Cantor spectrum for multidimensional quasi-periodic Schr\"odinger operators

TL;DR

This paper studies the complex Cantor spectrum of multidimensional quasi-periodic Schrödinger operators and shows that for certain irrational frequencies, the spectrum's fractal dimension can be positive but arbitrarily small.

Contribution

It proves the existence of Cantor spectrum in multidimensional quasi-periodic Schrödinger operators and analyzes the spectrum's fractal dimension for the critical almost Mathieu operator.

Findings

Spectrum exhibits Cantor structure in multidimensional cases.

Positive Hausdorff dimension for spectrum at certain frequencies.

Spectrum's fractal dimension can be made arbitrarily small.

Abstract

In this paper, we investigate the spectrum of a class of multidimensional quasi-periodic Schr\"odinger operators that exhibit a Cantor spectrum, which provides a resolution to a question posed by Damanik, Fillman, and Gorodetski \cite{DFG}. Additionally, we prove that for a dense set of irrational frequencies with positive Hausdorff dimension, the Hausdorff (and upper box) dimension of the spectrum of the critical almost Mathieu operator is positive, yet can be made arbitrarily small.