On the Failure and the Modification of Semiclassical Trace Formula in the Quantum Standard Map

We consider a failure of the semiclassical trace formula attributable to resonant periodic orbits and quasi-integrability of the system by using the quantum standard map. We derive Gutzwiller-type semiclassical expressions of TrU and TrU² lor the Floquet operator U, and compare them with quantum rigorous expressions. We demonstrate that the divergence of the formula due to resonant orbits is avoided by making use of the normal forms for the action. For a quasi-integrable case, we confirm that the semiclassical expression is also well modified.