Uniform observation of semiclassical Schr{\"o}dinger eigenfunctions on an interval
Analysis of PDEs
2023-04-26 v2 Spectral Theory
Abstract
We consider eigenfunctions of a semiclassical Schr{\"o}dinger operator on an interval, with a single-well type potential and Dirichlet boundary conditions. We give upper/lower bounds on the L^2 density of the eigenfunctions that are uniform in both semiclassical and high energy limits. These bounds are optimal and are used in an essential way in a companion paper in application to a controllability problem. The proofs rely on Agmon estimates and a Gronwall type argument in the classically forbidden region, and on the description of semiclassical measures for boundary value problems in the classically allowed region. Limited regularity for the potential is assumed.
Keywords
Cite
@article{arxiv.2203.03271,
title = {Uniform observation of semiclassical Schr{\"o}dinger eigenfunctions on an interval},
author = {Camille Laurent and Matthieu Léautaud},
journal= {arXiv preprint arXiv:2203.03271},
year = {2023}
}