Distribution of matrix elements and level spacings for classically chaotic systems

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1. [1] S. Agmon, Lectures on Exponential Decay of Solutions od Second Order Elliptic Equation. Bound on Eigenfunctions of N-Body Schrödinger operators, Mathematical Notes of Princeton University. Zbl0503.35001MR745286
2. [2] M.V. Berry, Classical Adiabatic Angles and Quantal Adiabatic Phase, J. Phys. A., Vol. 18, 1985, pp. 15-27. Zbl0569.70020MR777620
3. [3] R. Brummelhuis and A. Uribe, A Trace Formula for Schrödinger Operators, Comm. Math. Phys., Vol. 136, 1991, pp. 567-584. Zbl0729.35093MR1099696
4. [4] B.V. Chirikov, Time-Dependent Quantum Systems, in Chaos and Quantum Physics, Les Houches, 1989, North-Holland1991, pp. 518-519. MR1188424
5. [5] Y. Colin De Verdière, Ergodicité et fonctions propres du Laplacien, Comm. Math. Phys., Vol. 102, 1985, pp. 497-502. Zbl0592.58050MR818831
6. [6] M. Dauge and D. Robert, Weyl's Formula for a Class of Pseudodifferential Operators with Negative Order on L2 (Rn), Lecture Notes in Math, No. 1256, Springer-Verlag, 1986. Zbl0629.35098
7. [7] V. Donnay and C. Liverani, Potentials on the Two-Torus for which the Hamiltonian Flow is Ergodic, Comm. Math. Phys., Vol. 135, 1991, pp. 267-302. Zbl0719.58022MR1087385
8. [8] S. Dozias, Thesis, DMI, ENS, Paris (to appear). 
9. [9] J.J. Duistermaat and V. Guillemin, The Spectrum of Positive Elliptic Operators and Periodic Bicharacteristic, Inv. Math., Vol. 29, 1975, pp. 39-79. Zbl0307.35071MR405514
10. [10] A. Feingold and A. Peres, Distribution of Matrix Elements of Chaotic Systems, Physical Review A, Vol. 34, No. 1, 1986, pp. 591-595. MR848319
11. [11] V. Guillemin, Lectures on Spectral Theory of Elliptic Operators, Duke Math. J., 1977, pp. 485-517. Zbl0463.58024MR448452
12. [12] V. Guillemin and A. Uribe, Circular Symmetry and the Trace Formula, Invent. Math., Vol. 96, 1989, pp. 385-423. Zbl0686.58040MR989702
13. [13] M. Gutzwiller, Periodic Orbits and Classical Quantization Conditions, J. of Math. Phys., Vol. 12, No. 3, 1971, pp. 343-358. 
14. [14] B. Helffer, Semi-Classical Analysis for the Schrödinger Operator and Applications, Lecture Notes in Mathematics, No. 1336, Springer-Verlag. Zbl0647.35002MR960278
15. [15] B. Helffer, Remarks on Recent Results in Semi-Classical Analysis, Publ. of Technische Universitat, Berlin, 1991. 
16. [16] B. Helffer, A. Martinez and D. Robert, Ergodicité et limite semi-classique, Comm. Math. Phys., Vol. 109, 1987, pp. 313-326. Zbl0624.58039MR880418
17. [17] B. Helffer and D. Robert, Calcul fonctionnel par la transformée de Mellin, J. of Funct. Anal., Vol. 53, No. 3, 1983, pp. 246-268. Zbl0524.35103MR724029
18. [18] B. Helffer and D. Robert, Propriétés asymptotiques du spectre d'opérateurs pseudodifférentiels sur Rn, Comm. in PDE, Vol. 7, (7), 1982, pp. 795-882. Zbl0501.35081MR662451
19. [19] B. Helffer and J. Sjöstrand, Multiple Wells in the Semi-Classical limit I, Comm. in PDE, Vol. 9, (4), 1984, pp. 337-408. Zbl0546.35053MR740094
20. [20] J.W. Helton, An Operator Algebra Approach to Partial Differential Equations, Indiana Univ. Math. J., Vol. 26, 1977, pp. 997-1018. Zbl0373.35060MR494325
21. [21] L. Hörmander, The spectral Function of an Elliptic Operator, Acta. Math., Vol. 121, 1968, pp. 193-218. Zbl0164.13201MR609014
22. [22] V. Ivrii, Semi-Classical Analysis and Precise Spectral Asymptotics, Preprints, 1, 2, 3, École Polytechnique, 1990/1991. 
