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

Abstract

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.