We study the Anderson model with decaying randomness inv ≥ 3 dimensions and show that there is absolutely continuous spectrum in [−2v, 2v]. The distribution of the potentials is assumed to have finite variance and the coupling constants decay at infinity at a rate α > 1.

In this note we prove a relation between the Riemann Zeta function, ζ and the ξ function (Krein spectral shift) associated with the harmonic oscillator in one dimension. This gives a new integral representation of the zeta function and also a reformulation of the Riemann hypothesis as a question inL¹(ℝ).

In this paper we consider some Anderson type models, with free parts having long range tails and with the random perturbations decaying at different rates in different directions and prove that there is a.c. spectrum in the model which is pure. In addition, we show that there is pure point spectrum outside some interval. Our models include potentials decaying in all directions in which case absence of singular continuous spectrum is also shown.

In this paper we consider one dimensional random Jacobi operators with decaying independent randomness and show that under some condition on the decay vis-a-vis the distribution of randomness, that the distribution function of the average spectral measures of the associated operators are smooth.

In this paper we give some new criteria for identifying the components of a probability measure, in its Lebesgue decomposition. This enables us to give new criteria to identify spectral types of self-adjoint operators on Hilbert spaces, especially those of interest.

In this paper we discuss the continuity properties of the integrated density of states for random models based on that of the single site distribution. Our results are valid for models with independent randomness with arbitrary free parts. In particular in the case of the Anderson type models (with stationary, growing, decaying randomness) on the 𝑣 dimensional lattice, with or without periodic and almost periodic backgrounds, we show that if the single site distribution is uniformly 𝛼-Hölder continuous, $0 < \alpha \leq 1$, then the density of states is also uniformly 𝛼-Hölder continuous.

In this paper we consider the interband light absorption coefficient (ILAC), in a symmetric form, in the case of random operators on the 𝑑-dimensional lattice. We show that the symmetrized version of ILAC is either continuous or has a component which has the same modulus of continuity as the density of states.

In this paper we consider the interband light absorption coefficient (ILAC) for various models. We show that at the lower and upper edges of the spectrum the Lifshitz tails behaviour of the density of states implies similar behaviour for the ILAC at appropriate energies. The Lifshitz tails property is also exhibited at some points corresponding to the internal band edges of the spectrum.

In this paper we consider two classes of random Hamiltonians on $L^2(\mathbb{R}^d)$: one that imitates the lattice case and the other a Schrödinger operator with non-decaying, non-sparse potential both of which exhibit a.c. spectrum. In the former case we also know the existence of dense pure point spectrum for some disorder thus exhibiting spectral transition valid for the Bethe lattice and expected for the Anderson model in higher dimension.