Long time propagation and control on scarring for perturbed quantized hyperbolic toral automorphisms

Annales Henri Poincare

Volumn 6, Issue 5, 2005, Pages 885-913

  Bouclet, Jean Marc ^a   Bièvre, Stephan De ^a  

^a UNIV LILLE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 26844580828     PISSN: 14240637     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00023-005-0228-6     Document Type: Article

References (24)

1. N. Anantharaman, The eigenfunctions of the Laplacian do not concentrate on sets of small topological entropy, preprint June 2004.
2. M. Brin, G. Stuck, Introduction to dynamical systems, Cambridge Univ. Press (2002).
3. M. Blank, G. Keller, C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity 15, no. 6, 1905-1973 (2002).
4. F. Bonecchi, S. De Bièvre, Exponential mixing and | ln ℏ| time scales in quantized hyperbolic maps on the torus, Comm. Math. Phys. 211, 659-686 (2000).
5. _, Controlling strong scarring for quantized ergodic toral automorphisms, Duke Math. J117, No. 3, 571-587 (2003).
6. _, Controlling strong scarring for quantized ergodic toral automorphisms, Section 5, mp_arc 02-81 (2002).
7. A. Bouzouina, S. De Bièvre, Equipartition of the eigenfunctions of quantized ergodic maps on the torus, Comm. Math. Phys. 178, 83-105 (1996).
8. A. Bouzouina, D. Robert, Uniform semiclassical estimates for the propagation of quantum observables, Duke Math. J. 111, No. 2, 223-252 (2002).
9. Y. Colin de Verdière, Ergodicité et fonctions propres du Laplacien, Commun. Math. Phys. 102, 497-502 (1985).
10. Y.V. Egorov, On canonical transformations of pseudo-differential operators (in Russian), Uspekhi Mat. Nauk. 24, no. 5, 235-236 (1969).
11. F. Faure, S. Nonnenmacher, S. De Bièvre, Scarred eigenstates for quantum cats of minimal periods, Commun. Math. Phys. 239, 449-492 (2003).
12. F. Faure, S. Nonnenmacher, On the maximal scarring for quantum cat map eigenstates, Commun. Math. Phys. 245, 201-214 (2004).
13. F. Faure, S. Nonnenmacher, contribution at the Workshop on Random Matrix theory and Arithmetic Aspects of Quantum Chaos, Newton Institute, Cambridge, june 2004.
14. G.B. Folland, Harmonic analysis in phase space, Ann. Math. Studies, Princeton Univ. Press 122, (1989).
15. B. Helffer, A. Martinez, D. Robert, Ergodicité et limite semi-classique, Comm. Math. Phys. 109, 313-326 (1987).
16. L. Hörmander, The analysis of linear partial differential operators III, Springer-Verlag (1985).
17. E. Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity, Annals of Math., to appear.
18. D. Robert, Autour de l'approximation semi-classique, Progress in mathematics 68, Birkhäuser (1987).
19. Z. Rudnick, P. Sarnak, The behaviour of eigenstates of hyperbolic arithmetic manifolds, Commun. Math. Phys. 161, 1, 195-213 (1994).
20. A. Schnirelman, Ergodic properties of eigenfunctions, Usp. Math. Nauk. 29, 181-182 (1974).
21. R. Schubert, Semiclassical behaviour of expectation values in time evolved coherent states for large times, preprint january 2004.
22. M. Taylor, Pseudo-differential operators, Princeton Mathematical Series 34, Princeton University Press (1981).
23. F. Trèves, Introduction to pseudo-differential and Fourier integral operators, Vol. 2: Fourier integral operators, Univ. Ser. Math., Plenum, New-York (1980).
24. Zelditch, Uniform distribution of the eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55, 919-941 (1987).