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

### Julius Borcea

### Rikard Bøgvad

Stockholm University, Sweden### Boris Shapiro

Stockholm University, Sweden

## Abstract

Consider a homogenized spectral pencil of exactly solvable linear differential operators $T_{λ}=∑_{i=0}Q_{i}(z)λ_{k−i}dz_{i}d_{i} $, where each $Q_{i}(z)$ is a polynomial of degree at most $i$ and $λ$ is the spectral parameter. We show that under mild nondegeneracy assumptions for all sufficiently large positive integers $n$ there exist exactly $k$ distinct values $λ_{n,j}$, $1≤j≤k$, of the spectral parameter $λ$ such that the operator $T_{λ}$ 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 $Ψ_{j}(z)=lim_{n→∞}λ_{n,j}p_{n,j}(z)p_{n,j}(z) $ exist, are analytic and satisfy the algebraic equation $∑_{i=0}Qi(z)Ψ_{j}(z)=0$ almost everywhere in $CP_{1}$. As a consequence we obtain a class of algebraic functions possessing a branch near $∞∈CP_{1}$ which is representable as the Cauchy transform of a compactly supported probability measure.

## Cite this article

Julius Borcea, Rikard Bøgvad, Boris Shapiro, Homogenized Spectral Problems for Exactly Solvable Operators: Asymptotics of Polynomial Eigenfunctions. Publ. Res. Inst. Math. Sci. 45 (2009), no. 2, pp. 525–568

DOI 10.2977/PRIMS/1241553129