The limiting absorption principle for massless Dirac operators, properties of spectral shift functions, and an application to the Witten index of non-Fredholm operators

We derive a limiting absorption principle on any compact interval in $\mathbb{R}\mathrm{\backslash }\{0\}$ for the free massless Dirac operator, $H_0 = \alpha \cdot (-i \nabla)$in$[L^2($\mathbb{R}$^n)]^N$,$n \geq 2$,$N=2^{\lfloor(n+1)/2\rfloor}$, and then prove the absence of singular continuous spectrum of interacting massless Dirac operators $H=H_0+V$, where $V$ decays like $O(\mathrm{\mid }x{\mathrm{\mid }}^{-1-\epsilon })$. Expressing the spectral shift function $\xi (\cdot ;H,H_0)$ as normal boundary values of regularized Fredholm determinants, we prove that for sufficiently decaying $V$, $\xi(,\cdot,;H,H_0) \in C((-\infty,0) \cup (0,\infty))$,andthattheleftandrightlimitsatzero,$\xi(0_{\pm}; H,H_0)$, exist. Introducing the non-Fredholm operator $\boldsymbol{D}{\boldsymbol{A}} = \frac{d}{dt} + \boldsymbol{A}$in$L^2\big($\mathbb{R}$;[L^2($\mathbb{R}$^n)]^N\big)$,where$\boldsymbol{A} = \boldsymbol{A-} + \boldsymbol{B}$,$\boldsymbol{A_-}$,and$\boldsymbol{B}$aregeneratedintermsof$H, H_0$and$V$,via$A(t) = A_- + B(t)$,$A_- = H_0$,$B(t)=b(t) V$,$t \in $\mathbb{R}$$,assuming$b$issmooth,$b(-\infty) = 0$,$b(+\infty) = 1$,andintroducing$\boldsymbol{H_1} = \boldsymbol{D}{\boldsymbol{A}}^{*} \boldsymbol{D}{\boldsymbol{A}}$,$\boldsymbol{H_2} = \boldsymbol{D}{\boldsymbol{A}} \boldsymbol{D}{\boldsymbol{A}}^{*}$, one of the principal results in this manuscript expresses the $k$th resolvent regularized Witten index $W_{k,r}({\mathit{D}}_{\mathit{A}})$ ($k\in \mathbb{N}$, $k \geq \lceil n/2 \rceil$) in terms of spectral shift functions as $$W_{k,r}({\mathit{D}}_{\mathit{A}})=\xi (0_{+};,)=[\xi (0_{+};H,H_0)+\xi (0_{-};H,H_0)]\mathrm{/}2.$$ Here $L^2(\mathbb{R};\mathcal{H})={\int }_{\mathbb{R}}^{\oplus }dt\mathcal{H}$ and $\mathit{T}={\int }_{\mathbb{R}}^{\oplus }dtT(t)$ abbreviate direct integrals.