Detecting localized eigenstates of linear operators
This work addresses the problem of identifying localized eigenstates in large linear systems, which is important for applications in physics and numerical analysis, but the method is incremental as it builds on known power iteration concepts.
The authors propose a method to detect localized eigenvectors of linear operators by analyzing local maxima of a function based on powers of the operator applied to basis vectors. They provide theoretical justification and a fast randomized algorithm, demonstrating its effectiveness on random band matrices and discretized differential operators.
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 $(-Δ)^{3/4} + V$.