Browsing by Author "Demirel, Semra"

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Spectral theory of quantum graphs
(2012) Demirel, Semra; Weidl, Timo (Prof. TeknD)
We study some spectral problems for quantum graphs with a potential. On the one hand we analyze the quantitative dependence of bound states of the Schrödingeroperator on the potential. On the other hand we generalize certain basic identities from the one-dimensional scattering theory to quantum graphs. The first paper is concerned with the study of the discrete negative spectra of quantum graphs. We use the method of trace identities (sum rules) to derive inequalities of Lieb-Thirring, Payne-Polya-Weinberger, and Yang types, among others. 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. The second paper deals with the scattering problem for the Schrödinger equation on the half-line with the Robin boundary condition at the origin. We derive an expression for the trace of the difference of the perturbed and unperturbed resolvent in terms of a Wronskian. This leads to a representation for the perturbation determinant and to trace identities of Buslaev-Faddeev type. In the third paper we generalize results from the half-line case to the full graph case. More precisely, we consider the Schrödinger problem on a star shaped graph with n edges joined at a single vertex. A trace formula is derived for 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. Besides these three articles this thesis also contains some further results. The method of sum rules is applied to the modified Schrödinger operator with variable coefficients to obtain a Lieb-Thirring type inequality with optimal constant. Furthermore, Lieb-Thirring inequalities are studied for star shaped graphs by using variational arguments and the method of symmetry decomposition of the corresponding Hilbert space. In several cases this leads to optimal constants in the inequalities.