Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials

Yingte Sun

doi:10.3934/dcds.2021047

Article Contents

2021, Volume 41, Issue 10: 4531-4543. Doi: 10.3934/dcds.2021047

This issue Previous Article Counting finite orbits for the flip systems of shifts of finite type Next Article On the critical decay for the wave equation with a cubic convolution in 3D

Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials

Yingte Sun^,

School of Mathematical Sciences, Yangzhou University, Yangzhou 225009, China

^* Corresponding author
^* Corresponding author

Received: September 2020

Revised: February 2021

Early access: March 2021

Published: October 2021

The author is is supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 19KJB110025) and School Foundation of Yangzhou University(Grant No. 2019CXJ009)

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Abstract

In this paper, we study the one-dimensional stationary Schrödinger equation with quasi-periodic potential $ u(\omega t) $. We show that if the frequency vector $ \omega $ is sufficient large, the Schrödinger equation admits two linear independent Floquet solutions for a set of positive measure of energy $ E $. In contrast with previous results, the conditions of small potential $ u $ or large energy $ E $ are no longer needed.

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Mathematics Subject Classification: Primary: 37J40, 34D05.

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References

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