THE SMOOTH SPECTRAL COUNTING FUNCTION AND THE TOTAL PHASE-SHIFT FOR QUANTUM BILLIARDS

The interior-exterior duality provides a means to extract spectral inf ormation (for the interior problem) from the scattering matrix (which is relevant to the exterior problem). We study the smooth spectral cou nting function for the interior, and compare it to the smooth total ph ase shift in the exterior. To leading order in the semiclassical appro ximation these functions are known to coincide. Using various techniqu es, we study the higher-order corrections of the two functions and dis cuss the difference between them.