Non-localization of eigenfunctions for Sturm-Liouville operators and applications (Q1686080)

The paper deals with localization properties of eigenfunctions of Sturm-Liouville operators of the form: \[ A_a = -\partial_{xx} + a(\cdot) Id, \] where \(a(\cdot)\) is a nonnegative bounded potential defined on the interval \((0,L)\). Such problems arise when minimizing the \(L^2\)-norm of a function \(e(\cdot)\) on a measurable subset \(\omega\) of \((0,L)\), where \(e(\cdot)\) runs over the set of all eigenfunctions of \(A_a\) and, at the same time, \(\omega\) runs over all measurable subsets of \((0,L)\) with the prescribed measure. Problems of this kind play important role in Quantum Physics. The one-dimensional problem, that is studied in this paper, arises in control and stabilization of the linear wave equation. Applications of the main result to control and observation theory, along with some simulation results, are also discussed.