Diffusion bound and reducibility for discrete Schrödinger equations with tangent potential

Abstract

In this paper, we consider the lattice Schrödinger equations $i\dot q_n (t) = \tan \pi (n\alpha + x)q_n (t) + \varepsilon \left( {q_{n + 1} (t) + q_{n - 1} (t)} \right) + \delta v_n (t)\left| {q_n (t)} \right|^{2\tau - 2} q_n (t),$ with α satisfying a certain Diophantine condition, x ε ℝ/ℤ, and τ = 1 or 2, where v_n(t) is a spatial localized real bounded potential satisfying |v_n(t)| ⩾ Ce^−ρ|n|. We prove that the growth of H¹ norm of the solution {q_n(t)}_nεℤ is at most logarithmic if the initial data {q_n(0)}_nεℤ ε H¹ for ɛ sufficiently small and a.e. x fixed. Furthermore, suppose that the linear equation has a time quasi-periodic potential, i.e., $i\dot q_n (t) = \tan \pi (n\alpha + x)q_n (t) + \varepsilon \left( {q_{n + 1} (t) + q_{n - 1} (t)} \right) + \delta v_n \left( {\theta ^0 + t\omega } \right)q_n \left( t \right).$ Then the linear equation can be reduced to an autonomous equation for a.e. x and most values of the frequency vectors ω if ɛ and δ are sufficiently small.

Keywords

Tangent potential / reducibility / Sobolev norm / Birkhoff normal form

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