The Trace Formula for Quantum Graphs with General Self Adjoint Boundary Conditions |
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Authors: | Jens Bolte Sebastian Endres |
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Institution: | 1. Department of Mathematics, Royal Holloway, University of London, Egham, TW20 0EX, United Kingdom 2. Institut für Theoretische Physik, Universit?t Ulm, Albert-Einstein-Allee 11, D-89069, Ulm, Germany
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Abstract: | We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing
spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic
orbits on the graph. This includes trace formulae with, respectively, absolutely and conditionally convergent periodic orbit
sums; the convergence depending on properties of the test functions used. We also prove a trace formula for the heat kernel
and provide small-t asymptotics for the trace of the heat kernel.
Submitted: May 20, 2008., Accepted: January 6, 2009. |
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