# The Heun Equation and Generalized Sl(2) Algebra

Volume 16, Issue 1, pp 77-80
• 1859 Views

### Authors

J. Sadeghi - Department of Physics, Islamic Azad University, Ayatollah Amoli Branch, P. O. Box 678, Amol, Iran. A. Vaezi - Department of Mathematics, University of Mazandaran, P. O. Box 95447, Babolsar, Iran. F. Larijani - Department of Physics, Islamic Azad University, Ayatollah Amoli Branch, P. O. Box 678, Amol, Iran.

### Abstract

In this paper, first we introduce the Heun equation. In order to solve such equation we show the generators of generalized $sl(2)$. Second, we arrange the Heun equation in terms of new operators formed of generalized $sl(2)$ generators and it's commutator relation. Here, instead of $J^+(r), J^-(r)$ and $J^0$ we use the $P^+(r), P^-(r)$ and $P^0(r)$ as operators of generalized sl(2) algebra. This correspondence gives us opportunity to arrange the parameters $\alpha$ and $\beta$ in $P^0(r)$. Also, the commutator of such operators leads us to have generalized $sl(2)$ algebra. Also, we obtain the Casimir operators and show that it corresponds to $P^+, P^-$ and some constants. These operators lead to deform the structure of generalized $sl(2)$ algebra in the Heun equation. Finally, we investigate the condition for exactly and quasi-exactly solvable system with constraint on the corresponding operators $P^+$ and $P^-$.

### Keywords

• Heun equation
• generalized $sl(2)$ algebra
• commutative relation.

### References

• [1] A. Decarreau, M. C. Dumont-Lepage, P. Maroni, A. Robert, A. Ronveaux, Formes canoniques des equations confluentes de l'equation de Heun, Ann. Soc. Sci. Bruxelles, 92 (1978), 53-78.

• [2] A. Decarreau, P. Maroni, A. Robert, Sur les equations confluentes de l'equation de Heun, Ann. Soc. Sci. Bruxelles, 92 (1978), 151-189.

• [3] B. D. B. Figueiredo, On some solutions to generalized spheroidal wave equations and applications, J. Phys. A, 35 (2002), 2877-2906.

• [4] B. D. B. Figueiredo, Ince's limits for confluent and double-confluent Heun equations, J. Math. Phys., 46 (2005), 23 pages.

• [5] B. D. B. Figueiredo, Generalized spheroidal wave equation and limiting cases, J. Math. Phys., 48 (2007), 43 pages.

• [6] T. Kimura, On Fuchsian differential equations reducible to hypergeometric equation by linear transformations, Funkcial. Ekvac., 13 (1970/71), 213-232.

• [7] E. W. Leaver, Solutions to a generalized spheroidal wave equation: Teukolsky's equations in general relativity, and the two-center problem in molecular quantum mechanics, J. Math. Phys., 27 (1986), 1238-1265.

• [8] E. G. C. Poole, Introduction to the theory of linear differential equations, Dover Publications, New York (1936)

• [9] A. Ronveaux, Heuns Differential Equations, Oxford University Press, New York (1995)

• [10] A. H. Wilson, A generalised spheroidal wave equation, proc. Roy. Soc. London, 118 (1928), 617-635.

• [11] A. H. Wilson, The ionised hydrogen molecule, Proc. Roy. Soc., 118 (1928), 635-647.