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

Eva-Maria Hekkelman, Edward McDonald

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.

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

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.
Paper Structure (9 sections, 29 theorems, 121 equations)

This paper contains 9 sections, 29 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Operator theory
  4. Left-disjoint families of operators
  5. Preliminaries on manifolds
  6. Proof of Theorem \ref{['main_theorem']}
  7. Manifolds of bounded geometry
  8. Roe's index theorem
  9. An example with a random operator

Key Result

Theorem 1.3

Let $(X,g)$ be a non-compact Riemannian manifold of bounded geometry with Property $($D$)$. Let $P \in EBD^2(X)$ be self-adjoint and lower-bounded, and let $w$ be the function on $X$ defined by Then $f(P)M_w$ is an element of $\mathcal{L}_{1,\infty}$ for all compactly supported functions $f\in C_c(\mathbb{R})$. If $P$ admits a density of states $\nu_P$, we have for all extended limits $\omega$

Theorems & Definitions (40)

  • Definition 1.1: Property (D)
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Corollary 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • ...and 30 more