This article is cited in 6 papers

Quantization of Periodic Motions on Compact Surfaces of Constant Negative Curvature in a Magnetic Field

J. Brüning^a, R. V. Nekrasov^a, A. I. Shafarevich<sup>b

a</sup> M. V. Lomonosov Moscow State University
^b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We use the semiclassical approach to study the spectral problem for the Schrödinger operator of a charged particle confined to a two-dimensional compact surface of constant negative curvature. We classify modes of classical motion in the integrable domain $E<E_{\textup{cr}}$ and obtain a classification of semiclassical solutions as a consequence. We construct a spectral series (spectrum part approximated by semiclassical eigenvalues) corresponding to energies not exceeding the threshold value $E_{\textup{cr}}$; the degeneration multiplicity is computed for each eigenvalue.

Keywords: Schrödinger equation, eigenvalue asymptotics, semiclassical approximation, confined classical motion, surface of negative curvature, symplectic structure.

UDC: 517.958+530.145.6

Received: 17.05.2006
Revised: 28.06.2006

DOI: 10.4213/mzm3515

 English version: Mathematical Notes, 2007, 81:1, 28–36

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