Rotation Numbers, Boundary Forces and Gap labelling

TL;DR

This paper reviews various gap labelling methods for 1D Schrödinger operators, compares them, and proposes a higher-dimensional generalization of the Johnson-Moser rotation number based on $K$-theory insights.

Contribution

It introduces a new perspective on gap labelling by connecting rotation numbers with $K$-theory and extending the concept to higher dimensions.

Findings

Comparison of different gap labels for Schrödinger operators.

Identification of a $K_1$-theoretical gap label as a natural higher-dimensional generalization.

Insights into the relationship between boundary forces and spectral gaps.

Abstract

We review the Johnson-Moser rotation number and the $K_0$-theoretical gap labelling of Bellissard for one-dimensional Schr\"odinger operators. We compare them with two further gap-labels, one being related to the motion of Dirichlet eigenvalues, the other being a $K_1$-theoretical gap label. We argue that the latter provides a natural generalisation of the Johnson-Moser rotation number to higher dimensions.