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

Understand

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.