# Wannier basis method for KAM effect in quantum mechanics

## TL;DR

This paper introduces a Wannier basis method in phase space to distinguish regular and irregular eigenstates in quantum systems, providing quantitative measures and clarifying the KAM effect's relation to Anderson localization.

## Contribution

The paper presents a novel Wannier basis approach to analyze KAM effects in quantum systems, enabling quantitative distinction of eigenstates and clarification of localization phenomena.

## Key findings

- Defined area and effective dimension of eigenstates
- Quantitatively distinguished regular and irregular states
- Clarified the difference between KAM effect and Anderson localization

## Abstract

The effect of 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 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 KAM effect and Anderson localization.