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

Abstract

Consider a homogenized spectral pencil of exactly solvable linear differential operators $T_{\lambda }={\sum }_{i=0}^k{Q}_i(z){\lambda }^{k-i}\frac{d^i}{d{z}^i}$, 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 ${\lambda }_{n,j}$, $1\le j\le k$, 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 ${\Psi }_j(z)={\lim }_{n\to \infty }\frac{p_{n,j}^{\prime }(z)}{{\lambda }_{n,j}{p}_{n,j}(z)}$ exist, are analytic and satisfy the algebraic equation ${\sum }_{i=0}^{k}Qi(z){\Psi }_j^i(z)=0$ almost everywhere in $C{P}^1$. As a consequence we obtain a class of algebraic functions possessing a branch near $\infty \in C{P}^1$ which is representable as the Cauchy transform of a compactly supported probability measure.

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