Trace Class Properties of Resolvents of Callias Operators

TL;DR

This paper establishes conditions under which differences of resolvents of Callias operators are trace class, contributing to the understanding of their spectral and index properties in mathematical physics.

Contribution

It provides new criteria for the trace class property of resolvent differences of Callias operators with variable coefficients.

Findings

Identifies conditions for trace class differences of resolvents

Extends spectral analysis of Callias operators

Enhances understanding of operator index theory

Abstract

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 $D=ic\nabla+A\left(X\right)$ satisfies that $\left(D^{\ast}D+1\right)^{-N}-\left(DD^{\ast}+1\right)^{-N}$ is trace class in $L^2\left(\mathbb{R}^{d},H^{r}\right)$. Here, $c\nabla$ 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 $L^2\left(\mathbb{R}^{d},H^{r}\right)$.