Weyl's Law and Connes' Trace Theorem for Noncommutative Two Tori

TL;DR

This paper establishes noncommutative analogues of Weyl's law and Connes' trace theorem for the noncommutative two torus, linking spectral asymptotics with noncommutative geometry.

Contribution

It proves the noncommutative Weyl's law and Connes' trace theorem for the noncommutative two torus with general conformal structures.

Findings

Asymptotic eigenvalue distribution of the perturbed Laplacian matches Weyl's law.

Dixmier trace and noncommutative residue coincide on order -2 pseudodifferential operators.

Results extend classical spectral geometry to noncommutative settings.

Abstract

We prove the analogue of Weyl's law for a noncommutative Riemannian manifold, namely the noncommutative two torus $\mathbb{T}_\theta^2$ equipped with a general translation invariant conformal structure and a Weyl conformal factor. This is achieved by studying the asymptotic distribution of the eigenvalues of the perturbed Laplacian on $\mathbb{T}_\theta^2$. We also prove the analogue of Connes' trace theorem by showing that the Dixmier trace and a noncommutative residue coincide on pseudodifferential operators of order -2 on $\mathbb{T}_\theta^2$.