On the quantum cat and sawtooth maps—Return to generic behaviour

The quantization of the continuous cat maps on the torus has led to rather pathological quantum objects [1]. The non-generic behaviour of this model has led some to conclude that the Correspondence Principle fails in this case [2]. In this note we introduce the quantum sawtooth models, since this is the natural family to which the cap maps belong. Thus, a simple propagator depending on a parameter is constructed which for integer values of the parameter becomes pathological quantum cat maps, while away from integer values we find a return to the generic behaviour of non-integrable quantum systems.