On semiclassical and universal inequalities for eigenvalues of quantum graphs

Details On semiclassical and universal inequalities for eigenvalues of quantum graphs On semiclassical and universal inequalities for eigenvalues of quantum graphs

2009-11-10

arxiv.org/abs/0911.1896v1

Mathematics

Spectral Theory

18 pages

Scientific paper

We study the spectra of quantum graphs with the method of trace identities (sum rules), which are used to derive inequalities of Lieb-Thirring, Payne-P\'olya-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.

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