Discrete alloy-type models: Regularity of distributions and recent results

Original · EN

We consider discrete random Schrödinger operators on ℓ² (Zᵈ) with a potential of discrete alloy-type structure. That is, the potential at lattice site x ∈ Zᵈ is given by a linear combination of independent identically distributed random variables, possibly with sign-changing coefficients. In a first part we show that the discrete alloy-type model is not uniformly τ-Hölder continuous, a frequently used condition in the literature of Anderson-type models with general random potentials. In a second part we review recent results on regularity properties of spectral data and localization properties for the discrete alloy-type model.