Absolutely Continuous Spectrum in the Anderson Model on the Bethe Lattice

We prove that the spectrum of the Anderson Hamiltonian $\;H_\lambda=-\De +\lambda V$ on the Bethe Lattice is absolutely continuous inside the spectrum of the Laplacian, if the disorder $\lambda$ 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, we prove that for small disorder, $\;H_\lambda$ has purely absolutely continuous spectrum in $I$ with probability one (i.e., $\sigma_{ac}( H_\lambda) \cap I = I$ and $\sigma_{pp}( H_\lambda) \cap I =\sigma_{sc}( H_\lambda) \cap I= \emptyset$ with probability one).

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Published 1 January 1994