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

Consider a homogenized spectral pencil of exactly solvable linear differential operators $T_{\la}=\sum_{i=0}^k Q_{i}(z)\la^{k-i}\frac {d^i}{dz^i}$, where each $Q_{i}(z)$ is a polynomial of degree at most $i$ and $\la$ is the spectral parameter. We show that under mild nondegeneracy assumptions for all sufficiently large positive integers $n$ there exist exactly $k$ distinct values $\la_{n,j}$, $1\le j\le k$, of the spectral parameter $\la$ such that the operator $T_{\la}$ 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 $\Psi_{j}(z)=\lim_{n\to\infty} \frac{p_{n,j}'(z)}{\la_{n,j}p_{n,j}(z)}$ exist, are analytic and satisfy the algebraic equation $\sum_{i=0}^k Q_{i}(z) \Psi_{j}^i(z)=0$ almost everywhere in $\bCP$. As a consequence we obtain a class of algebraic functions possessing a branch near $\infty\in \bCP$ which is representable as the Cauchy transform of a compactly supported probability measure.