The smooth spectral counting function and the total phase shift for quantum billiards

The interior-exterior duality provides a means to extract spectral information (for the interior problem) from the scattering matrix (which is relevant to the exterior problem). We study the smooth spectral counting function for the interior, and compare it to the smooth total phase shift in the exterior. To leading order in the semiclassical approximation these functions are known to coincide. Using various techniques, we study the higher-order corrections of the two functions and discuss the difference between them.