**SMS scnews item created by Daniel Daners at Fri 13 Feb 2015 2208**

Type: Seminar

Distribution: World

Expiry: 27 Feb 2015

**Calendar1: 27 Feb 2015 1400-1500**

**CalLoc1: AGR Carslaw 829**

Auth: daners@d110-33-115-151.mas800.nsw.optusnet.com.au (ddan2237) in SMS-WASM

### PDE Seminar

# Eigenvalue Estimates on Quantum Graphs

### Kennedy

James Kennedy

University of Stuttgart, Germany

Friday 27th January 2017 14:05-14:55, Carslaw Room 829 (AGR)

## Abstract

A quantum graph is a metric graph -- a collection of intervals of
possibly varying lengths, connected at a set of vertices -- on which a
differential operator such as the Laplacian acts. Such objects have
enjoyed considerable and growing popularity in the last 20 years, not
only because of the obvious applications to modelling networks of
various kinds, but also because they exhibit many features typical of
higher-dimensional problems despite their essentially one-dimensional
nature, thus serving as useful ``toy'' problems.

We will start by introducing the Kirchhoff Laplacian, the prototypical
differential operator on a metric graph. This operator enjoys all the
same basic properties as the Neumann Laplacian (either in one or more
dimensions), and in particular has a discrete set of eigenvalues and
eigenfunctions, which decompose the operator.

Our goal is to understand how these eigenvalues -- in particular the
first non-trivial one (equal to the spectral gap, the smallest
eigenvalue being zero) -- depend on the structure of the graph: its
total length, diameter, number of edges/vertices, connectivity, the
presence of certain subgraphs, and so on. Despite, or perhaps because
of, the seemingly simple nature of quantum graphs, very little is
known in this area. We will present a number of results which
hopefully demonstrate that such dependencies can be surprisingly
complex and subtle, despite being treatable by (relatively) elementary
techniques.

This is based on ongoing joint work with Pavel Kurasov (University of
Stockholm), Gabriela Malenová (KTH, Stockholm) and Delio Mugnolo
(FernUniversität Hagen).

Check also the PDE
Seminar page. Enquiries to Daniel Hauer or Daniel Daners.

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