95-544 Klein A.

Absolutely Continuous Spectrum in Random Schr\"odinger Operators (31K, AMS-LaTeX 1.1) Dec 22, 95

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Abstract. The spectrum of the Anderson Hamiltonian $\;H_\lb=-\De +\lb V$ on the Bethe Lattice is absolutely continuous inside the spectrum of the Laplacian, if the disorder $\lb$ is sufficiently small. More precisely, given any closed interval $I$ contained in the interior of the spectrum of the (centered) Laplacian $\De$ on the Bethe lattice, for small disorder $H_\lb$ has purely absolutely continuous spectrum in $I$ with probability one (i.e., $\si_{ac}( H_\lb) \cap I = I$ and $\si_{pp}( H_\lb) \cap I =\si_{sc}( H_\lb) \cap I= \emptyset$ with probability one). The proof is discussed and regularity properties are proven for the spectral measures restricted to such intervals of absolute continuity.

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