Annales Henri Poincaré

, Volume 10, Issue 1, pp 189–223

The Trace Formula for Quantum Graphs with General Self Adjoint Boundary Conditions


DOI: 10.1007/s00023-009-0399-7

Cite this article as:
Bolte, J. & Endres, S. Ann. Henri Poincaré (2009) 10: 189. doi:10.1007/s00023-009-0399-7


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.

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Department of Mathematics, Royal HollowayUniversity of LondonEghamUnited Kingdom
  2. 2.Institut für Theoretische PhysikUniversität UlmUlmGermany

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