Homogenized Spectral Problems for Exactly Solvable Operators: Asymptotics of Polynomial Eigenfunctions

Abstract

Consider a homogenized spectral pencil of exactly solvable linear differential operators T_λ = Σki=0 Qi(z)λ_k−i di/dz__i, where each Qi(z) is a polynomial of degree at most i and λ is the spectral parameter. We show that under mild nondegeneracy assumptions for all sufficiently large positive integers n there exist exactly k distinct values λn,j, 1 ≤ j ≤ k, of the spectral parameter λ such that the operator T_λ has a polynomial eigenfunction pn,j(z) of degree n. These eigenfunctions split into k different families according to the asymptotic behavior of their eigenvalues. We conjecture and prove sequential versions of three fundamental properties: the limits Ψ_j (z) = lim_n_→∞ p'n,j(z)/λ_n,jpn,j_(z) exist, are analytic and satisfy the algebraic equation Σki=0 Qi(z)Ψi_j_(z) = 0 almost everywhere in ℂ ℙ1. As a consequence we obtain a class of algebraic functions possessing a branch near ∞ ∈ ℂ ℙ1 which is representable as the Cauchy transform of a compactly supported probability measure.