Localization and the landscape function for regular Sturm-Liouville operators

TL;DR

This paper investigates the localization properties of eigenfunctions in regular Sturm-Liouville operators, providing bounds, characterizations, and numerical illustrations to deepen understanding of their behavior.

Contribution

It introduces new bounds and characterizations of eigenfunction localization and the landscape function for Sturm-Liouville operators, supported by numerical experiments.

Findings

Derived non-asymptotic and asymptotic bounds on localization coefficients

Characterized the landscape function via the first eigenfunction

Numerical experiments validate theoretical results

Abstract

We consider the localization in the eigenfunctions of regular Sturm-Liouville operators. After deriving non-asymptotic and asymptotic lower and upper bounds on the localization coefficient of the eigenfunctions, we characterize the landscape function in terms of the first eigenfunction. Several numerical experiments are provided to illustrate the obtained theoretical results.