Interlacing and non-orthogonality of spectral polynomials for the Lam\'e operator

TL;DR

This paper investigates the zeros of Van Vleck polynomials related to the Lamé operator, revealing their interlacing properties and non-orthogonality, thus providing counterexamples to classical orthogonal polynomial theorems.

Contribution

It demonstrates that Van Vleck polynomial zeros interlace and that the associated spectral polynomials are not orthogonal, challenging existing assumptions in spectral theory.

Findings

Zeros of Van Vleck polynomials interlace for successive degrees

Spectral polynomials from Van Vleck zeros are not orthogonal

Counterexample to the converse of the orthogonal polynomial interlacing theorem

Abstract

Polynomial solutions to the generalized Lam\'e equation, the Stieltjes polynomials, and the associated Van Vleck polynomials have been studied since the 1830's in various contexts including the solution of Laplace equations on an ellipsoid. Recently there has been renewed interest in the distribution of the zeros of Van Vleck polynomials as the degree of the corresponding Stieltjes polynomials increases. In this paper we show that the zeros of Van Vleck polynomials corresponding to Stieltjes polynomials of successive degrees interlace. We also show that the spectral polynomials formed from the Van Vleck zeros are not orthogonal with respect to any weight. This furnishes a counterexample, coming from a second order differential equation, to the converse of the well known theorem that the zeros of orthogonal polynomials interlace.