@article{AIHPA_1994__61_4_443_0,
author = {Combescure, Monique and Robert, Didier},
title = {Distribution of matrix elements and level spacings for classically chaotic systems},
journal = {Annales de l'I.H.P. Physique th\'eorique},
volume = {61},
year = {1994},
pages = {443-483},
mrnumber = {1311538},
zbl = {0833.58018},
language = {en},
url = {http://dml.mathdoc.fr/item/AIHPA_1994__61_4_443_0}
}
Combescure, Monique; Robert, Didier. Distribution of matrix elements and level spacings for classically chaotic systems. Annales de l'I.H.P. Physique théorique, Tome 61 (1994) pp. 443-483. http://gdmltest.u-ga.fr/item/AIHPA_1994__61_4_443_0/
[1] , Lectures on Exponential Decay of Solutions od Second Order Elliptic Equation. Bound on Eigenfunctions of N-Body Schrödinger operators, Mathematical Notes of Princeton University. | MR 745286 | Zbl 0503.35001
[2] , Classical Adiabatic Angles and Quantal Adiabatic Phase, J. Phys. A., Vol. 18, 1985, pp. 15-27. | MR 777620 | Zbl 0569.70020
[3] and , A Trace Formula for Schrödinger Operators, Comm. Math. Phys., Vol. 136, 1991, pp. 567-584. | MR 1099696 | Zbl 0729.35093
[4] , Time-Dependent Quantum Systems, in Chaos and Quantum Physics, Les Houches, 1989, North-Holland 1991, pp. 518-519. | MR 1188424
[5] , Ergodicité et fonctions propres du Laplacien, Comm. Math. Phys., Vol. 102, 1985, pp. 497-502. | MR 818831 | Zbl 0592.58050
[6] and , Weyl's Formula for a Class of Pseudodifferential Operators with Negative Order on L2 (Rn), Lecture Notes in Math, No. 1256, Springer-Verlag, 1986. | Zbl 0629.35098
[7] and , Potentials on the Two-Torus for which the Hamiltonian Flow is Ergodic, Comm. Math. Phys., Vol. 135, 1991, pp. 267-302. | MR 1087385 | Zbl 0719.58022
[8] , Thesis, DMI, ENS, Paris (to appear).
[9] and , The Spectrum of Positive Elliptic Operators and Periodic Bicharacteristic, Inv. Math., Vol. 29, 1975, pp. 39-79. | MR 405514 | Zbl 0307.35071
[10] and , Distribution of Matrix Elements of Chaotic Systems, Physical Review A, Vol. 34, No. 1, 1986, pp. 591-595. | MR 848319
[11] , Lectures on Spectral Theory of Elliptic Operators, Duke Math. J., 1977, pp. 485-517. | MR 448452 | Zbl 0463.58024
[12] and , Circular Symmetry and the Trace Formula, Invent. Math., Vol. 96, 1989, pp. 385-423. | MR 989702 | Zbl 0686.58040
[13] , Periodic Orbits and Classical Quantization Conditions, J. of Math. Phys., Vol. 12, No. 3, 1971, pp. 343-358.
[14] , Semi-Classical Analysis for the Schrödinger Operator and Applications, Lecture Notes in Mathematics, No. 1336, Springer-Verlag. | MR 960278 | Zbl 0647.35002
[15] , Remarks on Recent Results in Semi-Classical Analysis, Publ. of Technische Universitat, Berlin, 1991.
