Trace and index of Dirac–Schrödinger operators on open space with operator potentials

Abstract

We develop a principal trace and generalized index formula for a Dirac–Schrödinger operator $D$ on open space of odd dimension $d\ge 3$ with a potential given by a family of self-adjoint unbounded operators acting on an infinite-dimensional Hilbert space $H$. The presented results generalize formulas surrounding the Callias index theorem to the case of unbounded operator potentials, for which the operator $D$ is not necessarily Fredholm. This is the principal novelty of this paper. As application, we include examples where the trace formula is used to calculate the Witten index of non-Fredholm massless $(d+1)$-Dirac–Schrödinger operators acting in $L^2({\mathbb{R}}^{d+1},H)$.