Diffusion bound and reducibility for discrete Schrödinger equations with tangent potential

Abstract

In this paper, we consider the lattice Schr¨odinger equations \mathrm{i}{\dot{\mathrm{q}}}_{\mathit{n}}(t)=\tan \mathrm{\pi }(\mathit{n}\alpha +x)q_n(t)+ϵ(q_{n+\mathrm{1}}(t)+q_{n-\mathrm{1}}(t))+\delta {\upsilon }_n(t){\left|q_n(t)\right|}^{\mathrm{2}\tau -\mathrm{2}}{q}_n(t), with α satisfying a certain Diophantine condition, x\in ℝ/\mathbb{Z}, and τ = 1 or 2, where {\upsilon }_n(t) is a spatial localized real bounded potential satisfying \left|{\upsilon }_n(t)\right|\le C{\mathrm{e}}^{-\rho \left|n\right|}. We prove that the growth of H¹ norm of the solution {\{q_n(t)\}}_{n\in \mathbb{Z}} is at most logarithmic if the initial data {\{q_n(\mathrm{0})\}}_{n\in \mathbb{Z}}\in H^{\mathrm{1}} for ϵ sufficiently small and a.e. x fixed. Furthermore, suppose that the linear equation has a time quasi-periodic potential, i.e., \mathrm{i}{\dot{\mathit{q}}}_{\mathit{n}}(t)=\tan \mathrm{\pi }(\mathit{n}\alpha +x)q_n(t)+ϵ(q_{n+\mathrm{1}}(t)+q_{n-\mathrm{1}}(t))+\delta {\upsilon }_n({\theta }^{\mathrm{0}}+tw)q_n(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 ϵ and δ are sufficiently small.

Key words： Tangent potential; reducibility; Sobolev norm; Birkhoff normal form

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