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 $mathbb{R} backslash {0}$ for the free massless Dirac operator, $H_0 = alpha cdot (-i abla)$ in $[L^2(mathbb{R}^n)]^N$, $n geq 2$, $N=2^{lfloor(n+1)/2rfloor}$, 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 $L^2big(mathbb{R};[L^2(mathbb{R}^n)]^Nbig)$, 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.