On spectral polynomials of the Heun equation

TL;DR

This paper investigates the asymptotic distribution of spectral polynomials associated with the Heun equation, proposing a conjecture on their limiting roots as the polynomial degree tends to infinity.

Contribution

It introduces a conjecture describing the limiting set of roots of spectral polynomials for the Heun equation as degree increases, expanding understanding of their asymptotic behavior.

Findings

Formulation of a conjecture on the limiting roots set

Analysis of spectral polynomial roots as degree approaches infinity

Provision of additional insights into the structure of these roots

Abstract

The classical Heun equation has the form {Q(z) d^2/dz^2 +P(z) d/dz +V(z)}S(z)=0 where Q(z) is a cubic, P(z) at most quadratic and V(z) linear polynomials resp. In the second half of the 19-th century E.Heine and T.STieltjes initiated the study of the set of all V(z) such that the above equation has a polynomial solution S(z) of a given degree n. The main goal of the present paper is to study the union of the roots of the latter set of V(z)*s when n->oo. We formulate an intriguing conjecture of K.Takemura describing the limiting set and give a substantial amount of additional information.