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

Consider a homogenized spectral pencil of exactly solvable linear differential operators $T_{\la}=\sum_{i=0}^k Q_{i}(z)\la^{k-i}\frac {d^i}{dz^i}$,whereeach$Q_{i}(z)$isapolynomialofdegreeatmost$i$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 ${\text{la}}_{n,j}$, $1\le j\le k$, of the spectral parameter $\text{la}$ such that the operator $T_{\text{la}}$ 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 ${\mathrm{\Psi }}_j(z)={\lim }_{n\to \mathrm{\infty }}\frac{p_{n,j}^{\mathrm{\prime }}(z)}{{\text{la}}_{n,j}{p}_{n,j}(z)}$ exist, are analytic and satisfy the algebraic equation $\sum_{i=0}^k Q_{i}(z) \Psi_{j}^i(z)=0$almosteverywherein$\bCP$. As a consequence we obtain a class of algebraic functions possessing a branch near $\mathrm{\infty }\in \text{bCP}$ which is representable as the Cauchy transform of a compactly supported probability measure.