Degeneration of the Superintegrable System with Potentials Described by the Sixth Painlevé Transcendents

DOI: 10.4236/jamp.2014.211113   PDF   HTML     3,100 Downloads   3,462 Views  


This article concerns the quantum superintegrable system obtained by Tremblay and Winternitz, which allows the separation of variables in polar coordinates and possesses three conserved quantities with the potential described by the sixth Painlevé equation. The degeneration procedure from the sixth Painlvé equation to the fifth one yields another new superintegrable system; however, the Hermitian nature is broken.

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Sasaki, Y. (2014) Degeneration of the Superintegrable System with Potentials Described by the Sixth Painlevé Transcendents. Journal of Applied Mathematics and Physics, 2, 996-999. doi: 10.4236/jamp.2014.211113.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Tremblay, F. and Winternitz, P. (2010) Third-Order Superintegrable Systems Separating in Polar Coordinates. Journal of Physics A: Mathematical and Theoretical, 43, Article ID: 175206 (17p).
[2] Cosgrove, C.M. (2006) Higher Order Painlevé Equations in the Polynomial Class II. Bureau Symbol P1. Studies in Applied Mathematics, 116, 321-413.
[3] Cosgrove, C.M. and Scoufis, G. (1993) Painlevé Classifica-tion of a Class of Differential Equations of the Second Order and Second Degree. Stud. Appl. Math., 88, 25-87.
[4] Conte, R., Grundland, A.M. and Musette, M. (2006) A Reduction of the Resonant Three-Wave Interaction to the Generic Sixth Painlevé Equation. Journal of Physics A: Mathematical and General, 39, 12115-12127.
[5] Conte, R. and Musette, M. (2008) The Painlevé Handbook. Springer Science+Business Media B.V., Dordrecht.
[6] Ince, E.L. (1956) Ordinary Differential Equations. Dover Publ., Inc., New York.
[7] Okamoto, K. (1986) Isomonodromic Deformation and Painlevé Equations, and the Garnier System. J. Fac. Sci. Univ. Tokyo, Sect. IA, 33, 575-618.
[8] Marquette, I. and Winternitz, P. (2008) Superintegrable Systems with Third-Order Integrals of Motion. Journal of Physics A: Mathematical and General, 41, Article ID: 304031 (10p).

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