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Arithmetic Spectral Transitions for the Maryland Model

Presented by: 
W Liu Fudan University
Tuesday 7th April 2015 - 15:30 to 15:55
INI Seminar Room 1
In this talk, I will give a precise description of spectra of the Maryland model $ (h_{\lambda,\alpha,\theta}u) _n=u_{n+1}+u_{n-1}+ \lambda \tan \pi(\theta+n\alpha)u_n$ for all values of parameters. For Almost Mathieu Operator (H_{\lambda,\alpha,\theta}u) _n=u_{n+1}+u_{n-1}+ \lambda \cos 2\pi(\theta+n\alpha)u_n, the Lyapunov exponent can almost determine its spectral types(A.Avila, S.Jitomirskaya, J.You, Q.Zhou). When turn to Maryland model, I introduce an arithmetically defined index $\delta (\alpha, \theta)$ and show that for $\alpha\notin\mathbb{Q},$ $\sigma_{sc}(h_{\lambda,\alpha,\theta})=\overline{\{e:\gamma_{\lambda}(e)
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons