Localization for Some Continuous, Random Hamiltonians in d-Dimensions

We prove the existence with probability one of an interval of pure point spectrum for some families of continuous random Schrödinger operators in d-dimensions. For Anderson-like models with positive, short-range, single-site potentials, we also prove that the corresponding eigenfunctions decay exponentially and that the integrated density of states is Lipschitz continuous. For the other families of random potentials that we study, we show that the corresponding eigenfunctions decay faster than an inverse power of x, which depends upon the decay rate of the single-site potential. To obtain these results, we develop an extension of the classical Aronszajn-Donoghue theory for a class of relatively compact perturbations and a spectral averaging method which extends Kotani′s trick to these more general families of operators.