Detecting localized eigenstates of linear operators

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 $f_{\alpha }:\left\{\mathrm{1.2.}\dots ,n\right\}\to {\mathbb{R}}_{}$ $$f_{\alpha }(k)=\log \left(\Vert A^{\alpha }{e}_k{\Vert }_{{\ell }^2}\right),$$ where $\alpha \ge 0$ 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 $(-\Delta {)}^{3/4}+V$.

References in corpus (4)

• Maximally localized Wannier functions: Theory and applications
• One single static measurement predicts wave localization in complex structures
• On the Diffusion Geometry of Graph Laplacians and Applications
• Dual hidden landscapes in Anderson localization on discrete lattices

Cited by in corpus (1)

• Localization of Neumann Eigenfunctions near Irregular Boundaries