Chaotic scattering on noncompact surfaces of constant negative curvature. (English) Zbl 1071.37503

Lévay, Péter

Mladenov, I. M. (ed.) et al., Proceedings of the international conference on geometry, integrability and quantization, Varna, Bulgaria, September 1–10, 1999. Sofia: Coral Press Scientific Publishing (ISBN 954-90618-1-7/pbk). 145-157 (2000).

Conclusions: In this paper we consider the problem of quantizing the geodesic motion on noncompact surfaces of constant negative curvature. This problem can be regarded as a model of multichannel quantum scattering. Knowing that the geodesic motion on such surfaces is chaotic, we examine how the chaos of the underlying classical dynamics manifests itself in the corresponding quantum system. We calculate the scattering matrix, and introduce the associated time delays. With the help of Selberg’s trace formula we establish a connection between the classical periodic orbits and the quantum resonances and energy eigenvalues. Illustrative examples for a class of \(\Sigma_{g,2}\) surfaces are given.
For the entire collection see [Zbl 0940.00039].

MSC:

37D40	Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
11F72	Spectral theory; trace formulas (e.g., that of Selberg)
37N20	Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
81Q50	Quantum chaos
81S10	Geometry and quantization, symplectic methods