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.

Large Deviation Estimates and Hölder Regularity of the Lyapunov Exponents for Quasi-periodic Schrödinger Cocycles

We consider one-dimensional quasi-periodic Schrödinger operators with analytic potentials. In the positive Lyapunov exponent regime, we prove large deviation estimates, which lead to refined Hölder continuity of the Lyapunov exponents and the integrated density of states, in both small Lyapunov exponent and large coupling regimes. Our results cover all the Diophantine frequencies and some Liouville frequencies.

On Asymptotics of the Density of States for Hypoelliptic Almost Periodic Systems

In this paper, we find the asymptotics of integrated density of states with remainder estimate for hypoelliptic systems with almost periodic (a.p.) coefficients. We use the approximate spectral projector method for matrix a.p. operators with continuous spectrum.

On the Continuity of the Integrated Density of States in the Disorder

Recently, Hislop and Marx studied the dependence of the integrated density of states on the underlying probability distribution for a class of discrete random Schrödinger operators and established a quantitative form of continuity in weak* topology. We develop an alternative approach to the problem, based on Ky Fan inequalities, and establish a sharp version of the estimate of Hislop and Marx. We also consider a corresponding problem for continual random Schrödinger operators on $\mathbb{R}^d$.