Semra Demirel: Lieb-Thirring inequalities and Levinson's formula for quantum graphs

We study the spectra of quantum graphs with the method of sum rules, which are used to derive inequalities of Lieb-Thirring. We show that the sharp constants of these inequalities and even their forms depend on the topology of the graph. Conditions are identified under which the sharp constants are the same as for the classical inequalities; in particular, this is true in the case of trees. We also provide some counterexamples where the classical form of the inequalities is false. Next, we consider the Schrödinger operator on a star shaped graph with n edges joined at a single vertex. We derive an expression for the trace of the difference of the perturbed and unperturbed resolvent in terms of a Wronskian. This leads to representations for the perturbation determinant and the spectral shift function, and to an analog of Levinson's formula. This talk is partly based on a joint work with Evans Harrell.