Computer-assisted Existence Proofs for One-dimensional Schrödinger-Poisson Systems
| Autor: | Michael Plum, Jonathan Wunderlich |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Information Systems and Management
010102 general mathematics Essential spectrum Inverse Management Science and Operations Research Mathematical proof 01 natural sciences Theoretical Computer Science 010101 applied mathematics Computer-assisted proof Exact solutions in general relativity Linearization Norm (mathematics) Computer Science (miscellaneous) Applied mathematics Computer Vision and Pattern Recognition 0101 mathematics Electrical and Electronic Engineering Software Eigenvalues and eigenvectors Mathematics |
| Zdroj: | Acta Cybernetica. 24:373-391 |
| ISSN: | 2676-993X 0324-721X |
| DOI: | 10.14232/actacyb.24.3.2020.6 |
| Popis: | Motivated by the three-dimensional time-dependent Schrödinger-Poisson system we prove the existence of non-trivial solutions of the one-dimensional stationary Schrödinger-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. |
| Databáze: | OpenAIRE |
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