Random Laplace operators with arbitrary random self-adjoint singular perturbations supported on random discrete subsets of the line exhibit a dichotomy between absolutely continuous spectrum and Anderson localization under minimal disorder assumptions.
Kato, Perturbation Theory for Linear Operators , Springer, Berlin, 1980
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Random Hamiltonians with Arbitrary Point Interactions
Random Laplace operators with arbitrary random self-adjoint singular perturbations supported on random discrete subsets of the line exhibit a dichotomy between absolutely continuous spectrum and Anderson localization under minimal disorder assumptions.