Generalised Dirac-Schrödinger operators and the Callias Theorem
Koen van den Dungen
TL;DR
The paper addresses the problem of computing the index of a generalized Dirac--Schrödinger operator $\mathcal{D}_M - i\mathcal{S}(\cdot)$, where $\mathcal{S}(\cdot)$ is a family of unbounded self-adjoint operators acting on a Hilbert $C^*$-module, by reducing the computation to data on a compact hypersurface $N$.It develops a unified Callias-type framework that covers both finite- and infinite-rank potentials and connects the index to the boundary operator via the Kasparov product, the relative index of spectral projections, and spectral flow concepts.The main contributions are (i) a general Callias-type index theorem for generalized Dirac--Schrödinger operators, (ii) a relative-index-based reduction to a hypersurface, and (iii) a rigorous Kasparov-product formulation showing the index equals a boundary pairing $\mathrm{rel-ind}(P_+(\mathcal{S}_N(\cdot)),P_+(\mathcal{T}(\cdot))) \otimes_{C(N)} [\mathcal{D}_N]$, independent of auxiliary cobordism data.The results extend classical Callias theorems and spectral flow identities to the setting of Hilbert $C^*$-modules, providing tools for index computations in noncompact and infinite-rank contexts with potential applications to noncommutative geometry.
Abstract
We consider generalised Dirac--Schrödinger 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. The index of such an operator represents the pairing (Kasparov product) of the K-theory class of the potential with the K-homology class of D. Our main result in this paper is a generalisation of the Callias Theorem: the index of the Dirac--Schrödinger operator can be computed on a suitable compact hypersurface. Our theorem simultaneously generalises (and is inspired by) the well-known result that the spectral flow of a path of relatively compact perturbations depends only on the endpoints.