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Abstract. We consider the inverse spectral problem for a class of reflectionless bounded Jacobi operators with empty singularly continuous spectra. Our spectral hypotheses admit countably many accumulation points in the set of eigenvalues as well as in the set of boundary points of intervals of absolutely continuous spectrum. The corresponding isospectral set of Jacobi operators is explicitly determined in terms of Dirichlet-type data.

In this paper we consider the Anderson model with decaying randomness a n q ω (n), a n > 0, n ∈ Z Z ν and q ω (n), i.i.d. random variables with an absolutely continuous distribution µ. For a class of µ we show the following results on a set ω of full measure. (i) If |a n | → 0 as |n| → ∞, then σ c (H ω ) ⊆ [−2ν, 2ν] (ii) σ(H ω ) = IR. (iii) If |a n | ≤ (|n| −1− ) for large |n| and ν ≥ 3, the mobility edges are the two points {−2ν, 2ν}.

We give a spectral and dynamical description of certain models of random Schrödinger operators on L 2 (R d ) for which a modified version of the fractional moment method of Aizenman and Molchanov [3] can be applied. One family of models includes Schrödinger operators with random nonlocal interactions constructed from multidimensional wavelet bases. The second family includes Schrödinger operators with random singular interactions randomly located on sublattices of Z d , for d = 1, 2, 3. We prove that these models are amenable to Aizenman-Molchanov-type analysis of the Green's function, thereby eliminating the use of multiscale analysis. The basic technical result is an estimate on the expectation of fractional moments of the Green's function. Among our results, we prove a Wegner estimate, Hölder continuity of the integrated density of states, and spectral and Hilbert-Schmidt dynamical localization at negative energies.

present here. Even for these models the random potentials need to satisfy a complete covering condition. The Anderson model on the lattice for which regularity results were known earlier also satisfies the complete covering condition.