Ray splitting with ghost orbits: explicit, analytical and exact solution for spectra of scaling step potentials with tunnelling

In the familiar concept of ray splitting, a ray incident on an interface splits into reflected and transmitted rays. Tunnelling adds something new to ray splitting: an incident Newtonian ray splits into a reflected Newtonian ray and a transmitted ray with complex classical action-a ghost orbit. Using novel periodic-orbit expansion techniques, we are able to compute explicitly and analytically energy eigenvalues of an energy-scaling one-dimensional step potential. Inclusion of all periodic orbits in our series expansions, in particular a class of orbits called pure ghosts with imaginary classical action, makes our formulae exact. We suggest an experiment that may be used to verify the importance of pure ghosts. (letter to the editor)