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Progress of Theoretical Physics

Hidden Symmetry in Quasi-Exactly Solvable Fractional Power Potentials

Axel Schulze‐Halberg

Published Dec 1, 2003• 4 citations
Publisher

Abstract

We show that certain fractional power potentials possess the hidden symmetry as defined by M. Znojil. Schrödinger 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.

DOI
10.1143/ptp.110.1235

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We show that certain fractional power potentials possess the hidden symmetry as defined by M. Znojil. Schrödinger 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.
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We suggest the symmetrized Schrödinger equation and propose a general complex solution which is characterized by the imaginary units iii and εεε. This symmetrized Schrödinger equation appears some interesting features.
We show that the list of analytically solvable potentials in nonrelativistic quantum mechanics can be considerably enlarged. In particular, we show that those noncentral potentials for which the Schrödinger equation is separable are analytically solvable provided the separated problem for each of the coordinates belongs to the class of exactly solvable one dimensional problems. As an illustration, we discuss in detail two examples, one in two and the other in three dimensions. A list of analytically solvable noncentral potentials in spherical polar coordinates is also given. Extension of these ideas to other standard orthogonal coordinate systems as well as to higher dimensions is straightforward.