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

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

2011-11-05

arxiv.org/abs/1111.1358v1

Mathematics

Quantum Algebra

13 pages

Scientific paper

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 metric, by studying the asymptotic growth of the eigenvalues of its Laplacian. We also prove the analogue of Connes' trace theorem for pseudodifferential operators of order -2 on $\mathbb{T}_\theta^2$ by showing that the Dixmier trace and a noncommutative residue coincide on these operators.

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