The energy function with respect to the zeros of the exceptional Hermite polynomials

TL;DR

This paper investigates the energy function related to the zeros of exceptional Hermite polynomials, analyzing eigenvalue localization, special arrangements, and behavior of the energy function at regular zeros.

Contribution

It provides new insights into the Hessian eigenvalue localization and detailed behavior of the energy function for exceptional Hermite polynomial zeros.

Findings

Eigenvalues of the Hessian are localized in the general case.

More precise results are obtained for special zero arrangements.

The energy function's behavior at regular zeros is characterized.

Abstract

We examine the energy function with respect to the zeros of exceptional Hermite polynomials. The localization of the eigenvalues of the Hessian is given in the general case. In some special arrangements we have a more precise result on the behavior of the energy function. Finally we investigate the energy function with respect to the regular zeros of the exceptional Hermite polynomials.