Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 007, 34 pages      arXiv:2304.13272      https://doi.org/10.3842/SIGMA.2024.007
Contribution to the Special Issue on Global Analysis on Manifolds in honor of Christian Bär for his 60th birthday

A General Dixmier Trace Formula for the Density of States on Open Manifolds

Eva-Maria Hekkelman a and Edward Mcdonald b
a) School of Mathematics and Statistics, University of New South Wales, Kensington, NSW 2052, Australia
b) Department of Mathematics, Penn State University, University Park, PA 16802, USA

Received April 27, 2023, in final form January 10, 2024; Published online January 17, 2024

Abstract
We give an abstract formulation of the Dixmier trace formula for the density of states. This recovers prior versions and allows us to provide a Dixmier trace formula for the density of states of second order elliptic differential operators on manifolds of bounded geometry satisfying a certain geometric condition. This formula gives a new perspective on Roe's index on open manifolds.

Key words: density of states; index theory; Dixmier trace; singular trace; bounded geometry; manifolds.

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