The Witten index and the spectral shift function

TL;DR

This paper extends the spectral flow and index theory to a broader class of operators, including differential operators on non-compact manifolds, by relaxing previous restrictions and providing a new trace formula.

Contribution

It generalizes spectral flow results to operators with higher Schatten class perturbations without requiring ellipticity or Fredholm properties.

Findings

Extended spectral flow to higher Schatten class perturbations.

Derived a trace formula applicable to non-elliptic operators.

Applied results to Dirac operators on Euclidean spaces of arbitrary dimension.

Abstract

In \cite{APSIII} Atiyah, Patodi and Singer introduced spectral flow for elliptic operators on odd dimensional compact manifolds. They argued that it could be computed from the Fredholm index of an elliptic operator on a manifold of one higher dimension. A general proof of this fact was produced by Robbin-Salamon \cite{RS95}. In \cite{GLMST}, a start was made on extending these ideas to operators with some essential spectrum as occurs on non-compact manifolds. The new ingredient introduced there was to exploit scattering theory following the fundamental paper \cite{Pu08}. These results do not apply to differential operators directly, only to pseudo-differential operators on manifolds, due to the restrictive assumption that spectral flow is considered between an operator and {its perturbation by a relatively trace-class operator}. In this paper we extend the main results of these earlier papers to spectral flow between an operator and a perturbation satisfying a higher $p^{th}$ Schatten class condition for $0\leq p<\infty$, thus allowing differential operators on manifolds of any dimension $d<p+1$. In fact our main result does not assume any ellipticity or Fredholm properties at all and proves an operator theoretic trace formula motivated by \cite{BCPRSW, CGK16}. We illustrate our results using Dirac type operators on $L^2(\bbR^d)$ for arbitrary $d\in\bbN$. In this setting our main result substantially extends \cite[Theorem 3.5]{CGGLPSZ16}, where the case $d=1$ was treated.