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

Consider a homogenized spectral pencil of exactly solvable linear differential operators T-lambda = Sigma(k)(i=0) Q(i)(z)lambda(k-i) d(i)/dz(i), where each Q(i)(z) is a polynomial of degree at most i and lambda is the spectral parameter. We show that under mild nondegeneracy assumptions for all sufficiently large positive integers n there exist exactly k distinct values lambda(n,j), 1 <= j <= k, of the spectral parameter lambda such that the operator T-lambda has a polynomial eigenfunction p(n,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 Psi(j)(Z) = lim(n ->infinity) P'(n,j) (z)/lambda(n,j)p(n,j)(z) exist, are analytic and satisfy the algebraic equation Sigma(k)(i=0)Q(i)(z)Psi(i)(j)(z) = 0 almost everywhere in CP1. As a consequence we obtain a class of algebraic functions possessing a branch near infinity is an element of CP1 which is representable as the Cauchy transform of a compactly supported probability measure