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

Heiko Gimperlein, Magnus Goffeng

Research output: Contribution to journalArticle

3 Citations (Scopus)
49 Downloads (Pure)

Abstract

We consider the spectral behavior and noncommutative geometry of commutators [P,f][P,f], where PP is an operator of order 0 with geometric origin and ff a multiplication operator by a function. When ff is Hölder 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 pseudodifferential calculus is available, variations of Connes’ residue trace theorem and related integral formulas continue to hold. On the circle, a large class of nonmeasurable Hankel operators is obtained from Hölder continuous functions ff, displaying a wide range of nonclassical spectral asymptotics beyond the Weyl law. The results extend from Riemannian manifolds to contact manifolds and noncommutative tori.
Original languageEnglish
Article numbere3
Number of pages57
JournalForum of Mathematics, Sigma
Volume5
Early online date23 Jan 2017
DOIs
Publication statusPublished - 2017

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