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 $[L^2({\mathbb{R}}^n){]}^N$, $n\ge 2$, $N=2^{\lfloor (n+1)/2\rfloor }$, and then prove the absence of singular continuous spectrum of interacting massless Dirac operators $H=H_0+V$, where $V$ decays like $O(|x{|}^{-1-\epsilon })$. Expressing the spectral shift function $\xi (\cdot ;H,H_0)$ as normal boundary values of regularized Fredholm determinants, we prove that for sufficiently decaying $V$, $\xi (\cdot ;H,H_0)\in C((-\infty ,0)\cup (0,\infty ))$, and that the left and right limits at zero, $\xi (0_{\pm };H,H_0)$, exist. Introducing the non-Fredholm operator ${\mathbf{D}}_{\mathbf{A}}=\frac{d}{dt}+\mathbf{A}$ in $L^2(\mathbb{R};[L^2({\mathbb{R}}^n){]}^N)$, where $\mathbf{A}={\mathbf{A}}_{-}+\mathbf{B}$, ${\mathbf{A}}_{-}$, and $\mathbf{B}$ 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 ${\mathbf{H}}_1={\mathbf{D}}_{\mathbf{A}}^{\ast }{\mathbf{D}}_{\mathbf{A}}$, ${\mathbf{H}}_2={\mathbf{D}}_{\mathbf{A}}{\mathbf{D}}_{\mathbf{A}}^{\ast }$, one of the principal results in this manuscript expresses the $k$th resolvent regularized Witten index $W_{k,r}({\mathbf{D}}_{\mathbf{A}})$ ($k\in \mathbb{N}$, $k\ge \lceil n/2\rceil$) in terms of spectral shift functions as \[ W_{k,r}(\boldsymbol{D}_{\boldsymbol{A}}) = ξ(0_+; \boldsymbol{H_2}, \boldsymbol{H_1}) = [ξ(0_+;H,H_0) + ξ(0_-;H,H_0)]/2. \] Here $L^2(\mathbb{R};\mathcal{H})={\int }_{\mathbb{R}}^{\oplus }dt\mathcal{H}$ and $\mathbf{T}={\int }_{\mathbb{R}}^{\oplus }dtT(t)$ abbreviate direct integrals.

158 pages