Geodesic
NONCLASSICAL SPECTRAL ASYMPTOTICS AND DIXMIER TRACES: FROM CIRCLES TO CONTACT MANIFOLDS
Forum of Mathematics, Sigma, Tome 5 (2017)

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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 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 $f$ , displaying a wide range of nonclassical spectral asymptotics beyond the Weyl law. The results extend from Riemannian manifolds to contact manifolds and noncommutative tori. 
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     author = {HEIKO GIMPERLEIN and MAGNUS GOFFENG},
     title = {NONCLASSICAL {SPECTRAL} {ASYMPTOTICS} {AND} {DIXMIER} {TRACES:} {FROM} {CIRCLES} {TO} {CONTACT} {MANIFOLDS}},
     journal = {Forum of Mathematics, Sigma},
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HEIKO GIMPERLEIN; MAGNUS GOFFENG. NONCLASSICAL SPECTRAL ASYMPTOTICS AND DIXMIER TRACES: FROM CIRCLES TO CONTACT MANIFOLDS. Forum of Mathematics, Sigma, Tome 5 (2017). doi: 10.1017/fms.2016.33

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