MPHA Seminar, Blocker 627, 2nd November 2007, 1:50pm

Speaker: Justin Wilson, Maryland
Title: Generalized Method of Images on Quantum Graphs

A finite quantum graph is a space of compact intervals joined at
vertices.  In this work the Laplacian is applied along each interval
(i.e. a bond) with appropriate self-adjoint boundary conditions at
the vertices.  Taking a solution which solves the Laplacian on the
whole real line, we construct a solution for an arbitrary quantum
graph with the method of images.  This amounts to a sum over all the
classical paths.  Applying this method to an integral kernel on the
whole real line, we can get the corresponding kernel on a quantum
graph.  This method has been used in previous work to find the
cylinder kernel on a quantum graph with scale-invariant boundary
conditions.  In this work this is extended to general self-adjoint
boundary conditions.  This form of the kernel can then be used to find
its trace. We find in general that the trace has non-vanishing,
non-constant terms from all closed orbits on the graph.  Agreement is
found with this trace and the trace previously found for
scale-invariant boundary conditions.  In addition, we reproduce the
case of Robin boundary conditions on the interval.  In principle, this
result can then be used to find the trace formula and the vacuum
energy (as well as the vacuum energy density).