Lifshitz tail for continuous Anderson models driven by Lévy operators

We investigate the behavior near zero of the integrated density of states for random Schrödinger operators [Formula: see text] in [Formula: see text], [Formula: see text], where [Formula: see text] is a complete Bernstein function such that for some [Formula: see text], one has [Formula: see text], [Formula: see text], and [Formula: see text] is a random nonnegative alloy-type potential with compactly supported single site potential [Formula: see text]. We prove that there are constants [Formula: see text] such that [Formula: see text] where [Formula: see text] is the common cumulative distribution function of the lattice random variables [Formula: see text]. For typical examples of [Formula: see text] the constants [Formula: see text] and [Formula: see text] can be eliminated from the statement above. We combine probabilistic and analytic methods which allow to treat, in a unified manner, the large class of operator monotone functions of the Laplacian. This class includes both local and nonlocal kinetic terms such as the Laplace operator, its fractional powers, the quasi-relativistic Hamiltonians and many others.

Morphological analysis of chiral rod clusters from a coarse-grained single-site chiral potential

We present a compuationally efficient single-site potential for modelling chiral particles.

Wegner estimate and localization for alloy-type models with sign-changing exponentially decaying single-site potentials

We study Schrödinger operators on L2(ℝd) and ℓ2(ℤd) with a random potential of alloy-type. The single-site potential is assumed to be exponentially decaying but not necessarily of fixed sign. In the continuum setting, we require a generalized step-function shape. Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. In the described situation, a Wegner estimate, which is polynomial in the volume of the box and linear in the size of the energy interval, holds. We apply the established Wegner estimate as an ingredient for a localization proof via multiscale analysis.