Semiclassical Weyl law and exact spectral asymptotics in noncommutative geometry

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 $$\lim_{h\downarrow 0} h^p\mathrm{Tr}(\chi_{(-\infty,0)}(h^2D^2+V)) = \int V_-^{\frac{p}{2}}|ds|^p.$$ where $\int$ is Connes' noncommutative integral.