Trace Class Properties of Resolvents of Callias Operators

We present conditions for a family ${\left(A\left(x\right)\right)}_{x\in {\mathbb{R}}^d}$ of self-adjoint operators in $H^r={\mathbb{C}}^r\otimes H$ for a separable complex Hilbert space $H$, such that the Callias operator $D=ic\mathrm{\nabla }+A\left(X\right)$ satisfies that ${(D^{\ast }D+1)}^{-N}-{(D{D}^{\ast }+1)}^{-N}$ is trace class in $L^2({\mathbb{R}}^d,H^r)$. Here, $c\mathrm{\nabla }$ is the Dirac operator associated to a Clifford multiplication $c$ of rank $r$ on ${\mathbb{R}}^d$, and $A\left(X\right)$ is fibre-wise multiplication with $A\left(x\right)$ in $L^2({\mathbb{R}}^d,H^r)$.