Poincare surface of section and quantum scattering

A very simple model for the quantum-mechanical scattering of a particle is studied with a dual goal: The chaotic nature of the corresponding classical problem should be quite obvious, and the method of solution should use an approach that is closely related to the surface of section in classical mechanics. Moreover, the mathematical operations should be elementary so that the errors in a semiclassical approximation or in any computational work have a chance of being controllable. Finally, the mode of presentation is such as to be understandable for a newcomer to the field of chaos. The model is a variation of the Sinai billiard where the circular hard wall inside a box (parallelogram) is replaced by a trombone-shaped surface for the particle to enter and exit the box. The rim (circular boundary between trombone and box) is the surface of section, with the total current at fixed energy in either direction providing the measure for the wave functions. The Poincare map then becomes the product of two unitary transformations, where the first is diagonal in angular momentum, while the second is diagonal in angle.