### The Floquet spectrum of a quantum graph

Rueckriemen, R *(Dartmouth College)*

Monday 26 July 2010, 17:30-17.45

Seminar Room 1, Newton Institute

#### Abstract

We define the Floquet spectrum of a quantum graph as the collection of all spectra of operators of the form
$D=(-i\frac{\partial}{\partial x}+\alpha(\frac{\partial}{\partial x}))^2$ where $\alpha$ is a closed $1$-form. We show that the Floquet
spectrum completely determines planar 3-connected graphs
(without any genericity assumptions on the graph). It determines whether or not a graph is planar. Given the combinatorial graph,
the Floquet spectrum uniquely determines all edge lengths of a quantum graph.

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