Entropy and localization of eigenfunctions

 Access to full text

 Full (PDF)

1. N. Anantharaman, Entropy and the localization of eigenfunctions, to appear in Ann. Math. Zbl1175.35036
2. N. Anantharaman, S. Nonnenmacher, Entropy of semiclassical measures of the Walsh-quantized baker’s map, to appear in Ann. H. Poincaré. Zbl1109.81035
3. N. Anantharaman, S. Nonnenmacher, Half–delocalization of eigenfunctions of the laplacian on an Anosov manifold,http://hal.archives-ouvertes.fr/hal-00104963 Zbl1145.81033
4. M.V. Berry, Regular and irregular semiclassical wave functions, J.Phys. A 10, 2083–2091 (1977) Zbl0377.70014MR489542
5. O. Bohigas, Random matrix theory and chaotic dynamics, in M.J. Giannoni, A. Voros and J. Zinn-Justin eds., Chaos et physique quantique, (École d’été des Houches, Session LII, 1989), North Holland, 1991 
6. J. Bourgain, E. Lindenstrauss, Entropy of quantum limits, Comm. Math. Phys. 233, 153–171 (2003). Zbl1018.58019MR1957735
7. A. Bouzouina and D. Robert: Uniform Semi-classical Estimates for the Propagation of Quantum Observables, Duke Math. J. 111, 223–252 (2002) Zbl1069.35061MR1882134
8. Y. Colin de Verdière, Ergodicité et fonctions propres du laplacien, Commun. Math. Phys. 102, 497–502 (1985) Zbl0592.58050MR818831
9. A. Connes, H. Narnhofer, W. Thirring, Dynamical entropy of $C^{*}$ algebras and von Neumann algebras, Comm. Math. Phys. 112 no. 4, 691–719 (1987). Zbl0637.46073MR910587
10. M. Dimassi, J. Sjöstrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, 268. Cambridge University Press, Cambridge, 1999. Zbl0926.35002MR1735654
11. L.C. Evans and M. Zworski, Lectures on semiclassical analysis (version 0.2), available at http://math.berkeley.edu/~zworski 
12. F. Faure, S. Nonnenmacher and S. De Bièvre, Scarred eigenstates for quantum cat maps of minimal periods, Commun. Math. Phys. 239, 449–492 (2003). Zbl1033.81024MR2000926
13. F. Faure and S. Nonnenmacher, On the maximal scarring for quantum cat map eigenstates, Commun. Math. Phys. 245, 201–214 (2004) Zbl1071.81044MR2036373
14. A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its applications vol.54, Cambridge University Press, 1995. Zbl0878.58020MR1326374
15. D. Kelmer, Arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus, preprint (2005) math-ph/0510079 
16. K. Kraus, Complementary observables and uncertainty relations, Phys. Rev. D 35, 3070–3075 (1987) MR897714
17. P. Kurlberg and Z. Rudnick, Hecke theory and equidistribution for the quantization of linear maps of the torus, Duke Math. J. 103, 47–77 (2000) Zbl1013.81017MR1758239
18. F. Ledrappier, L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula, Ann. of Math. (2) 122 (1985), no. 3, 509–539. Zbl0605.58028
19. E. Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity, Annals of Math. 163, 165-219 (2006) Zbl1104.22015MR2195133
20. H. Maassen and J.B.M. Uffink, Generalized entropic uncertainty relations, Phys. Rev. Lett. 60, 1103–1106 (1988) MR932170
21. Z. Rudnick and P. Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Commun. Math. Phys. 161, 195–213 (1994) Zbl0836.58043MR1266075
22. A. Schnirelman, Ergodic properties of eigenfunctions, Usp. Math. Nauk. 29, 181–182 (1974) MR402834
23. J. Sjöstrand and M. Zworski, Asymptotic distribution of resonances for convex obstacles, Acta Math. 183, 191–253 (1999) Zbl0989.35099MR1738044
24. A. Voros, Semiclassical ergodicity of quantum eigenstates in the Wigner representation, Lect. Notes Phys. 93, 326-333 (1979) in: Stochastic Behavior in Classical and Quantum Hamiltonian Systems, G. Casati, J. Ford, eds., Proceedings of the Volta Memorial Conference, Como, Italy, 1977, Springer, Berlin Zbl0404.70012MR550907
25. S.A. Wolpert, The modulus of continuity for ${\Gamma }_0\left(m\right)/ℍ$ semi-classical limits, Commun. Math. Phys. 216, 313–323 (2001) Zbl1007.11028MR1814849
26. S. Zelditch, Uniform distribution of the eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55, 919–941 (1987) Zbl0643.58029MR916129