Hidden Symmetry in Quasi-Exactly Solvable Fractional Power Potentials

Abstract

We show that certain fractional power potentials possess the hidden symmetry as defined by M. Znojil. Schrodinger equations for such symmetric potentials are shown to be related to each other by a simple change of coordinate which involves a fractional power of the imaginary unit i. Our result explains and generalizes a recent one on two particular fractional power potentials. Introduction Interactions for which the nonrelativistic Schrodinger equation can be completely solved by algebraic methods are very rare (p.e. harmonic oscillator, Coulomb potential). While for most potentials no exact solution can be given at all, for some at least a finite number of eigenstates and energies take an elementary form. Potentials of the latter class are called quasi-exactly solvable (QES) and ever since they were introduced about 25 years ago, 1)-3) numerous papers have been and are being devoted to methods of finding and classifying them (see Ref. 4) for an overview). Almost 20 years ago QES potentials V of the form