Localization of eigenfunctions via an effective potential

TL;DR

This paper introduces a new framework where the reciprocal of the landscape function acts as an effective potential, enabling explicit bounds on eigenfunction decay and eigenvalue distribution for operators on complex domains.

Contribution

It reveals that $1/u$ serves as an effective potential, providing detailed insights into eigenfunctions and eigenvalues from a single landscape measurement.

Findings

The reciprocal of the landscape function bounds eigenfunction decay.

Explicit estimates on eigenvalue distribution near the spectrum's bottom.

Numerical examples illustrate the theoretical results.

Abstract

We consider the localization of eigenfunctions for the operator $L=-\mbox{div} A \nabla + V$ on a Lipschitz domain $\Omega$ and, more generally, on manifolds with and without boundary. In earlier work, two authors of the present paper demonstrated the remarkable ability of the landscape, defined as the solution to $Lu=1$, to predict the location of the localized eigenfunctions. Here, we explain and justify a new framework that reveals a richly detailed portrait of the eigenfunctions and eigenvalues. We show that the reciprocal of the landscape function, $1/u$, acts as an effective potential. Hence from the single measurement of $u$, we obtain, via $1/u$, explicit bounds on the exponential decay of the eigenfunctions of the system and estimates on the distribution of eigenvalues near the bottom of the spectrum. (This version strengthens and simplifies the results of the first one by replacing a global bi-Lipschitz hypothesis on the domain with a local bi-Lipschitz hypothesis. It improves on the second version by adding pictures and numerical examples. This version is identical to the third version; all that is changed is to correct some tex mistakes in symbols in this abstract. There are no changes to the paper itself.)