The integral QHE can be explained either as resulting from bulk or
edge currents (or, in reality, as a combination of both). The equality
of the two conductances at zero temperature was recently established
for the case that the Fermi energy falls in the spectral gap of the bulk
system. We define the edge conductance via a suitable time averaging
procedure in the more general case of a bulk system which exhibits
dynamical localization in the vicinity of the Fermi energy, and show
that the two conductances are equal.
This is a joint work with G.-M. Graf and J. Schenker.

Bond-percolation graphs are random subgraphs of the d-dimensional
integer lattice generated by a standard Bernoulli bond-percolation
process. The
associated graph Laplacians, subject to Dirichlet or Neumann conditions at
cluster boundaries, represent bounded, self-adjoint, ergodic random
operators. They possess almost surely the
non-random spectrum [0,4d] and a self-averaging integrated density
of states. This integrated density of states is shown to exhibit Lifshits
tails at both spectral edges in the non-percolating phase. Depending
on the boundary condition and on the spectral edge, the Lifshits tail
discriminates between different cluster geometries (linear clusters
versus cube-like
clusters) which contribute the dominating eigenvalues. Lifshits tails
arising
from cube-like clusters continue to show up above the percolation
threshold.
In contrast, the other type of Lifshits tails cannot be observed in the
percolating
phase any more because they are hidden by van Hove singularities from the
percolating cluster.

We consider Jacobi matrices built on equilibrium measures of hyperbolic polynomials. We show their property, which, on one side, is related to almost periodicity of such matrices, and, on the other side, is a sort of noncommutative
Perron-Frobenius-Ruelle theorem. While proving these key property one is naturally brought to consider a two-weight Hilbert transform. Its boundedness can be proved in our situation, while the general two-weight Hilbert transform
boundedness criterion is not yet available.
We will mention other problems in spectral theory of Jacobi matrices, where this paradigm of nonhomogeneous harmonic analysis---two weight Hilbert transform---appears in the natural way.

Following a work by A. Melin and J. Sj\"ostrand, it has become
increasingly clear that non-selfadjoint operators in dimension two share many of the pleasant features of operators in dimension one. In particular, in the semiclassical limit, it is often possible to get complete asymptotics for individual eigenvalues of such operators in some domain in the complex plane, by means of a suitable Bohr-Sommerfeld quantization rule. In this talk, we would like to report on some recent results in this direction obtained together with Johannes Sj\"ostrand, as
a part of an ongoing program on small non-selfadjoint perturbations of selfadjoint operators. We shall also try to discuss applications to asymptotics of scattering poles for semiclassical Schr\"odinger operators, and to dissipative wave equations on compact manifolds.

We study the family of Hamiltonians which corresponds to the
adjacency operators on a percolation graph. We characterise the set of
energies which are almost surely eigenvalues with finitely supported
eigenfunctions. This set of energies is a dense subset of the algebraic
integers. The integrated density of states has discontinuities precisely
at this set of energies. We show that the convergence of the integrated
densities of states of finite box Hamiltonians to the one on the whole
space holds even at the points of discontinuity. For this we use an
equicontinuity-from-the-right argument. The same statements hold for the
restriction of the Hamiltonian to the infinite cluster. In this case we
prove that the integrated density of states can be constructed using local
data only. Finally we study some mixed Anderson-Quantum percolation models
and establish results in the spirit of Wegner, and Delyon and Souillard.