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

TL;DR

This paper proves that certain unbounded quasiperiodic Schrödinger operators exhibit purely singular continuous spectrum on specific energy sets and that, generically, their spectral measures have full spectral and packing dimensions.

Contribution

It establishes the presence of purely singular continuous spectrum for unbounded potentials and demonstrates that, for generic parameters, the spectral measure has full dimension.

Findings

Purely singular continuous spectrum on a specified energy set.

Spectral measure has full spectral and packing dimension for generic parameters.

Results hold under mild regularity and zero-finiteness assumptions on the potential functions.

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.