Homogenized Spectral Problems for Exactly Solvable Operators: Asymptotics of Polynomial Eigenfunctions
Julius Borcea
Stockholm University, SwedenRikard Bøgvad
Stockholm University, SwedenBoris Shapiro
Stockholm University, Sweden
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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.
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