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 \({\mathbb{T}_{\theta}^{2}}\) .
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Fathizadeh, F., Khalkhali, M. Weyl’s Law and Connes’ Trace Theorem for Noncommutative Two Tori. Lett Math Phys 103, 1–18 (2013). https://doi.org/10.1007/s11005-012-0593-2
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DOI: https://doi.org/10.1007/s11005-012-0593-2