Shape Invariant Rational Extensions And Potentials Related to Exceptional Polynomials

TL;DR

This paper develops a method to construct rational extensions of shape invariant potentials using isospectral shift deformation, leading to new potentials with exceptional polynomial solutions, exemplified by the radial oscillator and P"{o}schl-Teller potentials.

Contribution

It introduces a novel approach to generate infinite sets of rational extensions of shape invariant potentials with exceptional polynomials, including previously unexplored solutions involving third-type exceptional Laguerre polynomials.

Findings

Constructed new rational extensions of the radial oscillator potential.

Identified potential for third-type exceptional Laguerre polynomial solutions.

Suggested multiple rational extensions for the P"{o}schl-Teller potential.

Abstract

In this paper, we show that an attempt to construct shape invariant extensions of a known shape invariant potential leads to, apart from a shift by a constant, the well known technique of isospectral shift deformation. Using this, we construct infinite sets of generalized potentials with $X_m$ exceptional polynomials as solutions. These potentials are rational extensions of the existing shape invariant potentials. The method is elucidated using the radial oscillator and the trigonometric P\"{o}schl-Teller potentials. For the case of radial oscillator, in addition to the known rational extensions, we construct two infinite sets of rational extensions, which seem to be less studied. For one of the potential, we show that its solutions involve a third type of exceptional Laguerre polynomials. Explicit expressions of this generalized infinite set of potentials and the corresponding solutions are presented. For the trigonometric P\"{o}schl-Teller potential, our analysis points to the possibility of several rational extensions beyond those known in literature.