The subject of the talk is the spectrum of a two-dimensional
Schrodinger operator with constant magnetic field and a compactly supported electric potential. The eigenvalues of such an operator form clusters around the Landau levels.
The eigenvalues in these clusters accumulate towards the Landau levels super-exponentially fast. It appears that these eigenvalues can be related to a certain sequence of orthogonal polynomials in the complex domain. This allows one to accurately describe the rate of accumulation of eigenvalues towards the Landau levels. This description involves the logarithmic capacity of the support of the electric potential. The talk is based on a joint work with Nikolai FIlonov from St.Petersburg.

I will discuss the resonance counting function for Schrodinger operators with compactly-supported, $L^\infty$, real-, or complex-valued potentials, in odd dimensions $d \geq 3$. In joint work with T. Christiansen, we prove that the set of such potentials for which the resonance counting function has maximal order of growth $d$ is generic.

Dissipative transport in solids can be described by a Markov
semigroup of completely positive operators on the observable algebra of the charge carriers creation and annihilation operators. A model of generators of such semigroups, called the quantum jump model, will be presented. The linear response theory will be shown to provide the expression of transport coefficients through a Green-Kubo formula. This formula will be justified rigorously through the spectral property of the
generator of the quantum jump model in various situations. The case of aperiodic solids, such as strongly disordered systems will be emphasized, in view of its relevance in the theory of the Quantum Hall effect.