Arithmetic Spectral Transitions for the Maryland Model

Duration: 26 mins 39 secs

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Description:	Liu, W (Fudan University) Tuesday 07 April 2015, 15:30-15:55


Created:	2015-04-08 18:01
Collection:	Periodic and Ergodic Spectral Problems
Publisher:	Isaac Newton Institute
Copyright:	Liu, W
Language:	eng (English)


Abstract:	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 $\alpha\notin\mathbb{Q},$ $\sigma_{sc}(h_{\lambda,\alpha,\theta})=\overline{\{e:\gamma_{\lambda}(e) <\delta (\alpha, \theta) \}}$ and $\sigma_{pp}(h_{\lambda,\alpha,\theta})=\{e:\gamma_{\lambda}(e) \geq \delta (\alpha, \theta) \}$. Since $\sigma_{ac}(h_{\lambda,\alpha,\theta})=\emptyset,$(B.Simon and T.Spencer),this gives complete description of the spectral decomposition for all values of parameters $\lambda,\alpha,\theta$. This is a joint work with S.Jitomirskaya.

Abstract:

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 $\alpha\notin\mathbb{Q},$ $\sigma_{sc}(h_{\lambda,\alpha,\theta})=\overline{\{e:\gamma_{\lambda}(e) <\delta (\alpha, \theta) \}}$ and $\sigma_{pp}(h_{\lambda,\alpha,\theta})=\{e:\gamma_{\lambda}(e) \geq \delta (\alpha, \theta) \}$. Since $\sigma_{ac}(h_{\lambda,\alpha,\theta})=\emptyset,$(B.Simon and T.Spencer),this gives complete description of the spectral decomposition for all values of parameters $\lambda,\alpha,\theta$.
This is a joint work with S.Jitomirskaya.

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