Gaplessness of Landau Hamiltonians on hyperbolic half-planes via coarse geometry

TL;DR

This paper demonstrates that Landau Hamiltonians on hyperbolic half-planes lack spectral gaps, indicating that edge states fill all gaps between Landau levels, similar to Euclidean cases, using coarse index methods.

Contribution

It introduces a coarse geometric approach to prove the absence of spectral gaps for Landau Hamiltonians on hyperbolic half-spaces, extending known Euclidean results.

Findings

No spectral gaps in hyperbolic Landau Hamiltonians

Edge states fill all Landau level gaps

Method applies to general imperfect half-spaces

Abstract

We use coarse index methods to prove that the Landau Hamiltonian on the hyperbolic half-plane, and even on much more general imperfect half-spaces, has no spectral gaps. Thus the edge states of hyperbolic quantum Hall Hamiltonians completely fill up the gaps between Landau levels, just like those of the Euclidean counterpart.