Authors’ 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.
Reviewer’s comment: The operator in question is \[ P_\varepsilon:=-\varepsilon^2\partial_x^2+V_\varepsilon(x) . \]