Ballistic Transport in One-Dimensional Quasi-Periodic Continuous Schrödinger Equation

Abstract

For the solution $q(t)$ to the one-dimensional continuous Schrödinger equation $$\mathrm{i}{\partial }_{t}q(x,t)=-{\partial }_x^{2}q(x,t)+V(\omega x)q(x,t),x\in \mathbb{R},$$ with $\omega \in {\mathbb{R}}^d$ satisfying a Diophantine condition, and $V$ a real-analytic function on ${\mathbb{T}}^d$, we consider the growth rate of the diffusion norm $\Vert q(t){\Vert }_D:={\left({\int }_{\mathbb{R}}{x}^2|q(x,t){|}^{2}dx\right)}^{\frac{1}{2}}$ for any non-zero initial condition $q(0)\in H^1(\mathbb{R})$ with $\Vert q(0){\Vert }_D<\infty$. We prove that $\Vert q(t){\Vert }_D$ grows {\it linearly} with $t$ if $V$ is sufficiently small.

37 pages

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 (1)

• On time dependent Schr{ö}dinger equations: global well-posedness and growth of Sobolev norms