[16] , and , Ergodicité et limite semi-classique, Comm. Math. Phys., Vol. 109, 1987, pp. 313-326. | MR 880418 | Zbl 0624.58039
[17] and , Calcul fonctionnel par la transformée de Mellin, J. of Funct. Anal., Vol. 53, No. 3, 1983, pp. 246-268. | MR 724029 | Zbl 0524.35103
[18] and , Propriétés asymptotiques du spectre d'opérateurs pseudodifférentiels sur Rn, Comm. in PDE, Vol. 7, (7), 1982, pp. 795-882. | MR 662451 | Zbl 0501.35081
[19] and , Multiple Wells in the Semi-Classical limit I, Comm. in PDE, Vol. 9, (4), 1984, pp. 337-408. | MR 740094 | Zbl 0546.35053
[20] , An Operator Algebra Approach to Partial Differential Equations, Indiana Univ. Math. J., Vol. 26, 1977, pp. 997-1018. | MR 494325 | Zbl 0373.35060
[21] , The spectral Function of an Elliptic Operator, Acta. Math., Vol. 121, 1968, pp. 193-218. | MR 609014 | Zbl 0164.13201
[22] , Semi-Classical Analysis and Precise Spectral Asymptotics, Preprints, 1, 2, 3, École Polytechnique, 1990/1991.
[23] and , Spectrum and Eigenfunctions for a Hamiltonian with Stochastic Trajectories, Physical Review Letters, Vol. 42, No. 18, 1979, pp. 1189-1191.
[24] and , Sum Rules, Random-Phase-Approximations, and Constraint Self-Consistent Fields, Physical Review C., Vol. 7, No. 6, 1973, pp. 2281- 2293.
[25] , Semi-Classical Principal Symbols and Gutzwiller's Trace Formula, Publ. University of Freiburg, Nov. 1991.
[26] and , Sur la formula semi-classique des traces, C. R. Acad. Sci. Paris, T. 313, Series I, 1991, pp. 217-222. | MR 1126383 | Zbl 0738.58046
[27] , Distribution of Energy Eigenvalues in the Irregular Spectrum, Physical Review Letters, Vol. 51, No. 11, 1983, pp. 943-950. | MR 717920
[28] and , Asymptotique semi-classique du spectre d'hamiltoniens quantiques et trajectoires classiques périodiques, Comm. in PDE, Vol. 10, (4), 1985, pp. 365-390. | MR 784682 | Zbl 0574.35067
[29] , On the Rate of Mixing of Axiom A Flows, Invent. Math., Vol. 81, 1985, pp. 413-426. | MR 807065 | Zbl 0591.58025
[30] and , Distribution and Fluctuations of Transition Probabilities in a System Between Integrability and Chaos, J. Phys. A: Math. Gene., Vol. 26, 1993, pp. L319-L326. | Zbl 0772.58031
[31] and , Discordance Between Quantum and Classical Correlation Moments for Chaotic System, J. Phys. A: Math. Gene., Vol. 25, 1992, pp. L961-L965. | MR 1177543 | Zbl 0755.58068
[32] and , The Geometric Phase for Chaotic Systems, Proc. R. Soc. Lond. A., Vol. 436, 1991, pp. 631-661. | MR 1177580
[33] , Autour de l'approximation semi-classique, Birkhaüser, PM 68, 1987. | MR 897108 | Zbl 0621.35001
[34] , Resonances of Chaotic Dynamical Systems, Physical Review Letters, Vol. 56, No. 5, 1986, pp. 405-407. | MR 824170
[35] , Ergodic Properties of Eigenfunctions, Upehi Math. Nauk, Vol. 29, No. 6, 1974, pp. 181-182. | MR 402834
[36] , Développements semi-classiques, Thèse de doctorat, Paris-Orsay, 1977.
[37] , A Semi-Classical Sum Rule for Matrix Elements of Classically Chaotic Systems, J. Phys. A: Math. Gen., Vol. 20, 1987, pp. 2415-2423. | MR 895065
[38] , Uniform Distribution of Eigenfunctions on Compact Hyperbolic Surfaces, Duke Math. J., Vol. 55, 1987, pp. 919-941. | MR 916129 | Zbl 0643.58029
[39] , Quantum Transition Amplitude for Ergodic and for Completely integrable Systems, J. of Funct. Anal., Vol. 94, No. 2, 1990, pp. 415-436. | MR 1081652 | Zbl 0721.58051