The Method of Images on a Quantum Graph
Seminar Room 1, Newton Institute
We consider an arbitrary quantum graph with the Laplacian H = -d^2/dx^2 and the corresponding eigenvalue problem on the graph. Taking the free space solution to an associated PDE (such as non-stationary Schroedinger equation), we apply the method of images to obtain the corresponding solution for our graph. This method brings in periodic and non-periodic (bounce) orbits and gives insight into both types of orbits and the roles they play. Applying this method to different kernels, we obtain interesting spectral information including the trace formula for the density of the states and the regularized vacuum energy.