Ballistic Motion in One-Dimensional Quasi-Periodic Discrete Schrödinger Equation

Abstract

For the solution $q(t)=(q_n(t){)}_{n\in \mathbb{Z}}$ to one-dimensional discrete Schrödinger equation $$\mathrm{i}{\dot{q}}_n=-(q_{n+1}+q_{n-1})+V(\theta +n\omega )q_n,n\in \mathbb{Z},$$ with $\omega \in {\mathbb{R}}^d$ Diophantine, and $V$ a small real-analytic function on ${\mathbb{T}}^d$, we consider the growth rate of the diffusion norm $\Vert q(t){\Vert }_D:={\left({\sum }_n{n}^2|q_n(t){|}^2\right)}^{\frac{1}{2}}$ for any non-zero $q(0)$ with $\Vert q(0){\Vert }_D<\infty$. We prove that $\Vert q(t){\Vert }_D$ grows {\it linearly} with the time $t$ for any $\theta \in {\mathbb{T}}^d$ if $V$ is sufficiently small.

39 pages, a revised version

References in corpus (2)

• Quantum Dynamics of Periodic and Limit-Periodic Jacobi and Block Jacobi Matrices with Applications to Some Quantum Many Body Problems
• Absolutely continuous spectrum implies ballistic transport for quantum particles in a random potential on tree graphs

Cited by in corpus (2)

• On time dependent Schr{ö}dinger equations: global well-posedness and growth of Sobolev norms
• Ballistic Transport for Limit-Periodic Jacobi Matrices with Applications to Quantum Many-Body Problems