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

We consider discrete random Schr\"odinger operators on $\ell^2 (\mathbb{Z}^d)$ with a potential of discrete alloy-type structure. That is, the potential at lattice site $x \in \mathbb{Z}^d$ 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 $\tau$-H\"older 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.Comment: 20 pages, 0 figure

Uniform existence of the integrated density of states for models on \ZZ^d

We provide an ergodic theorem for certain Banach-space valued functions on structures over \ZZ^d, which allow for existence of frequencies of finite patterns. As an application we obtain existence of the integrated density of states for associated finite-range operators in the sense of convergence of the distributions with respect to the supremum norm. These results apply to various examples including periodic operators, percolation models and nearest-neighbour hopping on the set of visible points. Our method gives explicit bounds on the speed of convergence in terms of the speed of convergence of the underlying frequencies. It uses neither von Neumann algebras nor a framework of random operators on a probability space.Comment: 15 page

On spectral condition of J-Herminian operators

The spectral condition of a matrix H is the infimum of the condition numbers κ(Z) = ||Z|| ||Z -1||, taken over all Z such that Z -1HZ is diagonal. This number controls the sensitivity of the spectrum of H under perturbations. A matrix is called J-Hermitian if H* = JHJ for some J = J* = J -1. When diagonalizing J-Hermitian matrices it is natural to look at J-unitary Z, that is, those that satisfy Z*JZ = J. Our first result is: if there is such J-unitary Z, then the infimum above is taken on J-unitary Z, that is, the J unitary diagonalization is the most stable of all. For the special case when J-Hermitian matrix has definite spectrum, we give various upper bounds for the spectral condition, and show that all J-unitaries Z which diagonalize such a matrix have the same condition number. Our estimates are given in the spectral norm and the Hilbert-Schmidt norm. Our results are, in fact, formulated and proved in a general Hilbert space (under an appropriate generalization of the notion of \u27diagonalising\u27) and they are applicable even to unbounded operators. We apply our theory to the Klein-Gordon operator thus improving a previously known bound