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Quantitative Anderson localization of Schrödinger eigenstates under disorder potentials (1803.09950v4)
Published 27 Mar 2018 in math.NA, cs.NA, math-ph, math.MP, and math.SP
Abstract: This paper concerns spectral properties of linear Schr\"odinger operators under oscillatory high-amplitude potentials on bounded domains. Depending on the degree of disorder, we prove the existence of spectral gaps amongst the lowermost eigenvalues and the emergence of exponentially localized states. We quantify the rate of decay in terms of geometric parameters that characterize the potential. The proofs are based on the convergence theory of iterative solvers for eigenvalue problems and their optimal local preconditioning by domain decomposition.