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Path integral approach for superintegrable potentials on spaces of nonconstant curvature: I. Darboux spaces D I and D II

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Abstract

In this paper, the Feynman path integral technique is applied for superintegrable potentials on two-dimensional spaces of nonconstant curvature: these spaces are Darboux spaces D I and D II. On D I, there are three, and on D II four such potentials. We are able to evaluate the path integral in most of the separating coordinate systems, leading to expressions for the Green functions, the discrete and continuous wave-functions, and the discrete energy-spectra. In some cases, however, the discrete spectrum cannot be stated explicitly, because it is either determined by a transcendental equation involving parabolic cylinder functions (Darboux space I), or by a higher order polynomial equation. The solutions on D I in particular show that superintegrable systems are not necessarily degenerate. We can also show how the limiting cases of flat space (constant curvature zero) and the two-dimensional hyperboloid (constant negative curvature) emerge.

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Authors and Affiliations

  1. II. Institut für Theoretische Physik Universität Hamburg, Luruper Chaussee 149, 22761, Hamburg, Germany

    C. Grosche

  2. Laboratory of Theoretical Physics, Joint Institute for Nuclear Research (Dubna), Dubna, Moscow oblast, 141980, Russia

    G. S. Pogosyan

  3. Departamento de Matematicas CUCEI, Universidad de Guadalajara Guadalajara, Jalisco, Mexico

    G. S. Pogosyan & A. N. Sissakian

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  1. C. Grosche
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  2. G. S. Pogosyan
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Grosche, C., Pogosyan, G.S. & Sissakian, A.N. Path integral approach for superintegrable potentials on spaces of nonconstant curvature: I. Darboux spaces D I and D II . Phys. Part. Nuclei 38, 299–325 (2007). https://doi.org/10.1134/S1063779607030021

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