QUASI-EXACTLY SOLVABLE POTENTIALS WITH ARBITRARY TWO KNOWN EIGENSTATES FOR SYSTEMS WITH POSITION-DEPENDENT MASS

The number of exactly solvable potentials is rather limited. Therefore, much attention has been devoted to quasi exactly solvable (QES) problems for which it is possible to find exactly in an explicit form only a few energy levels and the corresponding wave functions. It was found that supersymmetric (SUSY) quantum mechanics is a powerful tool for this purpose. In our paper [V. M. Tkachuk, Phys. Lett. A 245, 177 (1998)] we proposed SUSY method for generating QES potentials with two known eigenstates. In the frame of this method the general expressions for the superpotential, the potential energy and two wavefunctions which correspond to the two energy levels were obtained. Using this method, we obtained QES potentials for which we found in the explicit form the energy levels and wave functions of the ground and first excited states. Later in [V. M. Tkachuk, J. Phys. A 34, 6339 (2001)] this method was generalized for the case of an arbitrary two energy level. In this paper the method of supersymmetric quantum mechanics has been used for discovering the quasiexactly solvable potentials for systems with position-dependent mass. We have established the conditions which provide regular potential energy in the case of a singular generating function. The examples of the quasi-exactly solvable potentials for a particle with position-dependent mass with two known eigenstates have been considered in the cases of regular and singular generating functions. We have found the exact eigenfunctions of a particle with position dependent mass which correspond to the two energy levels.