General Theory of the Quantum Kicked Rotator. I

This is the first of a series of two papers. We discuss some basic problems of the quantum kicked rotator (QKR) and review some important results in the literature. We point out the flaws in the inverse Cayley transform method to prove dynamic localization. When $\tau/2\pi$, where $\tau$ is the kick period, is very close to a rational number, the localization length is larger than the typical localization length. We analytically prove anomalous localization and confirm it by numerical calculations. We point out open problems that need further work.