Detecting localized eigenstates of linear operators

Lu, J

Steinerberger, S

2017-11-30T21:56:56Z

2017-11-30T21:56:56Z

2017-11-30

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

http://arxiv.org/abs/1709.03364v1

https://hdl.handle.net/10161/15780

Springer Science and Business Media LLC

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Detecting localized eigenstates of linear operators

Journal article

Lu, J|0000-0001-6255-5165

http://arxiv.org/abs/1709.03364v1

Chemistry

Duke

Mathematics

Physics

Temp group - logins allowed

Trinity College of Arts & Sciences