Construction of finite differentiable quasi-periodic Schr\"odinger operators with cantor spectrum

TL;DR

This paper introduces a method for explicitly constructing quasi-periodic Schrödinger operators with Cantor spectrum and smooth potentials, along with asymptotic estimates on spectral gap sizes.

Contribution

It provides a novel explicit construction of smooth quasi-periodic Schrödinger operators with Cantor spectrum and analyzes the spectral gap asymptotics.

Findings

Explicit construction of $C^k$ quasi-periodic Schrödinger operators with Cantor spectrum

Polynomial asymptotics for spectral gap sizes

Advances understanding of spectral properties of smooth quasi-periodic operators

Abstract

In this paper, we present an approach for explicitly constructing quasi-periodic Schr\"odinger operators with Cantor spectrum with $C^k$ potential. Additionally, we provide polynomial asymptotics on the size of spectral gaps.