Zeros of exceptional orthogonal polynomials and the maximum of the modulus of an energy function

TL;DR

This paper investigates the zeros of exceptional orthogonal polynomials, revealing that under certain conditions, the energy function of a related many-particle system reaches its maximum at these zeros, linking polynomial zeros to physical energy landscapes.

Contribution

It introduces a new property of the zeros of exceptional orthogonal polynomials and establishes conditions under which the energy function peaks at these zeros.

Findings

Energy function attains maximum at zeros of XOP under specific conditions

Fixed imaginary parts of complex zeros influence energy function maxima

Provides a sufficient condition related to weight function denominators

Abstract

We propose a new property of the zeros of exceptional orthogonal polynomials. It has been known that exceptional orthogonal polynomials (XOP) have both real and complex zeros. By fixing m variables at the imaginary parts of the complex zeros of XOP, we find that in some cases the modulus of the energy function of a many-particle system attains its maximum at the zeros of XOP. We give a sufficient condition for this result with respect to the denominators of the weight function of XOP.