Semiclassical Weyl law and exact spectral asymptotics in noncommutative geometry

Abstract

We prove a Tauberian theorem for singular values of noncommuting operators which allows us to prove exact asymptotic formulas in noncommutative geometry at a high degree of generality. We explain how, via the Birman--Schwinger principle, these asymptotics imply that a semiclassical Weyl law holds for many interesting noncommutative examples. In Connes' notation for quantized calculus, we prove that for a wide class of $p$-summable spectral triples $(\mathcal{A},H,D)$ and self-adjoint $V\in \mathcal{A}$, there holds \[\lim_{h\downarrow 0} h^p\mathrm{Tr}(χ_{(-\infty,0)}(h^2D^2+V)) = \int V_-^{\frac{p}{2}}|ds|^p.\] where $\int$ is Connes' noncommutative integral.

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• Index theory for locally compact noncommutative geometries
• Traces of compact operators and the noncommutative residue
• Measures from Dixmier Traces and Zeta Functions
• The Theory of Pseudo-differential Operators on the Noncommutative n-Torus
• Laplace-Beltrami Operators on Noncommutative Tori
• Noncommutative Residue and Canonical Trace on Noncommutative Tori. Uniqueness Results
• Connes Integration Formula without singular traces
• Mapping properties of operator-valued pseudo-differential operators

Cited by in corpus (2)

• Weyl's laws and Connes' integration formulas for matrix-valued $L\log L$-Orlicz potentials
• Cwikel Estimates and Negative Eigenvalues of Schroedinger Operators on Noncommutative Tori