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Uniform observation of semiclassical Schrödinger eigenfunctions on an interval

Camille Laurent and Matthieu Léautaud
Vol. 5 (2023), No. 1, 125–170
DOI: 10.2140/tunis.2023.5.125
Abstract

We consider eigenfunctions of a semiclassical Schrödinger operator on an interval, with a single-well type potential and Dirichlet boundary conditions. We give upper and lower bounds on the L2-density of the eigenfunctions that are uniform in both semiclassical and high energy limits. These bounds are optimal and are applied in an essential way in a companion paper 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.

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Keywords
semiclassical Schrödinger operator, eigenfunctions, observability
Mathematical Subject Classification
Primary: 35B60, 35P20, 47F05, 93B07, 93C73
Milestones
Received: 4 March 2022
Revised: 22 July 2022
Accepted: 11 September 2022
Published: 20 April 2023
Authors
Camille Laurent
Laboratoire Jacques-Louis Lions
Sorbonne Universités
Université Pierre et Marie Curie
Paris
France
Matthieu Léautaud
Laboratoire de Mathématiques d’Orsay
Université Paris-Saclay
Paris
France
© 2023 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers).