# Detecting localized eigenstates of linear operators

Research

First Online:

- 44 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 \(f_{\alpha }: \left\{ 1,2, \dots , n\right\} \rightarrow {\mathbb {R}}_{}\)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.

## References

- 1.Anderson, P.W.: Absence of diffusion in certain random lattices. Phys. Rev.
**109**, 1492–1505 (1958)CrossRefGoogle Scholar - 2.Benzi, M., Boito, P., Razouk, N.: Decay properties of spectral projectors with applications to electronic structure. SIAM Rev.
**55**, 3–64 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Benzi, M.: Localization in Matrix Computation: Theory and Applications, Lecture Notes in Mathematics, vol. 2173, pp. 211–317. Springer, Berlin (2016)zbMATHGoogle Scholar
- 4.Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability and low lying spectra in reversible Markov chains. Commun. Math. Phys.
**228**(2), 219–255 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times. J. Eur. Math. Soc.
**6**(4), 399–424 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Bovier, A., Gayrard, V., Klein, M.: Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues. J. Eur. Math. Soc.
**7**(1), 69–99 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Cheng, X., Rachh, M., Steinerberger, S.: On the diffusion geometry of graph Laplacians and applications. Preprint, arXiv:1611.03033
- 8.Filoche, M., Mayboroda, S.: Universal mechanism for Anderson and weak localization. Proc. Natl. Acad. Sci. USA
**109**(37), 14761–14766 (2012)MathSciNetCrossRefGoogle Scholar - 9.Filoche, M., Mayboroda, S.: The landscape of Anderson localization in a disordered medium. Contemp. Math.
**601**, 113–121 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Halko, N., Martinsson, P.G., Tropp, J.A.: Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev.
**53**(2), 217–288 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Kenney, C., Laub, A.: Small-sample statistical condition estimates for general matrix functions. SIAM J. Sci. Comput.
**15**(1), 36–61 (1994)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Lefebvre, G., Gondel, A., Dubois, M., Atlan, M., Feppon, F., Labbe, A., Gillot, C., Garelli, A., Ernoult, M., Mayboroda, S., Filoche, M., Sebbah, P.: One single static measurement predicts wave localization in complex structures. Phys. Rev. Lett.
**117**, 074301 (2016)CrossRefGoogle Scholar - 13.Lin, L., Lu, J.: Decay estimates of discretized Green’s functions for Schrödinger type operators. Sci. China Math.
**59**, 1561–1578 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Lyra, M., Mayboroda, S., Filoche, M.: Dual hidden landscapes in Anderson localization on discrete lattices. Eur. Phys. Lett.
**109**, 4700 (2015)CrossRefGoogle Scholar - 15.Marzari, N., Mostofi, A.A., Yates, J.R., Souza, I., Vanderbilt, D.: Maximally localized Wannier functions: theory and applications. Rev. Mod. Phys.
**84**, 1419–1475 (2012)CrossRefGoogle Scholar - 16.Steinerberger, S.: Localization of quantum states and landscape functions. Proc. Am. Math. Soc
**45**, 2895–2907 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Vempala, S.: The random projection method. With a foreword by Christos H. Papadimitriou. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 65. American Mathematical Society, Providence, RI (2004)Google Scholar
- 18.Vömel, Ch., Parlett, B.N.: Detecting localization in an invariant subspace. SIAM J. Sci. Comput.
**33**, 3447–3467 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 19.Wannier, G.: The structure of electronic excitation levels in insulating crystals. Phys. Rev.
**52**, 191 (1937)CrossRefzbMATHGoogle Scholar - 20.Weinan, E., Li, Tiejun, Lu, Jianfeng: Localized bases of eigensubspaces and operator compression. Proc. Natl. Acad. Sci. USA
**107**, 1273–1278 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 21.Zou, H., Hastie, T., Tibshirani, R.: Sparse principal component analysis. J. Comput. Graph. Stat.
**15**, 262–286 (2006)MathSciNetCrossRefGoogle Scholar

## Copyright information

© SpringerNature 2018