Disconjugacy, regularity of multi-indexed rationally-extended potentials, and Laguerre exceptional polynomials

TL;DR

This paper uses disconjugacy properties of second-order differential equations to analyze the regularity of rationally-extended quantum potentials related to exceptional orthogonal polynomials, focusing on the isotonic oscillator.

Contribution

It demonstrates how disconjugacy can verify the regularity of extended potentials and provides a polynomial-based proof of potential denominator properties in supersymmetric quantum mechanics.

Findings

Potential denominator polynomial has a nonvanishing constant term.

The sign of the constant term matches the highest-degree term.

Eigenfunctions and potentials are nodeless due to disconjugacy.

Abstract

The power of the disconjugacy properties of second-order differential equations of Schr\"odinger type to check the regularity of rationally-extended quantum potentials connected with exceptional orthogonal polynomials is illustrated by re-examining the extensions of the isotonic oscillator (or radial oscillator) potential derived in kth-order supersymmetric quantum mechanics or multistep Darboux-B\"acklund transformation method. The function arising in the potential denominator is proved to be a polynomial with a nonvanishing constant term, whose value is calculated by induction over k. The sign of this term being the same as that of the already known highest-degree term, the potential denominator has the same sign at both extremities of the definition interval, a property that is shared by the seed eigenfunction used in the potential construction. By virtue of disconjugacy, such a property implies the nodeless character of both the eigenfunction and the resulting potential.