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This paper concerns spectral properties of linear Schr\"odinger operatorsunder oscillatory high-amplitude potentials on bounded domains. Depending onthe degree of disorder, we prove the existence of spectral gaps amongst thelowermost eigenvalues and the emergence of exponentially localized states. Wequantify the rate of decay in terms of geometric parameters that characterizethe potential. The proofs are based on the convergence theory of iterativesolvers for eigenvalue problems and their optimal local preconditioning bydomain decomposition.

Quantitative Anderson localization of Schr\"odinger eigenstates under disorder potentials