# Detecting localized eigenstates of linear operators

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2017-11-30

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## 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 $(-\Delta)^{3/4} + V$.

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### Scholars@Duke

#### Jianfeng Lu

Jianfeng Lu is an applied mathematician interested in mathematical analysis and algorithm development for problems from computational physics, theoretical chemistry, materials science, machine learning, and other related fields.

More specifically, his current research focuses include:

High dimensional PDEs; generative models and sampling methods; control and reinforcement learning; electronic structure and many body problems; quantum molecular dynamics; multiscale modeling and analysis.

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