Evolution and exact eigenstates of a resonant quantum system

We consider the generalized quantum Chirikov map under a resonance condition 2{π}{}=M/N. This is the quantum-resonance condition discovered by Casati et al. At the resonance, the quantum system reduces to a set of independent N×{}N unitary matrix eigenequations. We can reduce the evolution operator of the original system as a direct sum of these N×{}N unitary matrices. We then obtain the eigenstates and eigenenergies (the pseudoenergies) of the quantum map and illustrate their dependences on a number of parameters. We plot these eigenstates in the coherent-state representation and show that they follow closely the Kolmogorov-Arnol'd-Moser curves and other classical orbits.