American Institute of Mathematical Sciences

Advanced
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

[1]

Dubi Kelmer. Quantum ergodicity for products of hyperbolic planes. Journal of Modern Dynamics, 2008, 2 (2) : 287-313. doi: 10.3934/jmd.2008.2.287
[2]

Zheng Yin, Ercai Chen. Conditional variational principle for the irregular set in some nonuniformly hyperbolic systems. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6581-6597. doi: 10.3934/dcds.2016085
[3]

Dmitry Dolgopyat. The work of Federico Rodriguez Hertz on ergodicity of dynamical systems. Journal of Modern Dynamics, 2016, 10: 175-189. doi: 10.3934/jmd.2016.10.175
[4]

Matteo Tanzi, Lai-Sang Young. Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 6015-6041. doi: 10.3934/dcds.2020257
[5]

Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785
[6]

Bernd Kawohl, Friedemann Schuricht. First eigenfunctions of the 1-Laplacian are viscosity solutions. Communications on Pure & Applied Analysis, 2015, 14 (1) : 329-339. doi: 10.3934/cpaa.2015.14.329
[7]

Gary Froyland. On Ulam approximation of the isolated spectrum and eigenfunctions of hyperbolic maps. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 671-689. doi: 10.3934/dcds.2007.17.671
[8]

Luis Barreira, Claudia Valls. Existence of stable manifolds for nonuniformly hyperbolic $c^1$ dynamics. Discrete & Continuous Dynamical Systems, 2006, 16 (2) : 307-327. doi: 10.3934/dcds.2006.16.307
[9]

F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi and R. Ures. A criterion for ergodicity for non-uniformly hyperbolic diffeomorphisms. Electronic Research Announcements, 2007, 14: 74-81. doi: 10.3934/era.2007.14.74
[10]

Mohammadreza Molaei. Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds. Electronic Research Announcements, 2018, 25: 8-15. doi: 10.3934/era.2018.25.002
[11]

P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1
[12]

Anatole Katok, Federico Rodriguez Hertz. Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups. Journal of Modern Dynamics, 2010, 4 (3) : 487-515. doi: 10.3934/jmd.2010.4.487
[13]

Keith Burns, Dmitry Dolgopyat, Yakov Pesin, Mark Pollicott. Stable ergodicity for partially hyperbolic attractors with negative central exponents. Journal of Modern Dynamics, 2008, 2 (1) : 63-81. doi: 10.3934/jmd.2008.2.63
[14]

C.P. Walkden. Stable ergodicity of skew products of one-dimensional hyperbolic flows. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 897-904. doi: 10.3934/dcds.1999.5.897
[15]

Carlos H. Vásquez. Stable ergodicity for partially hyperbolic attractors with positive central Lyapunov exponents. Journal of Modern Dynamics, 2009, 3 (2) : 233-251. doi: 10.3934/jmd.2009.3.233
[16]

Matteo Petrera, Yuri B. Suris. Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor. Journal of Computational Dynamics, 2019, 6 (2) : 401-408. doi: 10.3934/jcd.2019020
[17]

Nicolai T. A. Haydn, Kasia Wasilewska. Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2585-2611. doi: 10.3934/dcds.2016.36.2585
[18]

Yuri Kifer. Another proof of the averaging principle for fully coupled dynamical systems with hyperbolic fast motions. Discrete & Continuous Dynamical Systems, 2005, 13 (5) : 1187-1201. doi: 10.3934/dcds.2005.13.1187
[19]

David Burguet, Todd Fisher. Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2253-2270. doi: 10.3934/dcds.2013.33.2253
[20]

Xinsheng Wang, Weisheng Wu, Yujun Zhu. Local unstable entropy and local unstable pressure for random partially hyperbolic dynamical systems. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 81-105. doi: 10.3934/dcds.2020004

2020 Impact Factor: 0.848

Tools

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences