# Talk by Sabine Bögli

On December 1st, 2021, Prof. Dr. Sabine Bögli (Durham University) gave a talk about "**On the discrete eigenvalues of Schrödinger operators with complex potentials**" as part of the research seminar Analysis of the FernUniversität in Hagen. This lecture is partially supported by the COST action Mathematical models for interacting dynamics on networks.

## Abstract

In this talk I shall present constructions of Schrödinger operators with complex-valued potentials whose spectra exhibit interesting properties. One example shows that for sufficiently large $p$, namely $p>(d+1)/2$ where $d$ is the dimension, the discrete eigenvalues need not be bounded by the $L^p$ norm of the potential. This is a counterexample to the Laptev-Safronov conjecture (Comm. Math. Phys. 2009). Another construction proves optimality (in some sense) of generalisations of Lieb-Thirring inequalities to the non-selfadjoint case - thus giving us information about the accumulation rate of the discrete eigenvalues to the essential spectrum.

This talk is based on joint works with Jean-Claude Cuenin and Frantisek Stampach.