# Detecting localized eigenstates of linear operators

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## 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 \(f_{\alpha }: \left\{ 1,2, \dots , n\right\} \rightarrow {\mathbb {R}}_{}\)where \(\alpha \ge 0\) is a parameter and \(e_k = (0,0,\ldots , 0,1,0, \ldots , 0)\) is the

$$\begin{aligned} f_{\alpha }(k) = \log \left( \Vert A^{\alpha } e_k \Vert _{\ell ^2} \right) , \end{aligned}$$

*k*th 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 \((-\Delta )^{3/4} + V\).## Keywords

Eigenvectors Localization Power iteration Randomized numerical linear algebra Anderson localization## Mathematics Subject Classification

Primary 35P20 Secondary 82B44## Notes

### Acknowledgements

The research of J.L. was supported in part by the National Science Foundation under award DMS-1454939. He would also like to thank Yingzhou Li for helpful discussions.

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