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Heat kernels on metric graphs and a trace formula

Vadim Kostrykin·Jurgen Potthoff·Robert Schrader
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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.

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Vadim Kostrykin, Jurgen Potthoff, Robert Schrader. 2007-01-04. Heat kernels on metric graphs and a trace formula. https://arxiv.org/abs/math-ph/0701009

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