A scattering approach to the quantization of billiards- The inside-outside duality

We present some recent results on the semiclassical quantization of billiards using an approach which is based on the strong link between the billiard interior and exterior problems. That is, the spectrum of the interior problem is extracted from the scattering matrix of the exterior problem. Once this is put on a rigorous basis, the semiclassical approximation is used to derive the semiclassical zeta function and the spectral density. The duality between the inside and outside problems prevails also in the classical description and offers new insight into this quantization procedure. The relation between the present approach and the more standard quantization methods is also discussed and illustrated with some numerical results.