Singular continuous spectrum and generic full spectral/packing dimension for unbounded quasiperiodic Schr\"odinger operators

Abstract

We proved that Schr\"odinger operators with unbounded potentials $(H_{\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+ \frac{g(\theta+n\alpha)}{f(\theta+n\alpha)} u_n$ have purely singular continuous spectrum on the set $\{E: 0<L(E)<\delta{(\alpha,\theta;f,g)}\}$, where $\delta$ is an explicit function and $L$ is the Lyapunov exponent. We only require $f,g$ are H\"older continuous functions and $f$ has finitely many zeros with weak non-degenerate assumptions. Moreover, we show that for generic $\alpha$ and a.e. $\theta$, the spectral measure of $H_{\alpha,\theta}$ has full spectral/packing dimension.