Trace and Index of Dirac-Schr\"odinger Operators on Open Space with Operator Potentials

Abstract

We develop a principal trace and generalized index formula for a Dirac-Schr\"odinger operator $D$ on open space of odd dimension $d\geq 3$ with a potential given by a family of self-adjoint unbounded operators acting on a infinite dimensional Hilbert space $H$. The presented results generalize formulas surrounding the Callias index theorem to 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-Schr\"odinger operators acting in $L^2\left(\mathbb{R}^{d+1},H\right)$.