Detecting localized eigenstates of linear operators

Abstract

We describe a way of detecting the location of localized eigenvectors of the eigenvalue problem $$Ax = \lambda x$$ for eigenvalues $$\lambda $$ with $$|\lambda |$$ comparatively large. We define the family of functions $$f_{\alpha }: \left\{ 1,2, \dots , n\right\} \rightarrow {\mathbb {R}}_{}$$ $$\begin{aligned} f_{\alpha }(k) = \log \left( \Vert A^{\alpha } e_k \Vert _{\ell ^2} \right) , \end{aligned}$$ where $$\alpha \ge 0$$ is a parameter and $$e_k = (0,0,\ldots , 0,1,0, \ldots , 0)$$ is the kth standard basis vector. We prove that eigenvectors associated with 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 $$(-\Delta )^{3/4} + V$$ .