zbMATH — the first resource for mathematics

Flensted-Jensen’s functions attached to the Landau problem on the hyperbolic disc. (English) Zbl 1164.33301
Summary: We give an explicit expression of a two-parameter family of Flensted-Jensen’s functions \(\Psi _{\mu ,\alpha }\) on a concrete realization of the universal covering group of \(U(1,1)\). We prove that these functions are, up to a phase factor, radial eigenfunctions of the Landau Hamiltonian on the hyperbolic disc with a magnetic field strength proportional  to \(\mu \), and corresponding to the eigenvalue \(4\alpha ( \alpha -1)\).
33C05 Classical hypergeometric functions, \({}_2F_1\)
35J10 Schrödinger operator, Schrödinger equation
35Q40 PDEs in connection with quantum mechanics
43A90 Harmonic analysis and spherical functions
Full Text: DOI EuDML
[1] L. Landau, E. Lifschitz: Quantum Mechanics: Non-relativistic Theory. Pergamon Press, New York, 1965. · Zbl 0178.57901
[2] E.V. Ferapontov, A. P. Veselov: Integrable Schrödinger operators with magnetic fields: Factorization method on curved surfaces. J. Math. Phys. 42 (2001), 590–607. · Zbl 1013.81007
[3] A. Comtet: On the Landau levels on the hyperbolic plane. Ann. Phys. 173 (1986), 185–209. · Zbl 0635.58034
[4] J. Negro, M. A. del Olmo, and A. Rodriguez-Marco: Landau quantum systems: an approach based on symmetry. J. Phys. A, Math. Gen. 35 (2002), 2283–2307. · Zbl 1022.81062
[5] M. Flensted-Jensen: Spherical functions on a simply connected semisimple Lie group. Am. J. Math. 99 (1977), 341–361. · Zbl 0372.43005
[6] Z. Mouayn: Characterization of hyperbolic Landau states by coherent state transforms. J. Phys. A, Math. Gen. 36 (2003), 8071–8076. · Zbl 1058.81037
[7] S. A. Albeverio, P. Exner, and V. A. Geyler: Geometric phase related to point-interaction transport on a magnetic Lobachevsky plane. Lett. Math. Phys. 55 (2001), 9–16. · Zbl 0986.81031
[8] I. S. Gradshteyn, I. M. Ryzhik: Table of Integrals, Series and Products. Academic Press, New York-London-Toronto, 1980. · Zbl 0521.33001
[9] Analyse Harmonique (Ecole d’été, d’analyse harmonique, Université de Nancy I, Septembre 15 au Octobre 3, 1980). Les Cours du C.I.M.P.A. (P. Eymard, J. L. Clerc, J. Faraut, M. Raïs, and R. Takahashi, eds.). Centre International de Mathématiques Pures et Appliquées, C.I.M.P.A, 1980. (In French.)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.