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

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 < Ecr 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 Ecr; the degeneration multiplicity is computed for each eigenvalue