Computer-assisted Existence Proofs for One-dimensional Schrödinger-Poisson Systems

Keywords: computer-assisted proof, existence, enclosure, Schrödinger-Poisson system


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.


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How to Cite
Wunderlich, J., & Plum, M. (2020). Computer-assisted Existence Proofs for One-dimensional Schrödinger-Poisson Systems. Acta Cybernetica, 24(3), 373-391.
Uncertainty Modeling, Software, Verified Computing and Optimization