Geometric properties of quantum graphs and vertex scattering matrices
Abstract. Differential operators on metric graphs are investigated. It is proven that vertex matching (boundary) conditions can be successfully parameterized by the vertex scattering matrix. Two new families of matching conditions are investigated: hyperplanar Neumann and hyperplanar Dirichlet conditions. Using trace formula it is shown that the spectrum of the Laplace operator determines certain geometric properties of the underlying graph.
Keywords: scattering theory, quantum graphs, matching (boundary) conditions.
Mathematics Subject Classification: 35R30, 47A10, 81U40, 81Q10.