On Selberg's trace formula: chaos, resonances and time delays

The quantization of the chaotic geodesic motion on Riemann surfaces Sigma(g ,kappa) of constant negative curvature with genus g and a finite number of points kappa infinitely faraway (cusps) describing scattering channels is i nvestigated. It is shown that terms in Selberg's trace formula describing s cattering states can be expressed in terms of a renormalized time delay. Th is quantity is the time delay associated with the surface in question minus the time delay corresponding to the scattering problem on the Poincare upp er half-plane uniformizing our surface. Poles in these quantities give rise to resonances reflecting the chaos of the underlying classical dynamics. O ur results are illustrated for the surfaces Sigma(1,1) (Gutzwiller's leaky torus), Sigma(0,3) (pants), and a class of Sigma(g,2) surfaces. The general ization covering the inclusion of an integer B greater than or equal to 2 m agnetic field is also presented. It is shown that the renormalized time del ay is not dependent on the magnetic field. This shows that the semiclassica l dynamics with an integer magnetic field is the same as the free dynamics.