Abstract
We study perturbationsĤ of the quantized versionĤ 0 of integrable Hamiltonian systems by point interactions. We relate the eigenvalues ofĤ to the zeros of a certain meromorphic function ξ. Assuming the eigenvalues ofĤ 0 are Poisson distributed, we get detailed information on the joint distribution of the zeros of ξ and give bounds on the probability density for the spacings of eigenvalues of Ĥ. Our results confirm the “wave chaos” phenomenon, as different from the “quantum chaos” phenomenon predicted by random matrix theory.
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References
S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable models in quantum mechanics (Springer, Berlin, 1988).
V. I. Arnold,Mathematical Methods of Classical Mechanics (Springer, New York, 1978).
M. Berry, Semiclassical mechanics of regular and irregular motion, inLes Houches 1981, Comportement Chaotique des Systèmes Déterministes, G. Iooss, R. H. G. Helleman, and R. Stora, eds. (North-Holland, Amsterdam, 1983).
M. Berry, Some quantum-to-classical asymptotics, inProceedings Les Houches School on Chaos and Quantum Physics '89 (North-Holland, Amsterdam).
O. Bohigas, M. J. Giannoni, and Ch. Schmit, Characterization of chaotic quantum spectra and universality of local fluctuation law,Phys. Rev. Lett. 52:1 (1984).
O. Bohigas, M. J. Giannoni, and Ch. Schmit, Spectral fluctuations, random matrix theories and chaotic motion, inStochastic Processes in Classical and Quantum Systems, S. Albeverio, G. Casati, and D. Merlini, eds. (Springer, Berlin, 1986), pp. 118–138.
G. Casati, B. V. Chirikov, and I. Guarneri, Energy local statistic of integrable quantum systems,Phys. Rev. Lett. 54:1350 (1985).
V. L. Girko,Spectral Theory of Random Matrices (Nauka, Moscow, 1988).
F. Haake,Quantum Signatures of Chaos (Springer-Verlag, Berlin, 1991).
A. J. Lichtenberg and M. A. Lieberman,Regular and Stochastic Motion (Springer-Verlag, Berlin, 1983).
M. L. Mehta,Random Matrices and the Statistical Theory of Spectra (Academic Press, New York, 1965).
P. Šeba, Wave chaos in singular quantum billiard,Phys. Rev. Lett. 64:1855 (1990).
Ya. G. Sinai, Mathematical problems in the theory of quantum chaos, Raymaond and Beverly Sackler Distinguished Lectures in Mathematics, Tel Aviv University Preprint (1990).
J. Zorbas, Perturbation of self-adjoint operators by Dirac distributions,J. Math. Phys. 21(4):840–847 (1980).
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SFB 237 Essen-Bochum-Düsseldorf
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Albeverio, S., Šeba, P. Wave chaos in quantum systems with point interaction. J Stat Phys 64, 369–383 (1991). https://doi.org/10.1007/BF01057882
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DOI: https://doi.org/10.1007/BF01057882