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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, (2006), 57.
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.
doi: 10.1088/0305-4470/10/12/016. |
[3] |
N. Burq, Mesures semi-classiques et mesures de défaut,, Séminaire Bourbaki, 245 (1997), 167.
|
[4] |
Y. Colin de Verdière, Ergodicité et fonctions propres du Laplacien,, Comm. in Math. Phys., 102 (1985), 497.
doi: 10.1007/BF01209296. |
[5] |
V. Donnay, Geodesic flow on the two-sphere. I. Positive measure entropy,, Ergodic Theory Dynam. Systems, 8 (1988), 531.
doi: 10.1017/S0143385700004685. |
[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. |
[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. |
[9] |
B. Gutkin, Note on converse quantum ergodicity,, Proc. AMS, 137 (2009), 2795.
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, 18 (2005), 277.
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.
doi: 10.4171/JST/23. |
[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.
|
[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).
|
[16] |
A. Šnirel'man, Ergodic properties of eigenfunctions,, Usp. Math. Nauk., 29 (1974), 181.
|
[17] |
A. Šnirel'man, Asymptotic properties of eigenfunctions in the regions of chaotic motion,, Addendum to, (1993).
|
[18] |
J. Sjöstrand, Asymptotic distribution of eigenfrequencies for damped wave equations,, Publ. RIMS, 36 (2000), 573.
doi: 10.2977/prims/1195142811. |
[19] |
M. Taylor, "Pseudodifferential Operators,", Princeton Mathematical Series, 34 (1981).
|
[20] |
F. Trèves, "Topological Vector Spaces, Distributions and Kernels,", Academic Press, (1967).
|
[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. |
[22] |
S. Zelditch, Quantum ergodicity on the sphere,, Comm. in Mat. Phys., 146 (1992), 61.
doi: 10.1007/BF02099207. |
[23] |
S. Zelditch, Recent developments in mathematical quantum chaos,, in, (2010), 115.
|
[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).
|
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. |
[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. |
[3] |
N. Burq, Mesures semi-classiques et mesures de défaut,, Séminaire Bourbaki, 245 (1997), 167.
|
[4] |
Y. Colin de Verdière, Ergodicité et fonctions propres du Laplacien,, Comm. in Math. Phys., 102 (1985), 497.
doi: 10.1007/BF01209296. |
[5] |
V. Donnay, Geodesic flow on the two-sphere. I. Positive measure entropy,, Ergodic Theory Dynam. Systems, 8 (1988), 531.
doi: 10.1017/S0143385700004685. |
[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. |
[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. |
[9] |
B. Gutkin, Note on converse quantum ergodicity,, Proc. AMS, 137 (2009), 2795.
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, 18 (2005), 277.
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.
doi: 10.4171/JST/23. |
[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.
|
[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).
|
[16] |
A. Šnirel'man, Ergodic properties of eigenfunctions,, Usp. Math. Nauk., 29 (1974), 181.
|
[17] |
A. Šnirel'man, Asymptotic properties of eigenfunctions in the regions of chaotic motion,, Addendum to, (1993).
|
[18] |
J. Sjöstrand, Asymptotic distribution of eigenfrequencies for damped wave equations,, Publ. RIMS, 36 (2000), 573.
doi: 10.2977/prims/1195142811. |
[19] |
M. Taylor, "Pseudodifferential Operators,", Princeton Mathematical Series, 34 (1981).
|
[20] |
F. Trèves, "Topological Vector Spaces, Distributions and Kernels,", Academic Press, (1967).
|
[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. |
[22] |
S. Zelditch, Quantum ergodicity on the sphere,, Comm. in Mat. Phys., 146 (1992), 61.
doi: 10.1007/BF02099207. |
[23] |
S. Zelditch, Recent developments in mathematical quantum chaos,, in, (2010), 115.
|
[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).
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