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 singular continuous 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 normal boundary values of regularized Fredholm determinants, we prove that for sufficiently decaying $V$, $\xi(\,\cdot\,;H,H_0) \in C((-\infty,0) \cup (0,\infty))$, 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 this manuscript expresses the $k$th resolvent regularized Witten index $W_{ k,r}(\boldsymbol{ D}_{ \boldsymbol{ A}})$ ($k \in \mathbb{ N}$, $k \geq \lceil n/2 \rceil$) 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 direct integrals.
Categories: math.SP