Vacuum energy and spectral analysis for Robin boundaries and quantum graphs

In the simplest configurations a solution of a partial differential equation with a Robin boundary condition can be generated from a solution of the corresponding Dirichlet problem. In more general cases similar reasoning can be adapted to construct an approximate solution as a sum over classical paths, which may suffer delayed reflection at the boundary. This analysis provides a new approach to spectral densities and vacuum energy densities. A quantum graph is a network of one-dimensional domains joined at vertices, at which, typically, a close analogue of the Robin boundary condition applies. Our techniques, therefore, apply to such models