# Semiclassical Weyl law and exact spectral asymptotics in noncommutative geometry

@inproceedings{Mcdonald2021SemiclassicalWL, title={Semiclassical Weyl law and exact spectral asymptotics in noncommutative geometry}, author={Edward Mcdonald and Fedor Sukochev and Dmitriy Zanin}, year={2021} }

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 (A, H,D) and self-adjoint V ∈ A, there holds lim h…

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In this paper, we establish Cwikel-type estimates for noncommutative tori for any dimension n ≥ 2. We use them to derive Cwikel–Lieb–Rozenblum inequalities and Lieb–Thirring inequalities for negative…

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In the recent paper [32] the authors have considered the Birman-Schwinger (Cwikel) type operators in a domain Ω ⊆ R, having the form TP = A∗PA. Here A is a pseudodifferential operator in Ω of order…