Non-localization of eigenfunctions for Sturm-Liouville operators and applications
type: Item
instance of: scholarly article
title: Non-localization of eigenfunctions for Sturm-Liouville operators and applications
zbMATH Open document ID: 1382.49017
author: Thibault Liard
author: Pierre Lissy
author: Yannick Privat
published in: Journal of Differential Equations
publication date: 2017-12-20T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 49K15
Mathematics Subject Classification ID: 47A75
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 93B07
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 34B24
zbMATH DE Number: 6820627
zbMATH Keywords: Sturm-Liouville operators
zbMATH Keywords: eigenfunctions
zbMATH Keywords: extremal problems
zbMATH Keywords: calculus of variations
zbMATH Keywords: control theory
zbMATH Keywords: wave equation
reviewed by: Vladimir Jacimovic
MaRDI profile type: Publication
full work available at URL: j.jde.2017.09.043
OpenAlex ID: W2769840057
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DOI: 10.1016/J.JDE.2017.09.043
description: scientific article; zbMATH DE number 6820627