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]
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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 |
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