Mixed spectral types for the one frequency discrete quasi-periodic Schr\"odinger operator

TL;DR

This paper studies a specific class of one-frequency quasi-periodic Schrödinger operators, demonstrating the coexistence of different spectral types such as absolutely continuous, point, and singular continuous spectra under various parameters.

Contribution

It provides explicit examples of spectral coexistence phenomena in quasi-periodic Schrödinger operators, expanding understanding of their spectral complexity.

Findings

Existence of parameters with both absolutely continuous and point spectrum.

Existence of parameters with both absolutely continuous and singular continuous spectrum.

Illustration of spectral coexistence phenomena in quasi-periodic operators.

Abstract

We consider a family of one frequency discrete analytic quasi-periodic Schr\"odinger operators which appear in [Bjer]. We show that this family provides an example of coexistence of absolutely continuous and point spectrum for some parameters as well as coexistence of absolutely continuous and singular continuous spectrum for some other parameters.