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.
Article PDF
Similar content being viewed by others
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jens Marklof.
Submitted: May 20, 2008., Accepted: January 6, 2009.
Rights and permissions
About this article
Cite this article
Bolte, J., Endres, S. The Trace Formula for Quantum Graphs with General Self Adjoint Boundary Conditions. Ann. Henri Poincaré 10, 189–223 (2009). https://doi.org/10.1007/s00023-009-0399-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-009-0399-7