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

Consider a homogenized spectral pencil of exactly solvable linear differential operators Tλ = Pk i=0 Qi(z)λ k−i d i dzi , 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 Tλ 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 Ψj(z) = limn→∞ p′n,j(z) λn,jpn,j(z) exist, are analytic and satisfy the algebraic equation Pk i=0 Qi(z)Ψ i j(z) = 0 almost everywhere in CP. As a consequence we obtain a class of algebraic functions possessing a branch near ∞ ∈ CP which is representable as the Cauchy transform of a compactly supported probability measure.