A Dixmier trace formula for the density of states

TL;DR

This paper proves that the measure derived from Connes' trace formula for Schrödinger operators with bounded potential matches the density of states measure in solid state physics, enabling explicit calculations.

Contribution

It establishes the equivalence between the Connes trace measure and the density of states measure for Schrödinger operators, providing a new tool for explicit density of states computations.

Findings

The two measures coincide for bounded potential Schrödinger operators.

Explicit formulas for the density of states are derived in certain cases.

The result bridges noncommutative geometry and solid state physics measures.

Abstract

A version of Connes trace formula allows to associate a measure on the essential spectrum of a Schr\"odinger operator with bounded potential. In solid state physics there is another celebrated measure associated with such operators --- the density of states. In this paper we demonstrate that these two measures coincide. We show how this equality can be used to give explicit formulae for the density of states in some circumstances.