Detecting localized eigenstates of linear operators

TL;DR

This paper introduces a method to detect localized eigenvectors of linear operators by analyzing the behavior of functions derived from powers of the operator, enabling identification of eigenvector localization in various types of matrices.

Contribution

The paper presents a novel approach using functions based on powers of the operator to identify localization of eigenvectors, supported by a fast randomized algorithm and diverse examples.

Findings

Eigenvectors with large eigenvalues localize around maxima of the defined functions.

The method effectively predicts localization in random band matrices and discretized operators.

The approach slows down power iteration convergence, indicating metastable states.

Abstract

We describe a way of detecting the location of localized eigenvectors of a linear system $Ax = \lambda x$ for eigenvalues $\lambda$ with $|\lambda|$ comparatively large. We define the family of functions $f_{\alpha}: \left\{1.2. \dots, n\right\} \rightarrow \mathbb{R}_{}$ $$ f_{\alpha}(k) = \log \left( \| A^{\alpha} e_k \|_{\ell^2} \right),$$ where $\alpha \geq 0$ is a parameter and $e_k = (0,0,\dots, 0,1,0, \dots, 0)$ is the $k-$th standard basis vector. We prove that eigenvectors associated to eigenvalues with large absolute value localize around local maxima of $f_{\alpha}$: the metastable states in the power iteration method (slowing down its convergence) can be used to predict localization. We present a fast randomized algorithm and discuss different examples: a random band matrix, discretizations of the local operator $-\Delta + V$ and the nonlocal operator $(-\Delta)^{3/4} + V$.