SEMICLASSICAL QUANTIZATION OF CHAOTIC BILLIARDS - A SCATTERING-THEORY APPROACH

We derive a semiclassical secular equation which applies to quantized (compact) billiards of any shape. Our approach is based on the fact that the billiard boundary defines two dual problems: the `inside problem' of the bounded dynamics, and the `outside problem' which can be looked upon as a scattering from the boundary as an obstacle. This duality exists both on the classical and quantum mechanical levels, and is therefore very useful in deriving a semiclassical quantization rule. We obtain a semiclassical secular equation which is based on classical input from a finite number of classical periodic orbits. We compare our result to secular equations which were recently derived by other means, and provide some numerical data which illustrate our method when applied to the quantization of the Sinai billiard.