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

  • Julius Borcea

    Stockholm University, Sweden
  • Rikard Bøgvad

    Stockholm University, Sweden
  • Boris Shapiro

    Stockholm University, Sweden

Abstract

Consider a homogenized spectral pencil of exactly solvable linear differential operators , where each  is a polynomial of degree at most and is the spectral parameter. We show that under mild nondegeneracy assumptions for all sufficiently large positive integers there exist exactly distinct values , , of the spectral parameter such that the operator has a polynomial eigenfunction of degree . These eigenfunctions split into different families according to the asymptotic behavior of their eigenvalues. We conjecture and prove sequential versions of three fundamental properties: the limits exist, are analytic and satisfy the algebraic equation almost everywhere in . As a consequence we obtain a class of algebraic functions possessing a branch near which is representable as the Cauchy transform of a compactly supported probability measure.

A correction to this paper is available.

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