Vadim Kostrykin, Jurgen Potthoff, and Robert Schrader
Heat Kernels on Metric Graphs and a Trace Formula
(87K, LaTeX 2e)
ABSTRACT. We study heat semigroups generated by self-adjoint Laplace operators on
metric graphs characterized by the property that the local scattering
matrices associated with each vertex of the graph are independent from the
spectral parameter. For such operators we prove a representation for the
heat kernel as a sum over all walks with given initial and terminal edges.
Using this representation a trace formula for heat semigroups is proven.
Applications of the trace formula to inverse spectral and scattering
problems are also discussed.