Heat kernels on metric graphs and a trace formula

Jurgen Potthoff; Robert Schrader; Vadim Kostrykin

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Heat kernels on metric graphs and a trace formula

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arxiv math-ph/0701009 v1 pith:END4UPAM submitted 2007-01-04 math-ph math.MPmath.SP

Vadim Kostrykin , Jurgen Potthoff , Robert Schrader This is my paper · ORCID

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keywords heatformulatracegraphsmetricoperatorsrepresentationscattering

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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.

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