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 $L^2$-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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