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
(2022) Second order linear differential equations with a basis of solutions having only real zeros. [Preprint]

[thumbnail of real-zeros5.pdf] Text
real-zeros5.pdf - Author Accepted Manuscript

Download (626kB) | Preview

Abstract

Let $A$ be a transcendental entire function of finite order. We show that if the differential equation $w''+Aw=0$ has two linearly independent solutions with only real zeros, then the order of $A$ must be an odd integer or one half of an odd integer. Moreover, $A$ 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.

Item Type: Preprint
Additional Information: 46 pages, 5 figures. V2: some overall revision of v1
Uncontrolled Keywords: 34M10 (Primary) 34M05, 30D15 (Secondary), math.CA, math.CV, math.CV
Divisions: Faculty of Science and Engineering > School of Physical Sciences
Depositing User: Symplectic Admin
Date Deposited: 09 Jun 2022 07:55
Last Modified: 15 Mar 2024 05:05
DOI: 10.48550/arxiv.2204.08949
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3156081