# A reducibility result for Schr\"odinger operator with finite smooth and time quasi-periodic potential

## TL;DR

This paper proves a reduction theorem for a Schrödinger operator with finite smooth, time-quasi-periodic potential, showing it has pure point spectra and zero Lyapunov exponent, using KAM techniques.

## Contribution

It introduces a reduction theorem for Schrödinger operators with finite smooth quasi-periodic potentials, advancing the understanding of their spectral properties.

## Key findings

- Schrödinger operator has pure point spectra.
- Operator exhibits zero Lyapunov exponent.
- Reduction achieved via KAM technique.

## Abstract

In the present paper, we establish a reduction theorem for linear Schr\"odinger equation with finite smooth and time-quasi-periodic potential subject to Dirichlet boundary condition by means of KAM technique. Moreover, it is proved that the corresponding Schr\"odinger operator possesses the property of pure point spectra and zero Lyapunov exponent.