TY - JOUR
AU - Jonathan Wunderlich
AU - Michael Plum
PY - 2020/03/16
Y2 - 2024/10/08
TI - Computer-assisted Existence Proofs for One-dimensional Schrödinger-Poisson Systems
JF - Acta Cybernetica
JA - Acta Cybern
VL - 24
IS - 3
SE - Uncertainty Modeling, Software, Verified Computing and Optimization
DO - 10.14232/actacyb.24.3.2020.6
UR - https://cyber.bibl.u-szeged.hu/index.php/actcybern/article/view/4034
AB - 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.
ER -