# Oscillation theory for the density of states of high dimensional random operators

## TL;DR

This paper extends oscillation theory to high-dimensional random operators, linking the integrated density of states to spectral flow and Pruefer phase winding in a von Neumann algebra setting.

## Contribution

It introduces a novel approach connecting oscillation theory with spectral flow for high-dimensional Jacobi operators using von Neumann algebra techniques.

## Key findings

- Integrated density of states approximated by Pruefer phase winding
- Spectral flow interpreted as a rotation number in von Neumann algebra
- Extension of oscillation theory to high-dimensional random operators

## Abstract

Sturm-Liouville oscillation theory is studied for Jacobi operators with block entries given by covariant operators on an infinite dimensional Hilbert space. It is shown that the integrated density of states of the Jacobi operator is approximated by the winding of the Pruefer phase w.r.t. the trace per unit volume. This rotation number can be interpreted as a spectral flow in a von Neumann algebra with finite trace.