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arXiv · 2604.15008

Noncommutative Geometry, Spectral Asymptotics, and Semiclassical Analysis

Raphael Ponge
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Abstract

Semiclassical analysis and noncommutative geometry are two pillars of quantum theory. It is only recently that bridges between them have been emerging. In this monograph, we combine various techniques from functional analysis and spectral theory to obtain semiclassical Weyl laws and extensions of Connes' integration formula for a large class of noncommutative manifolds (i.e., spectral triples). These results generalize and simplify recent results of McDonald-Sukochev-Zanin. In particular, all the regularity assumptions and restrictions on dimension there are removed in our approach. Moreover, the Tauberian condition used by McDonald-Sukochev-Zanin is replaced by a weaker spectral theoretic condition, called Condition (W). That condition holds in fairly greater generality and significantly opens the scope of applicability of the main results. We also give Tauberian conditions that imply Condition (W). These Tauberian conditions are easier to check in practice than the Tauberian condition of McDonald-Sukochev-Zanin and are satisfied in numerous examples. The need for these conditions was highlighted by Alain Connes in an online seminar. The main results of this paper are illustrated by semiclassical Weyl laws and integration formulas in the settings of closed Riemannian manifolds and quantum tori. In the former settings we recover well-known semiclassical Weyl laws, as well as Weyl laws for Steklov eigenvalues. The only novelty is obtaining them from old results of Minakshisundaram-Pleijel on heat kernel asymptotics. In the setting of quantum tori, the semiclassical Weyl laws provide a positive answer to a conjecture of Edward McDonald and the author. The integration formulas are refinements of several previous analogues of Connes' integration formula for quantum tori.

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BibTeXRIS

Raphael Ponge. 2026-04-16. Noncommutative Geometry, Spectral Asymptotics, and Semiclassical Analysis. https://arxiv.org/abs/2604.15008

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