Uniform observation of semiclassical Schr{\"o}dinger eigenfunctions on an interval

TL;DR

This paper establishes uniform, optimal bounds on the L^2 density of semiclassical Schrödinger eigenfunctions on an interval with a single-well potential, applicable in high energy and semiclassical limits.

Contribution

It provides the first uniform bounds on eigenfunction densities for boundary value problems with minimal regularity assumptions, using Agmon estimates and semiclassical measure analysis.

Findings

Bounds are optimal and uniform in semiclassical and high energy limits.

Results are applied to controllability problems in a companion paper.

Analysis relies on Agmon estimates and semiclassical measures.

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.