The current-current correlation function plays a crucial role in the
Mott theory of conductivity for disordered systems. We prove a Pastur-Shubin-type formula for the current-current correlation function 02expressing it as a thermodynamic limit. We prove that the limit is
independent of the self-adjoint boundary conditions and independent of a
large family of expanding regions. We relate this ^nite-volume de^nition
to the de^nition obtained by using the in^nite-volume operators and the
trace-per-unit volume.

In 1986 A. Ancona showed, using the Koebe one-quarter Theorem, that for a simply-connected planar domain the constant in the Hardy inequality with the distance to the boundary is greater than or equal to 1/16. We consider classes of domains for which there is a stronger version of the Koebe Theorem. This implies better estimates for the constant appearing in the Hardy inequality.

We study the extended states regime of the discrete Anderson model. The perturbative approach requires precise estimates on the free propagator,
$(a- e(p)+i\eta)^{-1}$,$\eta>0$, $\alpha\in \bR$,
where $e(p)= \sum_{i=1}^3 [1- \cos (p_i)]$, $p=(p_1, p_2, p_3)$,
is the dispersion relation of the three dimensional cubic lattice. The level surfaces of the function $e(p)$ have vanishing curvature. We will present new bounds on the Fourier transform of such surfaces. This will yield estimates on the probability that
a quantum particle travelling in a weak random environment
recollides with obstacles visited earlier.

I will consider inverse scattering theory for Jacobi operators which
are
short range perturbation of a quasi-periodic finite-gap background
operator.
In particular I want to investigate the algebraic constraints on the
scattering
data in this situation.