# 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 = Σki=0 Qi(z)λ_k−i*

*di/dz__i*, where each

*Qi*(

*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 λn,j, 1 ≤

*j*≤

*k*, of the spectral parameter λ such that the operator

*T_λ has a polynomial eigenfunction*(

*pn*,*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_→∞

*p'n,j*(

*z*)/λ_n,jpn,j_(

*z*) exist, are analytic and satisfy the algebraic equation Σki=0

*Qi*(

*z*)Ψi_j_(

*z*) = 0 almost everywhere in ℂ ℙ1. As a consequence we obtain a class of algebraic functions possessing a branch near ∞ ∈ ℂ ℙ1 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