Singular continuous spectrum for singular potential

TL;DR

This paper establishes conditions under which Schrödinger operators with certain meromorphic potentials exhibit purely singular continuous spectra, extending previous results to a broader class of operators.

Contribution

It proves the singular continuous spectrum for a general family of meromorphic potentials, generalizing prior specific models like the Maryland and almost Mathieu operators.

Findings

Spectral type is purely singular continuous under specified conditions.

Extends previous spectral results to broader meromorphic potentials.

Provides explicit conditions involving Lyapunov exponents.

Abstract

We prove that Schr\"odinger operators with meromorphic 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 $\{E: L(E)<\delta{(\alpha,\theta)}\}$, where $\delta$ is an explicit function, and $L$ is the Lyapunov exponent. This extends results of Jitomirskaya and Liu for the Maryland model and of Avila, You and Zhou for the almost Mathieu operator, to the general family of meromorphic potentials.