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, (2006), 57.  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.  doi: 10.1088/0305-4470/10/12/016.  Google Scholar

[3]

N. Burq, Mesures semi-classiques et mesures de défaut,, Séminaire Bourbaki, 245 (1997), 167.   Google Scholar

[4]

Y. Colin de Verdière, Ergodicité et fonctions propres du Laplacien,, Comm. in Math. Phys., 102 (1985), 497.  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.  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 (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, (2012).   Google Scholar

[8]

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

[9]

B. Gutkin, Note on converse quantum ergodicity,, Proc. AMS, 137 (2009), 2795.  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, 18 (2005), 277.  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.  doi: 10.4171/JST/23.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

L. Schwartz, "Théorie des Distributions,", (French) Publications de l'Institut de Mathématique de l'Université de Strasbourg, (1966).   Google Scholar

[16]

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

[17]

A. Šnirel'man, Asymptotic properties of eigenfunctions in the regions of chaotic motion,, Addendum to, (1993).   Google Scholar

[18]

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

[19]

M. Taylor, "Pseudodifferential Operators,", Princeton Mathematical Series, 34 (1981).   Google Scholar

[20]

F. Trèves, "Topological Vector Spaces, Distributions and Kernels,", Academic Press, (1967).   Google Scholar

[21]

S. Zelditch, Uniform distribution of the eigenfunctions on compact hyperbolic surfaces,, Duke Math. Jour., 55 (1987), 919.  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.  doi: 10.1007/BF02099207.  Google Scholar

[23]

S. Zelditch, Recent developments in mathematical quantum chaos,, in, (2010), 115.   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 (2012).   Google Scholar

show all references

References:
[1]

L. Barreira and Y. Pesin, Smooth ergodic theory and nonuniformly hyperbolic dynamics,, in, (2006), 57.  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.  doi: 10.1088/0305-4470/10/12/016.  Google Scholar

[3]

N. Burq, Mesures semi-classiques et mesures de défaut,, Séminaire Bourbaki, 245 (1997), 167.   Google Scholar

[4]

Y. Colin de Verdière, Ergodicité et fonctions propres du Laplacien,, Comm. in Math. Phys., 102 (1985), 497.  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.  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 (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, (2012).   Google Scholar

[8]

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

[9]

B. Gutkin, Note on converse quantum ergodicity,, Proc. AMS, 137 (2009), 2795.  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, 18 (2005), 277.  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.  doi: 10.4171/JST/23.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

L. Schwartz, "Théorie des Distributions,", (French) Publications de l'Institut de Mathématique de l'Université de Strasbourg, (1966).   Google Scholar

[16]

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

[17]

A. Šnirel'man, Asymptotic properties of eigenfunctions in the regions of chaotic motion,, Addendum to, (1993).   Google Scholar

[18]

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

[19]

M. Taylor, "Pseudodifferential Operators,", Princeton Mathematical Series, 34 (1981).   Google Scholar

[20]

F. Trèves, "Topological Vector Spaces, Distributions and Kernels,", Academic Press, (1967).   Google Scholar

[21]

S. Zelditch, Uniform distribution of the eigenfunctions on compact hyperbolic surfaces,, Duke Math. Jour., 55 (1987), 919.  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.  doi: 10.1007/BF02099207.  Google Scholar

[23]

S. Zelditch, Recent developments in mathematical quantum chaos,, in, (2010), 115.   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 (2012).   Google Scholar

[1]

Mauricio Achigar. Extensions of expansive dynamical systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020399

[2]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[3]

The Editors. The 2019 Michael Brin Prize in Dynamical Systems. Journal of Modern Dynamics, 2020, 16: 349-350. doi: 10.3934/jmd.2020013

[4]

Nitha Niralda P C, Sunil Mathew. On properties of similarity boundary of attractors in product dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021004

[5]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[6]

Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268

[7]

Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129

[8]

Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331

[9]

Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021012

[10]

Stefan Siegmund, Petr Stehlík. Time scale-induced asynchronous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1011-1029. doi: 10.3934/dcdsb.2020151

[11]

Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020108

[12]

Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 0: 331-348. doi: 10.3934/jmd.2020012

[13]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[14]

Gökhan Mutlu. On the quotient quantum graph with respect to the regular representation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020295

[15]

Evan Greif, Daniel Kaplan, Robert S. Strichartz, Samuel C. Wiese. Spectrum of the Laplacian on regular polyhedra. Communications on Pure & Applied Analysis, 2021, 20 (1) : 193-214. doi: 10.3934/cpaa.2020263

[16]

Harrison Bray. Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds. Journal of Modern Dynamics, 2020, 16: 305-329. doi: 10.3934/jmd.2020011

[17]

Andreu Ferré Moragues. Properties of multicorrelation sequences and large returns under some ergodicity assumptions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020386

[18]

Shang Wu, Pengfei Xu, Jianhua Huang, Wei Yan. Ergodicity of stochastic damped Ostrovsky equation driven by white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1615-1626. doi: 10.3934/dcdsb.2020175

[19]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[20]

Giulio Ciraolo, Antonio Greco. An overdetermined problem associated to the Finsler Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021004

2019 Impact Factor: 0.465

Metrics

  • PDF downloads (38)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]