Sharp arithmetic spectral transitions and bounds on quantum dynamics for supercritical almost Mathieu operators.

Wencai Liu

We will discuss a popular discrete quasiperiodic model- almost Mathieu operator, which is given by \begin{equation*} (H_{\lambda,\alpha,\theta}u)(n)=u({n+1})+u({n-1})+ 2\lambda \cos 2\pi (\theta+n\alpha)u(n), % \text{ with } v(\theta)=2\cos2\pi \theta, \end{equation*} where $\lambda$ is the coupling, $\alpha $ is the frequency, and $\theta $ is the phase. Firstly, we study the spectral transitions, in both phase and frequency, for the almost Mathieu operator in the regime of positive Lyapunov exponents. Combining with some known results, we prove sharp transitions in both regimes: for all frequencies $\alpha$ and almost all phases $\theta$, and for all phases $\theta$ and almost all frequencies $\alpha$.

http://scgp.stonybrook.edu/video_portal/video.php?id=2688

Simons- Workshop: Between Dynamics and Spectral Theory