The Callias Index Formula Revisited

We revisit the Callias index formula for Dirac-type operators $L$ in odd space dimension $n$, and prove that \begin{align} \text{ind} \, (L) =\bigg(\frac{i}{8\pi}\bigg)^{\frac{n-1}{2}}\frac{1}{2(\frac{n-1}{2})!} \lim_{\Lambda \to\infty}\frac{1}{\Lambda }\sum_{i_{1},\dots,i_{n} = 1}^n \varepsilon_{i_{1}\dots i_{n}} \int_{\Lambda S_{n-1}}\text{tr}_{\mathbb{C}^d}\, (U(x)(\partial_{i_{1}}U)(x)\dots (\partial_{i_{n-1}}U)(x)) x_{i_{n}}\, d^{n-1} \sigma(x), \, (*) \end{align} where $U(x) = \text{sgn} \,(\Phi(x))$ and $L$ in $L^{2}(\mathbb{R}^{n})^{2^{\widehat n}d}$ is of the form $$L= \mathcal{Q} + \Phi,$$ where $$\mathcal{Q} = \bigg(\sum_{j=1}^{n}\gamma_{j,n}\partial_{j}\bigg) I_d,$$ with $\gamma_{j,n}$ elements of the Euclidean Dirac algebra, and $n=2{\widehat n}$ or $n=2{\widehat n}+1$. Here $\Phi$ is assumed to satisfy the following conditions: \begin{align} & \Phi\in C_{b}^{2}\big(\mathbb{R}^{n};\mathbb{C}^{d\times d}\big), \quad d \in \mathbb{N}, \\ & \Phi(x)=\Phi(x)^{*}, \end{align} there exists $c>0$, $R\geq0$ such that $$|\Phi(x)|\geq c I_d, \quad x\in\mathbb{R}^{n}\backslash B(0,R),$$ and there exists $\varepsilon> 1/2$ such that for all $\alpha\in\mathbb{N}_{0}^{n}$, $|\alpha|<3$, there is $\kappa>0$ such that $$\|(\partial^{\alpha}\Phi)(x)\|\leq \begin{cases} \kappa (1+|x|)^{-1}, & |\alpha|=1,\\ \kappa (1+ |x|)^{-1-\varepsilon}, & |\alpha|=2, \end{cases}\quad x\in\mathbb{R}^{n}.$$ These conditions on $\Phi$ render $L$ a Fredholm operator, and appear to be the most general conditions known to date for which Callias' index formula has been derived. Generalizations of the index formula $(*)$ to certain classes of non-Fredholm operators $L$ invoking the (generalized) Witten index are also discussed.