Singular continuous spectrum and generic full spectral/packing dimension for unbounded quasiperiodic Schrödinger operators

Abstract

We proved that Schrödinger 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 Hölder 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.