Classical-quantum correspondence for shape-invariant systems

TL;DR

This paper demonstrates that a recent quantization method preserves the exact solvability of shape-invariant quantum systems and explores solutions involving classical and exceptional orthogonal polynomials.

Contribution

It applies a novel quantization procedure to shape-invariant potentials, showing preservation of solvability and analyzing solutions with orthogonal polynomials.

Findings

Quantization preserves exact solvability of shape-invariant potentials.

Explicit solutions involve classical and exceptional orthogonal Laguerre and Jacobi polynomials.

The method involves solving a Gambier XXVII transcendental equation.

Abstract

A quantization procedure, which has recently been introduced for the analysis of Painlev\'e equations, is applied to a general time-independent potential of a Newton equation. This analysis shows that the quantization procedure preserves the exact solvability property for the class of shape-invariant potentials. When a general potential is considered the quantization procedure involves the solution of a Gambier XXVII transcendental equation. Explicit examples involving classical and exceptional orthogonal Laguerre and Jacobi polynomials are discussed.