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Quasi-classical expansions and the problem of quantum chaos

Part of the Lecture Notes in Mathematics book series (LNM,volume 1469)

Abstract

In the present paper we comment some problems discussed in [1]. Following [1] we consider the distribution of eigenvalues E mn of the Laplace-Beltrami operator on a two-dimensional revolution surface. We prove that the quasi-classical quantization rules give a correct asymptotic expansion for large E mn and show that for the problem of quantum chaos two first terms of the quasi-classical expansion are essential. We specify a little bit the geometric problem studied in [1,5] to prove the Poisson distribution for the number ξ of the eigenvalues in the segment [E,E + c], when E → ∞, and show that the main theorem of [5] implies that for ‘typical’ revolution surfaces, ξ = ξ + ξ+ where ξ ≡ 0 (mod 4), ξ+ ≡ 0 (mod 2) and both ξ/4 and ξ+/2 obey the Poisson distributions.

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References

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© 1991 Springer-Verlag

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Bleher, P.M. (1991). Quasi-classical expansions and the problem of quantum chaos. In: Lindenstrauss, J., Milman, V.D. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 1469. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0089215

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  • DOI: https://doi.org/10.1007/BFb0089215

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54024-3

  • Online ISBN: 978-3-540-47355-8

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