On Asymptotics of Polynomial Eigenfunctions for Exactly-Solvable Differential Operators

Abstract

In this paper we study the asymptotic zero distribution of eigenpolynomials for degenerate exactly-solvable operators. We present an explicit conjecture and partial results on the growth of the largest modulus of the roots of the unique and monic n:th degree eigenpolynomial of any such operator as the degree n tends to infinity. Based on this conjecture we deduce the algebraic equation satified by the Cauchy transform of the asymptotic root measure of the properly scaled eigenpolynomials, for which the union of all roots is conjecturally contained in a compact set.