Transfer matrix methods and intersection theory are used to calculate the bands of edge states for a wide class of periodic two-dimensional tight-binding models including a sublattice and spin degree of freedom. This allows to define topological invariants by considering the associated Bott-Maslov indices which can be easily calculated numerically. For time-reversal symmetric systems in the symplectic universality class this leads to a Z 2 -invariant for the edge states. It is shown that the edge state invariants are related to Chern numbers of the bulk systems and also to (spin) edge currents, in the spirit of the theory of topological insulators.

We consider the nonlinear Schrödinger equation $$ i\psi_t= -\psi_{xx}\pm 2\cos 2x \ |\psi|^2\psi,\quad x\in S^1,\ t\in \R$$ and we prove that the solution of this equation, with small initial datum $\psi_0=\e (\cos x+\sin x)$, will periodically exchange energy between the Fourier modes $e^{ix}$ and $e^{-ix}$. This beating effect is described up to time of order $\e^{-9/4}$ while the frequency is of order $\e^2$. We also discuss some generalizations

We consider the Landau Hamiltonian (i.e. the 2D Schrödinger operator with constant magnetic field) perturbed by an electric potential V which decays sufficiently fast at infinity. The spectrum of the perturbed Hamiltonian consists of clusters of eigenvalues which accumulate to the Landau levels. Applying a suitable version of the anti-Wick quantization, we investigate the asymptotic distribution of the eigenvalues within a given cluster as the number of the cluster tends to infinity. We obtain an explicit description of the asymptotic density of the eigenvalues in terms of the Radon transform of the perturbation potential V .

We study the weighted averages of resonance clusters for the hydrogen atom with a Stark electric field in the weak field limit. We prove a semiclassical Szegö-type theorem for resonance clusters showing that the limiting distribution of the resonance shifts concentrates on the classical energy surface corresponding to a rescaled eigenvalue of the hydrogen atom Hamiltonian. This result extends Szegö-type results on eigenvalue clusters to resonance clusters. There are two new features in this work: first, the study of resonance clusters requires the use of non self-adjoint operators, and second, the Stark perturbation is unbounded so control of the perturbation is achieved using localization properties of coherent states corresponding to hydrogen atom eigenvalues. Contents

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