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Remarks on quantum ergodicity
1. | Laboratoire Paul Painlevé (UMR CNRS 8524), UFR de Mathématiques, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France |
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. |
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. |
[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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