Eigenfunction expansions in the imaginary Lobachevsky space
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ELEMENTARY PARTICLES AND FIELDS Theory
Eigenfunction Expansions in the Imaginary Lobachevsky Space∗ Yu. A. Kurochkin1)** , V. S. Otchik1)*** , D. R. Petrosyan2)**** , and G. S. Pogosyan3), 4), 5)***** Received May 18, 2016
¨ Abstract—Eigenfunctions of the Schrodinger equation with the Coulomb potential in the imaginary Lobachevsky space are studied in two coordinate systems admitting solutions in terms of hypergeometric functions. Normalization and coefficients of mutual expansions for some sets of solutions are found. DOI: 10.1134/S1063778817040135
Geometry of spaces of constant curvature with (pseudo)-orthogonal group of motion provides a natural framework for treatment of numerous physical problems. For example, the velocity space in relativity is the Lobachevsky space, and the imaginary Lobachevsky space corresponds to the unphysical region of momentum variables, which is also of interest for the study of scattering processes. Metrics of spaces of constant curvature appear as simplest solutions of Einstein equations and are used as a background in models aimed at the study of various effects of curvature in quantum theory. Derivation of eigenfunction expansions is an essential element of such study. Eigenfunction expansions associated with invariant operators on various hyperboloids in arbitrary dimensions were derived in [1] and in the special case of four-dimensional pseudo-Euclidean space in [2]. The authors of these works used the canonical subgroup reduction and separation of variables associated with it. Harmonic analysis in homogeneous spaces based on the methods of integral geometry was developed in [3]. These methods were applied in [4] to obtain eigenfunction expansions in several coordinate systems in the Lobachevsky space (the upper sheet of a double-sheeted hyperboloid), and also in [5], ∗
The text was submitted by the authors in English. B. I. Stepanov Institute of Physics, Minsk, Belarus. 2) Joint Institute for Nuclear Research, Dubna, Moscow oblast, Russia. 3) Joint Institute for Nuclear Research, Dubna, Russia. 4) Departamento de Matematicas, CUCEI, Universidad de ´ Guadalajara, Mexico. 5) Yerevan State University, Armenia. ** E-mail: [email protected] *** E-mail: [email protected] **** E-mail: [email protected] ***** E-mail: [email protected] 1)
where separable bases on one- and two-sheeted hyperboloids were studied (In [3, 5] the imaginary Lobachevsky space is defined as the single-sheeted hyperboloid with geometrically opposed points identified). Study of quantum mechanical problems in spaces ¨ of constant curvature was initiated by Schrodinger [6], who solved the Kepler–Coulomb problem on a threedimensional sphere. In the Lobachevsky space, this problem was solved by Infeld and Schild [7]. Since then, a considerable number of works concerning the Kepler–Coulomb problem in these two spaces (see e.g. [8–18] and references therein) has appeared. On the other hand, Grosche [19] in his development of path integral in the imaginary Lobachevsky space incorporated in it an analog of the Cou
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