Wannier basis method for the Kolmogorov-Arnold-Moser effect in quantum mechanics.

Abstract

The effect of the Kolmogorov-Arnold-Moser (KAM) theorem in quantum systems is manifested in dividing eigenstates into regular and irregular states. We propose an effective method based on the Wannier basis in phase space to illustrate this division of eigenstates. The quantum kicked-rotor model is used to illustrate this method, which allows us to define the area and effective dimension of each eigenstate to distinguish quantitatively regular and irregular eigenstates. This Wannier basis method also allows us to define the length of a Planck cell in the spectrum that measures how many Planck cells the system will traverse if it starts at the given Planck cell. Moreover, with this Wannier approach, we are able to clarify the distinction between the KAM effect and Anderson localization.