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Detecting localized eigenstates of linear operators

  • Jianfeng Lu
  • Stefan Steinerberger
Research
  • 26 Downloads

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

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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Copyright information

© SpringerNature 2018

Authors and Affiliations

  1. 1.Department of MathematicsDuke UniversityDurhamUSA
  2. 2.Department of PhysicsDuke UniversityDurhamUSA
  3. 3.Department of ChemistryDuke UniversityDurhamUSA
  4. 4.Department of MathematicsYale UniversityNew HavenUSA

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