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Mathematics > Numerical Analysis

Title: 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 $f_{\alpha}: \left\{1.2. \dots, n\right\} \rightarrow \mathbb{R}_{}$ $$ f_{\alpha}(k) = \log \left( \| A^{\alpha} e_k \|_{\ell^2} \right),$$ where $\alpha \geq 0$ 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 $f_{\alpha}$: 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 $(-\Delta)^{3/4} + V$.
Subjects: Numerical Analysis (math.NA); Mathematical Physics (math-ph); Spectral Theory (math.SP)
Cite as: arXiv:1709.03364 [math.NA]
  (or arXiv:1709.03364v2 [math.NA] for this version)

Submission history

From: Stefan Steinerberger [view email]
[v1] Mon, 11 Sep 2017 13:20:53 GMT (79kb,D)
[v2] Fri, 16 Mar 2018 19:02:40 GMT (79kb,D)