On the spectrum of quasi-periodic Schr\"odinger operators on $\mathbb{Z}^d$ with $C^2$-cosine type potentials

TL;DR

This paper proves Anderson localization, dynamical localization, and regularity of the IDS for multi-dimensional quasi-periodic Schr"odinger operators with asymmetric $C^2$-cosine potentials, extending existing methods to handle complex spectral features.

Contribution

It extends iteration and interlacing methods to analyze asymmetric potentials with collapsed gaps in multi-dimensional quasi-periodic Schr"odinger operators.

Findings

Established Anderson localization for the model.

Proved strong dynamical localization.

Showed $(rac 12-)$-H"older continuity of the IDS.

Abstract

In this paper, we establish the Anderson localization, strong dynamical localization and the $(\frac 12-)$-H\"older continuity of the integrated density of states (IDS) for some multi-dimensional discrete quasi-periodic (QP) Schr\"odinger operators with asymmetric $C^2$-cosine type potentials. We extend both the iteration scheme of \cite{CSZ23a} and the interlacing method of \cite{FV21} to handle asymmetric Rellich functions with collapsed gaps.