Wunderlich Jonathan and Plum Michael: Computer-assisted existence proofs for one-dimensional Schrödinger-poisson systems. In: Acta cybernetica, (24) 3. pp. 373-391. (2020)
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Abstract
Motivated by the three-dimensional time-dependent Schr¨odinger-Poisson system we prove the existence of non-trivial solutions of the one-dimensional stationary Schr¨odinger-Poisson system using computer-assisted methods. Starting from a numerical approximate solution, we compute a bound for its defect, and a norm bound for the inverse of the linearization at the approximate solution. For the latter, eigenvalue bounds play a crucial role, especially for the eigenvalues “close to” zero. Therefor, we use the Rayleigh-Ritz method and a corollary of the Temple-Lehmann Theorem to get enclosures of the crucial eigenvalues of the linearization below the essential spectrum. With these data in hand, we can use a fixed-point argument to obtain the desired existence of a non-trivial solution “nearby” the approximate one. In addition to the pure existence result, the used methods also provide an enclosure of the exact solution.
| Item Type: | Article |
|---|---|
| Heading title: | Uncertainty modeling, software verified computing and optimization |
| Journal or Publication Title: | Acta cybernetica |
| Date: | 2020 |
| Volume: | 24 |
| Number: | 3 |
| ISSN: | 0324-721X |
| Page Range: | pp. 373-391 |
| Language: | English |
| Publisher: | University of Szeged, Institute of Informatics |
| Place of Publication: | Szeged |
| Event Title: | Summer Workshop on Interval Methods (11.) (2018) (Rostock) |
| Related URLs: | http://acta.bibl.u-szeged.hu/69263/ |
| DOI: | 10.14232/actacyb.24.3.2020.6 |
| Uncontrolled Keywords: | Számítástechnika, Kibernetika |
| Additional Information: | Bibliogr.: p. 390-391. ; összefoglalás angol nyelven |
| Subjects: | 01. Natural sciences 01. Natural sciences > 01.02. Computer and information sciences |
| Date Deposited: | 2020. Jul. 30. 13:12 |
| Last Modified: | 2022. Jun. 21. 09:55 |
| URI: | http://acta.bibl.u-szeged.hu/id/eprint/69282 |
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