Abstract:

Open quantum maps provide simple finite-dimensional models of open quantum chaos. They are families of $N \times N$ matrices that quantize a symplectic relation on a compact phase space, and their eigenvalues model resonances of certain open quantum systems in the semiclassical limit as $N \to \infty$. This makes them especially conducive to numerical experimentation and thus appealing in the study of scattering resonances. We consider a particular toy model that arises from quantizing the classical baker’s map. We find a Weyl upper bound in the semiclassical limit for the number of eigenvalues in a fixed annulus, and derive an explicit dependence on the radius of the annulus given Gevrey regularity. These results are accompanied by numerical experiments.