Second order linear differential equations with a basis of solutions having only real zeros
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866913449177513984 |
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| author | Bergweiler, Walter Eremenko, Alexandre Rempe, Lasse |
| author_facet | Bergweiler, Walter Eremenko, Alexandre Rempe, Lasse |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_08949 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Second order linear differential equations with a basis of solutions having only real zeros Bergweiler, Walter Eremenko, Alexandre Rempe, Lasse Complex Variables Classical Analysis and ODEs 34M10 (Primary) 34M05, 30D15 (Secondary) 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. |
| title | Second order linear differential equations with a basis of solutions having only real zeros |
| topic | Complex Variables Classical Analysis and ODEs 34M10 (Primary) 34M05, 30D15 (Secondary) |
| url | https://arxiv.org/abs/2204.08949 |