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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
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.

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