@article{Wunderlich_Plum_2020, title={Computer-assisted Existence Proofs for One-dimensional Schrödinger-Poisson Systems}, volume={24}, url={https://cyber.bibl.u-szeged.hu/index.php/actcybern/article/view/4034}, DOI={10.14232/actacyb.24.3.2020.6}, abstractNote={<p>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.</p> <p>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.</p> <p>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.</p>}, number={3}, journal={Acta Cybernetica}, author={Wunderlich, Jonathan and Plum, Michael}, year={2020}, month={Mar.}, pages={373-391} }