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 $U(x)=\text{sgn}(\mathrm{\Phi }(x))$ and $L$ in $L^2({\mathbb{R}}^n{)}^{2^{\widehat{n}}d}$ is of the form $$L=\mathcal{Q}+\mathrm{\Phi },$$ where $$\mathcal{Q}=\left(\sum _{j=1}^n{\gamma }_{j,n}{\mathrm{\partial }}_j\right)I_d,$$ with ${\gamma }_{j,n}$ elements of the Euclidean Dirac algebra, and $n=2\widehat{n}$ or $n=2{\widehat n}+1$\mathrm{.}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 $c>0$, $R\ge 0$ such that $$\mathrm{\mid }\mathrm{\Phi }(x)\mathrm{\mid }\ge c{I}_d,x\in {\mathbb{R}}^n\mathrm{\backslash }B(0,R),$$ and there exists $\epsilon >1\mathrm{/}2$ such that for all $\alpha \in {\mathbb{N}}_0^n$, $\mathrm{\mid }\alpha \mathrm{\mid }<3$, there is $\kappa >0$ such that $$\mathrm{\parallel }({\mathrm{\partial }}^{\alpha }\mathrm{\Phi })(x)\mathrm{\parallel }\le \{\begin{array}{ll}\kappa (1+\mathrm{\mid }x\mathrm{\mid }{)}^{-1},& \mathrm{\mid }\alpha \mathrm{\mid }=1,\\ \kappa (1+\mathrm{\mid }x\mathrm{\mid }{)}^{-1-\epsilon },& \mathrm{\mid }\alpha \mathrm{\mid }=2,\end{array}x\in {\mathbb{R}}^n\mathrm{.}$$ These conditions on $\mathrm{\Phi }$ 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 $(\ast )$ to certain classes of non-Fredholm operators $L$ invoking the (generalized) Witten index are also discussed.