An algorithm for identifying eigenvectors exhibiting strong spatial localization

TL;DR

This paper presents an algorithm to identify eigenvectors with strong spatial localization in selfadjoint operators, supported by theoretical analysis and numerical experiments, offering a new method for exploring eigenvector localization phenomena.

Contribution

The paper introduces a novel algorithm for detecting localized eigenvectors in selfadjoint operators, combining theoretical insights with practical implementation and testing.

Findings

Algorithm successfully identifies localized eigenvectors within specified regions.

Theoretical results support the accuracy and reliability of the method.

Numerical illustrations demonstrate the algorithm's effectiveness in various scenarios.

Abstract

We introduce an approach for exploring eigenvector localization phenomena for a class of (unbounded) selfadjoint operators. More specifically, given a target region and a tolerance, the algorithm identifies candidate eigenpairs for which the eigenvector is expected to be localized in the target region to within that tolerance. Theoretical results, together with detailed numerical illustrations of them, are provided that support our algorithm. A partial realization of the algorithm is described and tested, providing a proof of concept for the approach.