Second order linear differential equations with a basis of solutions having only real zeros

Bergweiler, Walter, Eremenko, Alexandre and Rempe, Lasse ORCID: 0000-0001-8032-8580 (2023) Second order linear differential equations with a basis of solutions having only real zeros. Journal d'Analyse Mathématique. pp. 1-56.

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Official URL: http://dx.doi.org/10.1007/s11854-023-0294-z

Abstract

<jats:title>Abstract</jats:title><jats:p>Let <jats:italic>A</jats:italic> be a transcendental entire function of finite order. We show that, if the differential equation <jats:italic>w</jats:italic>″ + <jats:italic>Aw</jats:italic> = 0 has two linearly independent solutions with only real zeros, then the order of <jats:italic>A</jats:italic> must be an odd integer or one half of an odd integer. Moreover, <jats:italic>A</jats:italic> has completely regular growth in the sense of Levin and Pfluger. These results follow from a more general geometric theorem, which classifies symmetric local homeomorphisms from the plane to the sphere for which all zeros and poles lie on the real axis, and which have only finitely many singularities over finite non-zero values.</jats:p>

Item Type:	Article
Divisions:	Faculty of Science and Engineering > School of Physical Sciences
Depositing User:	Symplectic Admin
Date Deposited:	02 Oct 2023 15:06
Last Modified:	15 Mar 2024 05:05
DOI:	10.1007/s11854-023-0294-z
Open Access URL:	https://doi.org/10.1007/s11854-023-0294-z
Related URLs:	Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3173316