Homogenized spectral problems for exactly solvable operators: asymptotics of polynomial eigenfunctions

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Consider a homogenized spectral pencil of exactly solvable linear differential operators T=∑ᵢ₌₀ᵏ Qᵢ(z)ᵏ⁻ⁱdⁱ/dzⁱ, where each Qᵢ(z) is a polynomial of degree at most i and is the spectral parameter. We show that under mild nondegeneracy assumptions for all sufficiently large positive integers n there exist exactly k distinct values ₙ,ⱼ, 1≤ j≤ k, of the spectral parameter such that the operator T 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)=ₙ→∞ pₙ,ⱼ'(z)ₙ,ⱼpₙ,ⱼ(z) exist, are analytic and satisfy the algebraic equation ∑ᵢ₌₀ᵏ Qᵢ(z) Ψⱼⁱ(z)=0 almost everywhere in. As a consequence we obtain a class of algebraic functions possessing a branch near ∞∈ which is representable as the Cauchy transform of a compactly supported probability measure.