Geodesic
Quantization of Periodic Motions on Compact Surfaces of Constant Negative Curvature in a Magnetic Field
Matematičeskie zametki, Tome 81 (2007) no. 1, pp. 32-42

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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 $E$ 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. 
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     author = {J. Br\"uning and R. V. Nekrasov and A. I. Shafarevich},
     title = {Quantization of {Periodic} {Motions} on {Compact} {Surfaces} of {Constant} {Negative} {Curvature} in a {Magnetic} {Field}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {32--42},
     publisher = {mathdoc},
     volume = {81},
     number = {1},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2007_81_1_a2/}
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J. Brüning; R. V. Nekrasov; A. I. Shafarevich. Quantization of Periodic Motions on Compact Surfaces of Constant Negative Curvature in a Magnetic Field. Matematičeskie zametki, Tome 81 (2007) no. 1, pp. 32-42. http://geodesic.mathdoc.fr/item/MZM_2007_81_1_a2/