23. [23] S. Mc Donald and A. Kaufman, Spectrum and Eigenfunctions for a Hamiltonian with Stochastic Trajectories, Physical Review Letters, Vol. 42, No. 18, 1979, pp. 1189-1191. 
24. [24] E. Marshalek and J. Da Providência, Sum Rules, Random-Phase-Approximations, and Constraint Self-Consistent Fields, Physical Review C., Vol. 7, No. 6, 1973, pp. 2281- 2293. 
25. [25] E. Meirenken, Semi-Classical Principal Symbols and Gutzwiller's Trace Formula, Publ. University of Freiburg, Nov. 1991. 
26. [26] T. Paul and A. Uribe, Sur la formula semi-classique des traces, C. R. Acad. Sci. Paris, T. 313, Series I, 1991, pp. 217-222. Zbl0738.58046MR1126383
27. [27] P. Pechukas, Distribution of Energy Eigenvalues in the Irregular Spectrum, Physical Review Letters, Vol. 51, No. 11, 1983, pp. 943-950. MR717920
28. [28] V. Petrov and D. Robert, Asymptotique semi-classique du spectre d'hamiltoniens quantiques et trajectoires classiques périodiques, Comm. in PDE, Vol. 10, (4), 1985, pp. 365-390. Zbl0574.35067MR784682
29. [29] M. Pollicott, On the Rate of Mixing of Axiom A Flows, Invent. Math., Vol. 81, 1985, pp. 413-426. Zbl0591.58025MR807065
30. [30] T. Prosen and M. Robnik, Distribution and Fluctuations of Transition Probabilities in a System Between Integrability and Chaos, J. Phys. A: Math. Gene., Vol. 26, 1993, pp. L319-L326. Zbl0772.58031
31. [31] J.M. Robbins and M.V. Berry, Discordance Between Quantum and Classical Correlation Moments for Chaotic System, J. Phys. A: Math. Gene., Vol. 25, 1992, pp. L961-L965. Zbl0755.58068MR1177543
32. [32] J.M. Robbins and M.V. Berry, The Geometric Phase for Chaotic Systems, Proc. R. Soc. Lond. A., Vol. 436, 1991, pp. 631-661. MR1177580
33. [33] D. Robert, Autour de l'approximation semi-classique, Birkhaüser, PM 68, 1987. Zbl0621.35001MR897108
34. [34] D. Ruelle, Resonances of Chaotic Dynamical Systems, Physical Review Letters, Vol. 56, No. 5, 1986, pp. 405-407. MR824170
35. [35] A.I. Shnirelman, Ergodic Properties of Eigenfunctions, Upehi Math. Nauk, Vol. 29, No. 6, 1974, pp. 181-182. Zbl0324.58020MR402834
36. [36] A. Voros, Développements semi-classiques, Thèse de doctorat, Paris-Orsay, 1977. 
37. [37] M. Wilkinson, A Semi-Classical Sum Rule for Matrix Elements of Classically Chaotic Systems, J. Phys. A: Math. Gen., Vol. 20, 1987, pp. 2415-2423. MR895065
38. [38] S. Zelditch, Uniform Distribution of Eigenfunctions on Compact Hyperbolic Surfaces, Duke Math. J., Vol. 55, 1987, pp. 919-941. Zbl0643.58029MR916129
39. [39] S. Zelditch, Quantum Transition Amplitude for Ergodic and for Completely integrable Systems, J. of Funct. Anal., Vol. 94, No. 2, 1990, pp. 415-436. Zbl0721.58051MR1081652

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