On the Absolutely Continuous Subspace for Non - Selfadjoint Operators

On the Preservation of Absolutely Continuous Spectrum for Schrödinger Operators

We present general principles for the preservation of a.c. spectrum under weak perturbations. The Schrödinger operators on the strip and on the Caley tree (Bethe lattice) are considered. In this paper, we consider the problem of preservation of a.c. spectrum for Schrödinger operators when the perturbation (potential) is decaying slowly at infinity. In a recent years, lots of results in this dir...

Spectral instability for non-selfadjoint operators∗

We describe a recent result of M. Hager, stating roughly that for nonselfadjoint ordinary differential operators with a small random perturbation we have a Weyl law for the distribution of eigenvalues with a probability very close to 1.

Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I

This is the first in a series of works devoted to small non-selfadjoint perturbations of selfadjoint h-pseudodifferential operators in dimension 2. In the present work we treat the case when the classical flow of the unperturbed part is periodic and the strength ǫ of the perturbation is ≫ h (or sometimes only ≫ h2) and bounded from above by hδ for some δ > 0. We get a complete asymptotic descri...

Non-absolutely Continuous Foliations

We consider a partially hyperbolic diffeomorphism of a compact smooth manifold preserving a smooth measure. Assuming that the central distribution is integrable to a foliation with compact smooth leaves we show that this foliation fails to have the absolute continuity property provided that the sum of Lyapunov exponents in the central direction is not zero on a set of positive measure. We also ...

On the Spectral Properties of Degenerate Non-selfadjoint Elliptic systems of Differential Operators