Some examples of matrix-valued orthogonal functions having a differential and an integral operator as eigenfunctions

TL;DR

This paper presents examples of matrix-valued orthogonal functions that are eigenfunctions of both differential and integral operators, providing new structural insights and visualizations related to Hermite and wave functions.

Contribution

It introduces specific matrix-valued orthogonal functions that serve as simultaneous eigenfunctions of differential and integral operators, expanding the understanding of such functions.

Findings

Derived integral representations of the functions

Established structural formulas for the functions

Visualized relationships with Hermite and wave functions

Abstract

The aim of this paper is to show some examples of matrix-valued orthogonal functions on the real line which are simultaneously eigenfunctions of a second-order differential operator of Schr\"{o}dinger type and an integral operator of Fourier type. As a consequence we derive integral representations of these functions as well as other useful structural formulas. Some of these functions are plotted to show the relationship with the Hermite or wave functions.