Multidimensional Almost-Periodic Schr\"odinger Operators with Cantor Spectrum

TL;DR

This paper constructs multidimensional almost-periodic Schrödinger operators with spectra that are generalized Cantor sets of zero Lebesgue measure, revealing complex spectral structures with minimal measure.

Contribution

It introduces a method to construct multidimensional almost-periodic Schrödinger operators with spectra of zero lower box counting dimension, expanding understanding of spectral properties.

Findings

Spectrum is a generalized Cantor set of zero Lebesgue measure

Spectrum has zero lower box counting dimension

Spectral complexity in multidimensional operators

Abstract

We construct multidimensional almost-periodic Schr\"odinger operators whose spectrum has zero lower box counting dimension. In particular, the spectrum in these cases is a generalized Cantor set of zero Lebesgue measure.