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We derive a limiting absorption principle on any compact interval in$\mathbb{R} \backslash \{0\}$ 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 singularcontinuous spectrum of interacting massless Dirac operators $H = H_0 +V$, where$V$ decays like $O(|x|^{-1 - \varepsilon})$.  Expressing the spectral shift function $\xi(\,\cdot\,; H,H_0)$ as normalboundary values of regularized Fredholm determinants, we prove that forsufficiently decaying $V$, $\xi(\,\cdot\,;H,H_0) \in C((-\infty,0) \cup(0,\infty))$, and that the left and right limits at zero, $\xi(0_{\pm};H,H_0)$, 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 $\boldsymbol{B}$are generated in terms of $H, H_0$ and $V$, via $A(t) = A_- + B(t)$, $A_- =H_0$, $B(t)=b(t) V$, $t \in \mathbb{R}$, assuming $b$ is smooth, $b(-\infty) =0$, $b(+\infty) = 1$, and introducing $\boldsymbol{H_1} =\boldsymbol{D}_{\boldsymbol{A}}^{*} \boldsymbol{D}_{\boldsymbol{A}}$,$\boldsymbol{H_2} = \boldsymbol{D}_{\boldsymbol{A}}\boldsymbol{D}_{\boldsymbol{A}}^{*}$, one of the principal results in thismanuscript expresses the $k$th resolvent regularized Witten index$W_{k,r}(\boldsymbol{D}_{\boldsymbol{A}})$ ($k \in \mathbb{N}$, $k \geq \lceiln/2 \rceil$) in terms of spectral shift functions as \[W_{k,r}(\boldsymbol{D}_{\boldsymbol{A}}) = \xi(0_+; \boldsymbol{H_2},\boldsymbol{H_1}) = [\xi(0_+;H,H_0) + \xi(0_-;H,H_0)]/2. \] Here$L^2(\mathbb{R};\mathcal{H}) = \int_{\mathbb{R}}^{\oplus} dt \, \mathcal{H}$and $\boldsymbol{T} = \int_{\mathbb{R}}^{\oplus} dt \, T(t)$ abbreviate directintegrals.

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