The trace formula for quantum graphs with general self adjoint boundary conditions

Details The trace formula for quantum graphs with general self adjoint boundary conditions The trace formula for quantum graphs with general self adjoint boundary conditions

2008-05-20

arxiv.org/abs/0805.3111v2

Physics

Mathematical Physics

36 pages; improved, final version to appear in Ann. H. Poincare

Scientific paper

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.

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