra, J. Phys. B: At. Mol. Opt. Phys., 6 (1973), L229-L232. Google Scholar

[13]

Y. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, (Russian) Uspehi Mat. Nauk, 32 (1977), 55-112, 287.  Google Scholar

[14]

R. Schubert, "Semiclassical Localization in Phase Space," Ph.D Thesis, Ulm, (2001). Google Scholar

[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.  Google Scholar

[16]

A. Šnirel'man, Ergodic properties of eigenfunctions, Usp. Math. Nauk., 29 (1974), 181-182.  Google Scholar

[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.  Google Scholar

[18]

J. Sjöstrand, Asymptotic distribution of eigenfrequencies for damped wave equations, Publ. RIMS, 36 (2000), 573-611. doi: 10.2977/prims/1195142811.  Google Scholar

[19]

M. Taylor, "Pseudodifferential Operators," Princeton Mathematical Series, 34, Princeton University Press, Princeton, N.J., 1981.  Google Scholar

[20]

F. Trèves, "Topological Vector Spaces, Distributions and Kernels," Academic Press, New York-London, 1967.  Google Scholar

[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.  Google Scholar

[22]

S. Zelditch, Quantum ergodicity on the sphere, Comm. in Mat. Phys., 146 (1992), 61-71. doi: 10.1007/BF02099207.  Google Scholar

[23]

S. Zelditch, Recent developments in mathematical quantum chaos, in "Current Developments in Mathematics, 2009," Int. Press, Somerville, MA, (2010), 115-204.  Google Scholar

[24]

S. Zelditch, Random orthonormal bases of spaces of high dimension, arXiv:1210.2069, (2012). Google Scholar

[25]

M. Zworski, "Semiclassical Analysis," Graduate Studies in Mathematics, 138, AMS, Providence, RI, 2012.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

N. Burq, Mesures semi-classiques et mesures de défaut, Séminaire Bourbaki, Vol. 1996/97, Astérisque, 245 (1997), 167-195.  Google Scholar

[4]

Y. Colin de Verdière, Ergodicité et fonctions propres du Laplacien, Comm. in Math. Phys., 102 (1985), 497-502. doi: 10.1007/BF01209296.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. Galkowski, Quantum ergodicity for a class of mixed systems, to appear in J. of Spectral Theory, arXiv:1209.2968, (2012). Google Scholar

[8]

P. Gérard, Microlocal defect measures, CPDE, 16 (1991), 1761-1794. doi: 10.1080/03605309108820822.  Google Scholar

[9]

B. Gutkin, Note on converse quantum ergodicity, Proc. AMS, 137 (2009), 2795-2800. doi: 10.1090/S0002-9939-09-09849-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

I. C. Percival, Regular and irregular spectra, J. Phys. B: At. Mol. Opt. Phys., 6 (1973), L229-L232. Google Scholar

[13]

Y. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, (Russian) Uspehi Mat. Nauk, 32 (1977), 55-112, 287.  Google Scholar

[14]

R. Schubert, "Semiclassical Localization in Phase Space," Ph.D Thesis, Ulm, (2001). Google Scholar

[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.  Google Scholar

[16]

A. Šnirel'man, Ergodic properties of eigenfunctions, Usp. Math. Nauk., 29 (1974), 181-182.  Google Scholar

[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.  Google Scholar

[18]

J. Sjöstrand, Asymptotic distribution of eigenfrequencies for damped wave equations, Publ. RIMS, 36 (2000), 573-611. doi: 10.2977/prims/1195142811.  Google Scholar

[19]

M. Taylor, "Pseudodifferential Operators," Princeton Mathematical Series, 34, Princeton University Press, Princeton, N.J., 1981.  Google Scholar

[20]

F. Trèves, "Topological Vector Spaces, Distributions and Kernels," Academic Press, New York-London, 1967.  Google Scholar

[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.  Google Scholar

[22]

S. Zelditch, Quantum ergodicity on the sphere, Comm. in Mat. Phys., 146 (1992), 61-71. doi: 10.1007/BF02099207.  Google Scholar

[23]

S. Zelditch, Recent developments in mathematical quantum chaos, in "Current Developments in Mathematics, 2009," Int. Press, Somerville, MA, (2010), 115-204.  Google Scholar

[24]

S. Zelditch, Random orthonormal bases of spaces of high dimension, arXiv:1210.2069, (2012). Google Scholar

[25]

M. Zworski, "Semiclassical Analysis," Graduate Studies in Mathematics, 138, AMS, Providence, RI, 2012.  Google Scholar

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