The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects

Abstract

This work provides a first step towards the construction of a noncommutative geometry for the quantum Hall effect in the continuum. Taking inspiration from the ideas developed by Bellissard during the 80's we build a spectral triple for the $C^{\ast }$-algebra of continuous magnetic operators based on a Dirac operator with compact resolvent. The metric aspects of this spectral triple are studied, and an important piece of Bellissard's theory (the so-called first Connes' formula) is proved.

Keywords: Landau Hamiltonian; spectral triple; Dixmier trace; first Connes' formula

References in corpus (3)

• Algebras of distributions suitable for phase-space quantum mechanics. I
• Algebras of distributions suitable for phase-space quantum mechanics. II. Topologies on the Moyal algebra
• Non-compact spectral triples with finite volume

Cited by in corpus (2)

• Bulk-edge correspondence for unbounded Dirac-Landau operators
• The noncommutative geometry of the Landau Hamiltonian: Differential aspects