Abstract:
We study Schr\"odinger operators on $L^2 (\RR^d)$ and $\ell^2(\ZZ^d)$ 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.

Abstract:
We study Anderson and alloy type random Schr\"odinger operators on $\ell^2(\ZZ^d)$ and $L^2(\RR^d)$. Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. For a certain class of models we prove a Wegner estimate which is linear in the volume of the box and the length of the considered energy interval. The single site potential of the Anderson/alloy type model does not need to have fixed sign, but it needs be of a generalised step function form. The result implies the Lipschitz continuity of the integrated density of states.

Abstract:
One of the fundamental results in the theory of localization for discrete Schr\"odinger operators with random potentials is the exponential decay of Green's function and the absence of continuous spectrum. In this paper we provide a new variant of these results for one-dimensional alloy-type potentials with finitely supported sign-changing single-site potentials using the fractional moment method.

Abstract:
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.

Abstract:
In this article we study the Kirchhoff equation $$ -Big(a+b int_{mathbb{R}^N}| abla u|^2dxBig)Delta u+V(x)u = K(x)|u|^{q-1}u, quadhbox{in }mathbb{R}^N, $$ where $Ngeq 3$, $00$ are constants and $K(x), V(x)$ both change sign in $mathbb{R}^N$. Under appropriate assumptions on V(x), K(x), the existence of infinitely many solutions is proved by using the symmetric Mountain Pass Theorem.

Abstract:
We study the spectral minimum and Lifshitz tails for continuum random Schr\"{o}dinger operators of the form \begin{equation*} H_{\om}=-\De+V_{0}+\sum_{i\in\Z^{d}}\om_{i}u(\cdot-i), \end{equation*} where $V_{0}$ is the periodic potential, $\{\om_{i}\}_{i\in\Z^{d}}$ are i.i.d random variables and $u$ is the sign-indefinite impurity potential. Recently, this model has been proven to exhibit Lifshitz tails near the bottom of the spectrum under the small support assuption of $u$ and the reflection symmetry assumption of $V_{0}$ and $u$. We here drop the reflection symmetry assumption of $V_{0}$ and $u$. We first give characterizations of the bottom of the spectrum. Then, we show the existence of Lifshitz tails in the regime where the characterization of the bottom of the spectrum is explicit. In particular, this regime covers the reflection symmetry case.

Abstract:
We consider a family of self-adjoint operators [H_\omega = - \Delta + \lambda V_\omega, \quad \omega \in \Omega = \bigtimes_{k \in \ZZ^d} \RR,] on the Hilbert space $\ell^2 (\ZZ^d)$ or $L^2 (\RR^d)$. Here $\Delta$ denotes the Laplace operator (discrete or continuous), $V_\omega$ is a multiplication operator given by the function $$V_\omega (x) = \sum_{k \in \ZZ^d} \omega_k u(x-k) on $\ZZ^d$, or \quad V_\omega (x) = \sum_{k \in \ZZ^d} \omega_k U(x-k) on $\RR^d$,$$ and $\lambda > 0$ is a real parameter modeling the strength of the disorder present in the model. The functions $u:\ZZ^d \to \RR$ and $U:\RR^d \to \RR$ are called single-site potential. Moreover, there is a probability measure $\PP$ on $\Omega$ modeling the distribution of the individual configurations $\omega \in \Omega$. The measure $\PP = \prod_{k \in \ZZ^d} \mu$ is a product measure where $\mu$ is some probability measure on $\RR$ satisfying certain regularity assumptions. The operator on $L^2 (\RR^d)$ is called alloy-type model, and its analogue on $\ell^2 (\ZZ^d)$ discrete alloy-type model. This thesis refines the methods of multiscale analysis and fractional moments in the case where the single-site potential is allowed to change its sign. In particular, we develop the fractional moment method and prove exponential localization for the discrete alloy-type model in the case where the support of $u$ is finite and $u$ has fixed sign at the boundary of its support. We also prove a Wegner estimate for the discrete alloy-type model in the case of exponentially decaying but not necessarily finitely supported single-site potentials. This Wegner estimate is applicable for a proof of localization via multiscale analysis.

Abstract:
In this note we prove Minami's estimate for a class of discrete alloy-type models with a sign-changing single-site potential of finite support. We apply Minami's estimate to prove Poisson statistics for the energy level spacing. Our result is valid for random potentials which are in a certain sense sufficiently close to the standard Anderson potential (rank one perturbations coupled with i.i.d. random variables).

Abstract:
We study Schroedinger operators with a random potential of alloy type. The single site potentials are allowed to change sign. For a certain class of them we prove a Wegner estimate. This is a key ingredient in an existence proof of pure point spectrum of the considered random Schroedinger operators. Our estimate is valid for all bounded energy intervals and all space dimensions and implies the existence of the density of states.

Abstract:
We review recent and give some new results on the spectral properties of Schroedinger operators with a random potential of alloy type. Our point of interest is the so called Wegner estimate in the case where the single site potentials change sign. The indefinitness of the single site potential poses certain difficulties for the proof of the Wegner estimate which are still not fully understood. The Wegner estimate is a key ingredient in an existence proof of pure point spectrum of the considered random Schroedinger operators. Under certain assumptions on the considered models additionally the existence of the density of states can be proven.