Diffusion bound and reducibility for discrete Schr?dinger equations with tangent potential

doi:10.1007/s11464-012-0241-2

Front Math Chin

2012, Vol. 7

Issue (6) : 1213-1235 https://doi.org/10.1007/s11464-012-0241-2

RESEARCH ARTICLE

Diffusion bound and reducibility for discrete Schr?dinger equations with tangent potential

Shiwen ZHANG(

), Zhiyan ZHAO

Department of Mathematics, Nanjing University, Nanjing 210093, China

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Abstract

In this paper, we consider the lattice Schr&Dot;odinger equations iq ˙n(t)=tan?π(nα+x)qn(t)+?(qn+1(t)+qn-1(t))+δυn(t)|qn(t)|2τ-2qn(t), with α satisfying a certain Diophantine condition, x∈?/?, and τ = 1 or 2, where υn(t) is a spatial localized real bounded potential satisfying |υn(t)|≤Ce-ρ|n|. We prove that the growth of H¹ norm of the solution {qn(t)}n∈? is at most logarithmic if the initial data {qn(0)}n∈?∈H1 for ? sufficiently small and a.e. x fixed. Furthermore, suppose that the linear equation has a time quasi-periodic potential, i.e., iq ˙n(t)=tan?π(nα+x)qn(t)+?(qn+1(t)+qn-1(t))+δυn(θ0+tw)qn(t). Then the linear equation can be reduced to an autonomous equation for a.e. x and most values of the frequency vectors ω if ? and δ are sufficiently small.

Keywords Tangent potential reducibility Sobolev norm Birkhoff normal form

Corresponding Author(s): ZHANG Shiwen,Email:zhangshiwennju@163.com

Issue Date: 01 December 2012

Cite this article:

Zhiyan ZHAO,Shiwen ZHANG. Diffusion bound and reducibility for discrete Schr?dinger equations with tangent potential[J]. Front Math Chin, 2012, 7(6): 1213-1235.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-012-0241-2
https://academic.hep.com.cn/fmc/EN/Y2012/V7/I6/1213

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