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