Detecting localized eigenstates of linear operators

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⁽‖A^α 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.

Duke Scholars

Author Jianfeng Lu Mathematics