  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.
