Stephen A. Fulling
Local Spectral Density and Vacuum Energy near a Quantum Graph Vertex
(42K, AMS-LaTeX)

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 initial-value 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 spectral-projection 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 much-studied heat-kernel expansion.