The index of generalised Dirac–Schrödinger operators

Abstract

We study the relation between spectral flow and index theory within the framework of (unbounded) $K$K-theory. In particular, we consider a generalised notion of ‘Dirac–Schrödinger operators’, consisting of a self-adjoint elliptic first-order differential operator $\mathcal{D}$ with a skew-adjoint ‘potential’ given by a (suitable) family of unbounded operators on an auxiliary Hilbert module. We show that such Dirac–Schrödinger 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–Schrödinger operator represents the pairing (Kasparov product) of the $K$-theory class of the potential with the $K$-homology class of $\mathcal{D}$. 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.