  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.

Spectral Theory