Interrelation among Solvable Potentials and Extensions of SWKB Quantization Condition

TL;DR

This paper extends the SWKB quantization condition to a broader class of solvable quantum systems, including shape-invariant and position-dependent mass potentials, revealing deeper interrelations and exact quantization rules.

Contribution

It introduces extended SWKB formulas for Natanzon potentials and applies them to systems with position-dependent masses, linking classical orthogonal polynomials to exact quantization.

Findings

Extended SWKB quantization conditions derived for Natanzon potentials.

Exact quantization rule established for position-dependent mass systems.

Conjecture on the role of classical orthogonal polynomials in SWKB exactness.

Abstract

The exactly solvable Schr\"{o}dinger equations with the conventional shape-invariant potentials are known to be related with each other through point cannonical transformations. In this paper, we extend the idea to integral formulae called the SWKB integrals. By virtue of this, we derive extended forms of the SWKB quantization condition for certain classes of Natanzon potentials. We further demonstrate that the same idea can also be applied to obtain an exact quantization rule for a subclass of quantum systems with position-dependent effective masses, provided their solutions involve the classical orthogonal polynomials. Based on the findings, we conjecture about the implication of the exactness of the SWKB formula in relation to the classical orthogonal polynomials.