Computer-assisted existence proofs for one-dimensional Schrödinger-poisson systems

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
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: https://doi.org/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: 2020. Jul. 30. 13:12
URI: http://acta.bibl.u-szeged.hu/id/eprint/69282

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