Detecting localized eigenstates of linear operators

Abstract

We describe a way of detecting the location of localized eigenvectors of a linear system $Ax = λx$ for eigenvalues $λ$ with $|λ|$ comparatively large. We define the family of functions $f_α: \left\{1.2. \dots, n\right\} \rightarrow \mathbb{R}_{}$ $$ f_α(k) = \log \left( \| A^α e_k \|_{\ell^2} \right),$$ where $α\geq 0$ 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 $f_α$: 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 $-Δ+ V$ and the nonlocal operator $(-Δ)^{3/4} + V$.