Trace class properties of resolvents of Callias operators

Abstract

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