Singular value asymptotics on compact smooth Riemaniann manifolds
Fedor Sukochev, Fulin Yang, Dmitriy Zanin
TL;DR
The paper proves a sharp singular value asymptotic formula for operators S in the Civita-Pi_X algebra on a compact Riemannian manifold, linking t^{1/p}μ(t,S(1+Δ_G)^{-d/(2p)}) to the L_p-norm of the principal symbol sym_X(S) over the cosphere bundle with canonical weight. It develops a novel blend of double operator integrals, differentiability concepts inspired by Connes–Moscovici, and the notion of a good atlas to transfer local Euclidean estimates to the curved manifold setting, avoiding classical Complex Powers machinery. The main result unifies Weyl-type Weyl law asymptotics with Connes' noncommutative integral, providing explicit constants and weights in terms of the Laplace-Beltrami geometry and the symbol. This yields a broad, operator-algebraic version of Weyl's law on manifolds, with a clear path to noncommutative integration interpretations and potential applications to spectral estimates of quantum derivatives and related operator families.
Abstract
Let $(X,G)$ be a $d$-dimensional compact smooth Riemannian manifold equipped with Laplace-Beltrami operator $Δ_{G}$, and let $Π_{X}$ be the $C^{\ast}$-algebra obtained by locally transferring the $C^{\ast}$-algebra generated by multiplication operators and Riesz transforms on $\mathbb{R}^{d}$. Denote ${\rm sym}_{X}$ the principal symbol mapping of $Π_{X}$. For any $S\inΠ_{X}$, we prove that, in the framework of $C^{\ast}$-algebra, \begin{align*} \lim_{t\rightarrow\infty}t^{\frac{1}{p}}μ(t,S(1+Δ_G)^{-\frac{d}{2p}}) =(2π\sqrt[d]{d})^{-\frac{1}{p}}\Big\|{\rm sym}_{X}(S)\Big\|_{L_{p}(T^{\ast}X,e^{-q_{G}}dλ)}, \end{align*} where $0<p<\infty$, $e^{-q_{G}}$ is the canonical weight on $X$, and $dλ$ is the Liouville measure on the cotangent bundle $T^{\ast}X$.