## Localization for some continuous, random Hamiltonians in $$d$$-dimensions.(English)Zbl 0801.60054

Summary: 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.

### MSC:

 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics 60H25 Random operators and equations (aspects of stochastic analysis) 47B80 Random linear operators 47A11 Local spectral properties of linear operators 47A10 Spectrum, resolvent
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