Singular Continuous Spectrum and Generic Full Spectral/Packing Dimension for Unbounded Quasiperiodic Schrödinger Operators

We proved that quasiperiodic Schrödinger operators with unbounded potentials (Hα,θu)n=un+1+un-1+g(θ+nα)f(θ+nα)un\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(H_{\alpha ,\theta }u)_n=u_{n+1}+u_{n-1}+ \frac{g(\theta +n\alpha )}{f(\theta +n\alpha )} u_n$$\end{document} have purely singular continuous spectrum on the set {E:0<L(E)<δ(α,θ;f,g)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{E: 0<L(E)<\delta {(\alpha ,\theta ;f,g)}\}$$\end{document}, where δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} is an explicit function and L is the Lyapunov exponent. We assume that f, g are Hölder continuous functions and f has finitely many zeros with weak non-degenerate assumptions. Moreover, we show that for generic α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} and a.e. θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}, the spectral measure of Hα,θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\alpha ,\theta }$$\end{document} has full spectral dimension and packing dimension.