Schr{\"o}dinger eigenfunctions sharing the same modulus and applications to the control of quantum systems
Abstract
In this paper we investigate when linearly independent eigenfunctions of the Schr\''odinger operator may have the same modulus. General properties are established and the one-dimensional case is treated in full generality. The study is motivated by its application to the bilinear control of the Schr{\"o}dinger equation. By assuming that the potentials of interaction satisfy a saturation property and by adapting a strategy recently proposed by Duca and Nersesyan, we discuss when the system can be steered arbitrarily fast between energy levels. Extensions of the previous results to quantum graphs are finally presented.
Cite
@article{arxiv.2403.08341,
title = {Schr{\"o}dinger eigenfunctions sharing the same modulus and applications to the control of quantum systems},
author = {Ugo Boscain and Kévin Le Balc'H and Mario Sigalotti},
journal= {arXiv preprint arXiv:2403.08341},
year = {2025}
}
Comments
Updated version from June, 2025 after publication in MCSS,https://doi.org/10.1007/s00498-025-00406-y. We clarified the followingpoints: Properties of the eigenfunctions on the disk, Definition of thesaturation cones, Proof of the small-time isomodulus approximatecontrollability result, Proof of the verification of the saturationhypothesis