Non-monotonic Random Schr¨odinger Operators: The Anderson Model

We prove exponential localization at all energies for one-dimensional continuous Anderson-type models with single site potentials of changing sign. A periodic background potential is allowed. The main problem arises from non-monotonicity; i.e., the operator does not depend monotonically in the form sense on the random parameters. We show that the method of ‘‘two-parameter spectral averaging,’’ recently devised by Buschmann and Stolz to prove localization for Poisson and random displacement models, can be modified to work for the type of Anderson model considered here.