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Homogenized spectral problems for exactly solvable operators: asymptotics of polynomial eigenfunctions. (English) Zbl 1182.30008
Publ. Res. Inst. Math. Sci. 45, No. 2, 525-568 (2009); corrigendum 48, No. 1, 229-233 (2012).
Authors’ abstract: Consider a homogenized spectral pencil of exactly solvable linear differential operators \(T_{\lambda }=\sum_{i=0}^k Q_{i}(z) \lambda ^{k-i}\frac {d^i}{dz^i}\), where each \(Q_{i}(z)\) is a polynomial of degree at most \(i\) and \(\lambda \) is the spectral parameter. We show that under mild nondegeneracy assumptions for all sufficiently large positive integers \(n\) there exist exactly \(k\) distinct values \(\lambda _{n,j}, 1\leq j\leq k\), of the spectral parameter \(\lambda \) such that the operator \(T_{\lambda }\) 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 \(\Psi _{j}(z)=\lim _{n\to \infty } \frac {p^{\prime }_{n,j}(z)}{\lambda _{n,j}p_{n,j}(z)}\) exist, are analytic and satisfy the algebraic equation \(\sum _{i=0}^k Q_{i}(z) \Psi _{j}^i(z)=0\) almost everywhere in \(\mathbb {CP}^1\). As a consequence we obtain a class of algebraic functions possessing a branch near \(\infty \in \mathbb {CP}^1\) which is representable as the Cauchy transform of a compactly supported probability measure.

MSC:
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
31A35 Connections of harmonic functions with differential equations in two dimensions
34E05 Asymptotic expansions of solutions to ordinary differential equations
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