Higher-order SUSY, exactly solvable potentials, and exceptional orthogonal polynomials

TL;DR

This paper constructs exactly solvable rational extensions of radial oscillator potentials using higher-order supersymmetric quantum mechanics, linking them to Laguerre-type exceptional orthogonal polynomials, and classifies the potentials for specific polynomial degrees.

Contribution

It introduces a systematic method to generate and classify rationally-extended potentials and their associated exceptional orthogonal polynomials within higher-order SUSY quantum mechanics.

Findings

Existence of exactly $$, 2, and 3 potentials for $$, 2, and 3, respectively.

Construction of associated families of exceptional orthogonal polynomials.

Classification of potentials based on the degree of polynomial $g_{}$.

Abstract

Exactly solvable rationally-extended radial oscillator potentials, whose wavefunctions can be expressed in terms of Laguerre-type exceptional orthogonal polynomials, are constructed in the framework of $k$th-order supersymmetric quantum mechanics, with special emphasis on $k=2$. It is shown that for $\mu=1$, 2, and 3, there exist exactly $\mu$ distinct potentials of $\mu$th type and associated families of exceptional orthogonal polynomials, where $\mu$ denotes the degree of the polynomial $g_{\mu}$ arising in the denominator of the potentials.