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

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{sub I} and D{sub II}. On D{sub I}, there are three, and on D{sub II} four such potentials. We are able to evaluate the pa...

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Bibliographic Details
Published in:Physics of particles and nuclei 2007-05, Vol.38 (3)
Main Authors: Grosche, C., Pogosyan, G. S., Departamento de Matematicas CUCEI, Universidad de Guadalajara Guadalajara, Jalisco, Sissakian, A. N.
Format: Article
Language:English
Subjects:
ENERGY SPECTRA
FEYNMAN PATH INTEGRAL
GREEN FUNCTION
MATHEMATICAL SOLUTIONS
PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
POLYNOMIALS
POTENTIALS
TWO-DIMENSIONAL CALCULATIONS
WAVE FUNCTIONS
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Summary: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{sub I} and D{sub II}. On D{sub I}, there are three, and on D{sub 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{sub 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.
ISSN:1063-7796
1531-8559