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

We prove the analogue of Weyl's law for a noncommutative Riemannian manifold, namely the noncommutative two torus 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 . 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 .