Nonclassical spectral asymptotics and Dixmier traces: From circles to contact manifolds

TL;DR

This paper investigates the spectral asymptotics of certain geometric operators with singularities, extending noncommutative geometric tools like Dixmier traces from circles to contact manifolds and noncommutative tori.

Contribution

It develops methods to compute Dixmier traces for operators with singular spectral behavior, extending classical results to more singular and noncommutative settings.

Findings

Spectral asymptotics governed by singularities for Hölder continuous functions

Extension of residue trace formulas beyond classical pseudo-differential calculus

Identification of non-measurable Hankel operators with nonclassical spectral behavior

Abstract

We consider the spectral behavior and noncommutative geometry of commutators $[P,f]$, where $P$ is an operator of order $0$ with geometric origin and $f$ a multiplication operator by a function. When $f$ is H\"{o}lder continuous, the spectral asymptotics is governed by singularities. We study precise spectral asymptotics through the computation of Dixmier traces; such computations have only been considered in less singular settings. Even though a Weyl law fails for these operators, and no pseudo-differential calculus is available, variations of Connes' residue trace theorem and related integral formulas continue to hold. On the circle, a large class of non-measurable Hankel operators is obtained from H\"older continuous functions $f$, displaying a wide range of nonclassical spectral asymptotics beyond the Weyl law. The results extend from Riemannian manifolds to contact manifolds and noncommutative tori.