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The Floquet spectrum of a quantum graph

Monday 26th July 2010 - 17:30 to 17:45
INI Seminar Room 1
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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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons