Quasi-classical expansions and the problem of quantum chaos

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.