Abstract
The dynamical symmetry of a three-dimensional oscillator in a space of constant curvature is described by three operators formed from the components of the Fradkin-Higgs tensor and the generators of the quadratic Racah algebraQR(3). This algebra makes it possible to find all dynamical characteristics of the problem: the spectrum, degeneracy of the energy levels, and the overlap coefficients of the wave functions in different coordinate systems. The algebra that generates the spectrum is constructed and found to be the quadratic Jacobi algebraQJ(3).
Similar content being viewed by others
References
Ya. I. Granovskii, A. S. Zhedanov, and I. M. Lutsenko,Zh. Eksp. Teor. Fiz.,99, 353 (1991).
Ya. I. Granovskii and A. S. Zhedanov, “Exactly solvable problems and their quadratic algebras,” Preprint 89-7 [in Russian], Donetsk Physicotechnical Institute, Donetsk (1989).
Ya. I. Granovskii, A. S. Zhedanov, and I. M. Lutsenko,J. Phys. A (1991) (in print).
Ya. I. Granovskii and A. S. Zhedanov,Zh. Eksp. Teor. Fiz.,94, 49 (1988).
P. Higgs,J. Phys. A,12, 309 (1979); H. Leemon,J. Phys. A,12, 489 (1979).
E. K. Sklyanin,Funktsional Analiz i Ego Prilozhen.,16, 27 (1982);17, 34 (1983).
J. Wilson,SIAM J. Math. Anal.,11, 690 (1980).
A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov,Classical Orthogonal Polynomials of a Discrete Variable [in Russian], Nauka, Moscow (1985).
C. Quesne,J. Phys. A,21, 3093 (1988); O. J. Gal'bert, Ya. I. Granovskii, and A. S. Zhedanov,Phys. Lett. A,153, 177 (1991).
Additional information
Donetsk State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 2, pp. 207–216, May, 1992.
Rights and permissions
About this article
Cite this article
Granovskii, Y.I., Zhedanov, A.S. & Lutsenko, I.M. Quadratic algebras and dynamics in curved spaces. I. Oscillator. Theor Math Phys 91, 474–480 (1992). https://doi.org/10.1007/BF01018846
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01018846