Ballistic Transport and Absolute Continuity of One-Frequency Schrödinger Operators

Abstract

For the solution $u(t)$ to the discrete Schrödinger equation $$\mathrm{i}\frac{d}{dt}{u}_n(t)=-(u_{n+1}(t)+u_{n-1}(t))+V(\theta +n\alpha )u_n(t),n\in \mathbb{Z},$$ with $α\in\R\setminus\Q$ and $V\in C^ω(\T,\R)$, we consider the growth rate with $t$ of its diffusion norm $⟨u(t){⟩}_p:={\left({\sum }_{n\in \mathbb{Z}}(n^p+1)|u_n(t){|}^2\right)}^{\frac{1}{2}}$, and the (non-averaged) transport exponents $${\beta }_u^{+}(p):=\underset{t\to \infty }{\mathrm{limsup}}\frac{2\log ⟨u(t){⟩}_p}{p\log t},{\beta }_u^{-}(p):=\underset{t\to \infty }{\mathrm{liminf}}\frac{2\log ⟨u(t){⟩}_p}{p\log t}.$$ We will show that, if the corresponding Schrödinger operator has purely absolutely continuous spectrum, then ${\beta }_u^{\pm }(p)=1$, provided that $u(0)$ is well localized.

arXiv admin note: text overlap with arXiv:1001.2878 by other authors

References in corpus (5)

• The absolutely continuous spectrum of the almost Mathieu operator
• Almost reducibility and absolute continuity I
• 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
• On transport properties of isotropic quasiperiodic $XY$ spin chains

Cited by in corpus (1)

• Ballistic Transport for Limit-Periodic Jacobi Matrices with Applications to Quantum Many-Body Problems