The index of generalised Dirac-Schr\"odinger operators

TL;DR

This paper explores the relationship between spectral flow and index theory for generalized Dirac-Schr"odinger operators within KK-theory, establishing their Fredholm property, a relative index theorem, and the pairing of K-theory and K-homology classes.

Contribution

It introduces a broad class of Dirac-Schr"odinger operators, proves their Fredholm property, and relates their index to K-theory and K-homology pairings without requiring differentiability of the potential.

Findings

Operators are shown to be Fredholm.

A relative index theorem is established.

Index equals spectral flow in the real line case.

Abstract

We study the relation between spectral flow and index theory within the framework of (unbounded) KK-theory. In particular, we consider a generalised notion of 'Dirac-Schr\"odinger operators', consisting of a self-adjoint elliptic first-order differential operator D with a skew-adjoint 'potential' given by a (suitable) family of unbounded operators on an auxiliary Hilbert module. We show that such Dirac-Schr\"odinger operators are Fredholm, and we prove a relative index theorem for these operators (which allows cutting and pasting of the underlying manifolds). Furthermore, we show that the index of a Dirac-Schr\"odinger operator represents the pairing (Kasparov product) of the K-theory class of the potential with the K-homology class of D. We prove this result without assuming that the potential is differentiable; instead, we assume that the 'variation' of the potential is sufficiently small near infinity. In the special case of the real line, we recover the well-known equality of the index with the spectral flow of the potential.