Estimates on Green's functions, localization and the quantum kicked rotor model

Annals of Mathematics

Volumn 156, Issue 1, 2002, Pages 249-294

  Bourgain, J ^a  

^a INSTITUTE FOR ADVANCED STUDY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036662707     PISSN: 0003486X     EISSN: None     Source Type: Journal    
DOI: 10.2307/3597190     Document Type: Article

References (23)

1. [Ba1] S. Basu, New results on quantifier elimination over real closed fields and applications to constraint databases, J. ACM 46 (1999), 537-555.
2. [Ba2] _, On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets, Discrete Comput. Geom. 22 (1999), 1-18.
3. [Bel] J. Bellissard, Transition to chaos in classical and quantum mechanics (Montecatini Terme, 1991), Lecture Notes in Math. 1589 (1994), 1-64, Springer-Verlag, New York.
4. [B1] J. Bourgain, Growth of Sobolev norms in linear Schrödinger equations with quasiperiodic potential. Comm. Math. Phys. 204 (1999), 207-247.
5. [B2] _, On growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential, J. Analyse Math. 77 (1999), 315-348.
6. [B3] _, Positive Lyapounov exponents for most energies, in Geometric Aspects of Functional Analysis, Lecture Notes in Math. 1745 (2000), 37-66.
7. [B4] _, Hölder regularity of integrated density of states for the almost Mathieu operator in a perturbative regime, Lett. Math. Physics 51 (2000), 83-118.
8. [BBG] J. Bellissard, A. Bovier, and J.M. Ghez, Gap labelling theorems for one-dimensional discrete Schrödinger operators, Rev. Math. Phys. 4 (1992), 1-37.
9. [BCR] J. Bochnak, M. Coste, and M.-F. Roy, Real Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 36, Springer-Varlag, New York, 1998.
10. [BG] J. Bourgain and M. Goldstein, On nonperturbative localization with quasiperiodic potential, Ann. of Math. 152 (2000), 835-879.
11. [BGS1] J. Bourgain, M. Goldstein, and W. Schlag, Anderson localization for Schrödinger operators on ℤ with potentials given by the skew-shift, Comm. Math. Phys. 220 (2001), 583-621.
12. 2 with quasi-periodic potential Acta Math., to appear.
13. [BJ] J. Bourgain and S. Jitomirskaya, Anderson localization for the band model, in Geometric Aspects of Functional Analysis, Lectures Notes in Math. 1745 (2000), 67-79.
14. [BPR] S. Basu, R. Pollack, and M.-F. Roy, On the combinatorial and algebraic complexity of quantifier elimination, J. ACM 43 (1996), 1002-1045.
15. [CCFI] G. Casati, B. Chirikov, J. Ford, and F. Izraelev, Stochastic behavior of a quantum pendulum under a periodic perturbation, in Stochastic Behavior in Classical and Quantum Hamiltonian Systems, Lecture Notes in Physics 93 (1979), 334-352, Springer-Verlag, New York.
16. [CIS] B. Chirikov, F. Izrailev, and D. Shepelyanskii, Dynamical stochasticity in classical and quantum mechanics, Math. Phys. Reviews 2, 209-267, Harwood Academic, Chur, 1981.
17. [Ga] J. Garnett, Bounded Analytic Functions, Academic Press, Inc., New York, 1981.
18. [Gr] M. Gromov, Entropy, homology and semialgebraic geometry, Sem. Bourbaki 1985/86, no. 663, Astérisque 145-146 (1987), 225-240.
19. [Gr] M. Gromov, Entropy, homology and semialgebraic geometry, Sem. Bourbaki 1985/86, no. 663, Astérisque 145-146 (1987), 225-240.
20. [FGP] S. Fishman, D. Grempel, and R. Prange, Chaos, quantum recurrences, and Anderson localization, Phys. Rev. Lett. 49 (1982), 509-512.
21. [LS] Y. Last and B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math. 135 (1999), 329-367.
22. [Sin] Y. Sinai, Mathematical problems in the theory of quantum chaos, in Geometric Aspects of Functional Analysis (1989-1990), Lecture Notes in Math. 1469 (1991), 41-59.
23. k-resolution of semialgebraic mappings, Israel J. Math. 57 (1987), 301-317.