On non-round points of the boundary of the numerical range and an application to non-selfadjoint Schr\"odinger operators

We show that non-round boundary points of the numerical range of an unbounded operator (i.e. points where the boundary has infinite curvature) are contained in the spectrum of the operator. Moreover, we show that non-round boundary points, which are not corner points, lie in the essential spectrum. This generalizes results of H\"ubner, Farid, Spitkovsky and Salinas and Velasco for the case of bounded operators. We apply our results to non-selfadjoint Schr\"odinger operators, showing that in this case the boundary of the numerical range can be non-round only at points where it hits the essential spectrum.Comment: Shortened version. To appear in Journal of Spectral Theor

On the discrete spectrum of non-selfadjoint operators

We prove quantitative bounds on the eigenvalues of non-selfadjoint unbounded operators obtained from selfadjoint operators by a perturbation that is relatively-Schatten. These bounds are applied to obtain new results on the distribution of eigenvalues of Schroedinger operators with complex potentials