Eigenvalues of singular measures and Connes' noncommutative integration.
Journal article, 2022

In a domain Ω⊂R^N we consider compact, Birman–Schwinger type operators of the form T​=A^∗P A with P being a Borel measure in Ω, containing a singular part, andA being an order −N/2 pseudodifferential operator. Operators are defined by means of quadratic forms. For a class of such operators, we obtain a proper version of H. Weyl's law for eigenvalues, with order not depending on dimensional characteristics of the measure. These results lead to establishing measurability, in the sense of Dixmier–Connes, of such operators and the noncommutative version of integration over Lipschitz surfaces and rectifiable sets.

Weyl law

eigenvalue asymptotics

Connes integration

Design for manufacturing, welding, tolerance, robust design

singular measures

Author

Grigori Rozenblioum

Euler International Mathematical Institute

Chalmers, Mathematical Sciences

Journal of Spectral Theory

1664-039X (ISSN) 1664-0403 (eISSN)

Vol. 12 1 259-300

Subject Categories

Mathematical Analysis

DOI

10.4171/JST/401

More information

Latest update

10/25/2023