Math @ Duke
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Publications [#348058] of Jianfeng Lu
Papers Published
- Lu, J; Steinerberger, S, Detecting localized eigenstates of linear operators,
Research in Mathematical Sciences, vol. 5 no. 3
(September, 2018), Springer Science and Business Media LLC [doi]
(last updated on 2024/04/19)
Abstract: We describe a way of detecting the location of localized eigenvectors of the eigenvalue problem Ax = λx for eigenvalues λ with |λ| comparatively large. We define the family of functions fα: {1, 2, …,n} → R fα (k) = log(‖Aα ek ‖ℓ2), where α ≥ 0 is a parameter and ek = (0, 0, …, 0, 1, 0, …, 0) is the kth standard basis vector. We prove that eigenvectors associated with eigenvalues with large absolute value localize around local maxima of fα: 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 −Δ + V, and the nonlocal operator (−Δ)3/4 + V.
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