Detecting localized eigenstates of linear operators

We describe a way of detecting the location of localized eigenvectors of a linear system $Ax=\lambda x$ for eigenvalues $\lambda$ with $\mathrm{\mid }\lambda \mathrm{\mid }$ comparatively large. We define the family of functions $f_{\alpha }:\{\mathrm{1.2.}\dots ,n\}\to {\mathbb{R}}_{}$ $$f_{\alpha }(k)=\log \left(\mathrm{\parallel }{A}^{\alpha }{e}_k{\mathrm{\parallel }}_{{\mathrm{\ell }}^2}\right),$$ 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 $-\mathrm{\Delta }+V$ and the nonlocal operator $(-\mathrm{\Delta }{)}^{3\mathrm{/}4}+V$.