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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Entire functions of finite order as solutions to certain complex linear differential equations


Author: N. Anghel
Journal: Proc. Amer. Math. Soc. 140 (2012), 2319-2332
MSC (2010): Primary 30D15, 34M05; Secondary 33C10, 34L40
Published electronically: October 3, 2011
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Abstract: When is an entire function of finite order a solution to a complex 2nd order homogeneous linear differential equation with polynomial coefficients? In this paper we will give two (equivalent) answers to this question. The starting point of both answers is the Hadamard product representation of a given entire function of finite order. While the first answer involves certain Stieltjes-like relations associated to the function, the second one requires the vanishing of all but finitely many suitable expressions constructed via the Gil' sums of the zeros of the function. Applications of these results will also be given, most notably to the spectral theory of one-dimensional Schrödinger operators with polynomial potentials.


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Additional Information

N. Anghel
Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203
Email: anghel@unt.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11055-4
PII: S 0002-9939(2011)11055-4
Keywords: Complex differential equations, polynomial coefficients, entire functions, finite order, zeros, Stieltjes relations, Gil’ sums, Schrödinger operators
Received by editor(s): September 29, 2010
Received by editor(s) in revised form: February 4, 2011
Published electronically: October 3, 2011
Communicated by: Walter Van Assche
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.