Orthogonal functions related to Lax pairs in Lie algebras

TL;DR

This paper investigates Lax pairs within 2-parameter Lie algebras, revealing that eigenfunction overlaps relate to orthogonal polynomials, and explores eigenfunctions in specific representations of rak{sl}(d+1,\u00bb) Lie algebras.

Contribution

It introduces a connection between eigenfunctions of Lax pairs in Lie algebras and orthogonal functions, expanding understanding of their spectral properties.

Findings

Eigenfunctions overlap coefficients are expressed via orthogonal polynomials.

Analysis of eigenfunctions for rak{sl}(d+1,\u00bb) in certain representations.

Extension of Lax pair analysis to 2-parameter Lie algebras.

Abstract

We study a Lax pair in a $2$-parameter Lie algebra in various representations. The overlap coefficients of the eigenfunctions of $L$ and the standard basis are given in terms of orthogonal polynomials and orthogonal functions. Moreover, eigenfunctions for the operator $L$ for a Lax pair for $\mathfrak{sl}(d+1,\mathbb C)$ are studied in certain representations.