Eigenvalue bounds for the Laplacian on planar metric graphs *Giovedì 8 agosto 2019 *

*Luogo: *Dipartimento di Matematica, via Sommarive, 14 - Povo (TN) - Sala Seminari "-1" *Ore*: 11:00

*Relatore:*

- *Marvin Plümer* (FernUniversität Hagen, Germany)

*Abstract:*

Self-adjoint differential operators – in particular Schrödinger-type operators – acting on metric graphs have been studied very actively, in recent years. Here, a metric graph is a collection of intervals whose endpoints are identified in a graph-like fashion. In this talk, we present some new developments in the theory of Laplacians on metric graphs: in particular, we discuss the role played by planarity in spectral theory and derive some lower and upper bounds for the eigenvalues of the Laplacian. To derive upper bounds we make use of a geometric representation for planar graphs provided by the classical Circle Packing Theorem of Koebe, Andreev and Thurston. For the derivation of lower bounds, we extend and apply a recently developed transference principle by Amini and Cohen-Steiner that compares eigenvalues of continuous and discrete models in a very convenient way. In this context we give a brief introduction to metric graphs and differential operators on metric graphs. This is joint work with Delio Mugnolo. *Referente*: Stefano Bonaccorsi