 05277 Stephen A. Fulling
 Local Spectral Density and Vacuum Energy near a Quantum Graph Vertex
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Aug 17, 05

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Abstract. The delta interaction at a vertex generalizes the Robin
(generalized Neumann) boundary condition on an interval. Study
of a single vertex with N infinite leads suffices to determine
the localized effects of such a vertex on densities of states,
etc. For all the standard initialvalue problems, such as that
for the wave equation, the pertinent integral kernel (Green
function) on the graph can be easily constructed from the
corresponding elementary Green function on the real line. From the
results one obtains the spectralprojection kernel, local spectral
density, and local energy density. The energy density, which
refers to an interpretation of the graph as the domain of a
quantized scalar field, is a coefficient in the asymptotic
expansion of the Green function for an elliptic problem involving
the graph Hamiltonian; that expansion contains spectral/geometrical
information beyond that in the muchstudied heatkernel expansion.
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