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.

Consider a Schr"odinger operator on L_2(0,\infty),
and suppose that the potential V is known on an initial interval
[0,N]. We then prove bounds on the spectral measure \rho(I)
of intervals I. This extends (very) classical work of Chebyshev
and Markov on orthogonal polynomials.

We consider the discrete one-dimensional quasi-periodic
Schroedinger operator with potential defined by a Gevrey-class function.
We show - in the perturbative regime - that the operator satisfies
Anderson localization and that the Lyapunov exponent is positive and
continuous for all energies. We also mention a partial nonperturbative
result valid for some particular Gevrey classes. These results extend
some recent work by J. Bourgain, M. Goldstein, W. Schlag to a more general
class of potentials.