An algebraic approach for the Dunkl–Killingbeck problem from the bi-confluent Heun equation

Abstract

In this paper, we study the Dunkl–Killingbeck problem in two dimensions. We apply the Lie algebraic approach within the framework of quasi-exact solvability to the radial part of the Dunkl–Killingbeck problem to find the general exact expressions for the energies and corresponding wave functions. The allowed values of the potential parameters are the representation space of sl(2) Lie algebra. In addition, we discuss that the effective potential of the Dunkl–Killingbeck is the same as the obtained from the bi-confluent Heun equation by a suitable variable transformation. Following earlier results, we follow the explicit solutions of this differential equation expressed as a series expansion of Hermite functions and obtain the expansion coefficients from a three-term recurrence relation. In the sequel, we present that this construction leads to the known quasi-exactly solvable (QES) form of the Dunkl–Killingbeck problem. Therefore, we find that the expressions for the energy eigenvalues and wave functions of the corresponding potential term are in agreement with those from the QES formalism. Then, we derive the ladder operators for the Dunkl–Killingbeck problem within the algebraic approach. It seems that this method is the Dunkl–Killingbeck rotation problem solved by operators of the su[Formula: see text] Lie algebra in a specific way.