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The spectra of the surface Maryland model for all phases


Author: Wencai Liu
Journal: Proc. Amer. Math. Soc. 144 (2016), 5035-5047
MSC (2010): Primary 11F72, 37A30, 47A10
DOI: https://doi.org/10.1090/proc/13093
Published electronically: August 5, 2016
MathSciNet review: 3556250
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Abstract: We study the discrete Schrödinger operators $ H_{\lambda ,\alpha ,\theta }$ on $ \ell ^2(\mathbb{Z}^{d+1})$ with surface potential of the form $ V(n,x)=\lambda \delta (x)\tan \pi (\alpha \cdot n+\theta )$, and $ H_{\lambda ,\alpha ,\theta }^{+}$ on $ \ell ^2(\mathbb{Z}^{d}\times \mathbb{Z}_+)$ with the boundary condition $ \psi _{(n,-1)}=\lambda \tan \pi (\alpha \cdot n+\theta )\psi _{(n,0)} $, where $ \alpha \in \mathbb{R}^d$ is rationally independent. We show that the spectra of $ H_{\lambda ,\alpha ,\theta }$ and $ H_{\lambda ,\alpha ,\theta }^{+}$ are $ (-\infty ,\infty )$ for all parameters. We can also determine the absolutely continuous spectra and Hausdorff dimension of the spectral measures if $ d=1$.


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Additional Information

Wencai Liu
Affiliation: School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China
Address at time of publication: Department of Mathematics, University of California, Irvine, California 92697-3875
Email: liuwencai1226@gmail.com

DOI: https://doi.org/10.1090/proc/13093
Received by editor(s): November 18, 2015
Received by editor(s) in revised form: January 11, 2016
Published electronically: August 5, 2016
Communicated by: Michael Hitrik
Article copyright: © Copyright 2016 American Mathematical Society