Supersymmetric approach for generating quasi-exactly solvable potentials

Abstract:
Recently much attention has been given to the quasi-exactly solvable (QES) potentials, for which a finite number of energy levels and corresponding wave functions are known in an explicit form. In the frame of supersymmetric (SUSY)quantum mechanics we propose the inverse method for constructing the QES potentials with arbitrary two known energy levels and corresponding wave functions. The QES potential and the wave functions of the two energy levels are expressed by some generating function the properties of which determine the state numbers of these levels. Choosing different generating functions we present the explicit examples of the QES potentials. Multidimensional QES potentials are also discussed.