The Limiting Absorption Principle for Massless Dirac Operators, Properties of Spectral Shift Functions, and an Application to the Witten Index of Non-Fredholm Operators

Applying the theory of strongly smooth operators, we derive a limiting absorption principle (LAP) on any compact interval in $\mathbb{R}\backslash \{0\}$ for the free massless Dirac operator,

in $[L^2({\mathbb{R}}^n){]}^N$, $n\in \mathbb{N}$, $n\ge 2$, $N=2^{\lfloor (n+1)/2\rfloor }$. We then use this to demonstrate the absence of singular continuous spectrum of interacting massless Dirac operators $H=H_0+V$, where the entries of the (essentially bounded) matrix-valued potential $V$ decay like $O(|x{|}^{-1-\epsilon })$ as $|x|\to \infty$ for some $\epsilon >0$. This includes the special case of electromagnetic potentials decaying at the same rate. In addition, we derive a one-to-one correspondence between embedded eigenvalues of $H$ in $\mathbb{R}\backslash \{0\}$ and the eigenvalue $-1$ of the (normal boundary values of the) Birman–Schwinger-type operator

Upon expressing $\xi (\cdot ;H,H_0)$ as normal boundary values of regularized Fredholm determinants to the real axis, we then prove that in the concrete case $(H,H_0)$, under appropriate hypotheses on $V$ (implying the decay of $V$ like $O(|x{|}^{-n-1-\epsilon })$ as $|x|\to \infty$), the associated spectral shift function satisfies $\xi (\cdot ;H,H_0)\in C((-\infty ,0)\cup (0,\infty ))$, and that the left and right limits at zero, $\xi (0_{\pm };H,H_0)={\lim }_{\epsilon \downarrow 0}\xi (\pm \epsilon ;H,H_0)$, exist.

This fact is then used to express the resolvent regularized Witten index of the non-Fredholm operator ${\mathbf{D}}_{\mathbf{A}}^{}$ in $L^2(\mathbb{R};[L^2({\mathbb{R}}^n){]}^N)$ given by

where

in terms of $\xi (\cdot ;H,H_0)$. Here $\mathbf{A}$, ${\mathbf{A}}_{-}$, ${\mathbf{A}}_{+}$, $\mathbf{B}$, and ${\mathbf{B}}_{+}$ in $L^2(\mathbb{R};[L^2({\mathbb{R}}^n){]}^N)$ are generated with the help of the Dirac-type operators $H,H_0$ and potential matrices $V$, via

in $[L^2({\mathbb{R}}^n){]}^N$, assuming

In particular, $A_{\pm }$ are the asymptotes of the family $A(t)$, $t\in \mathbb{R}$, as $t\to \pm \infty$ in the norm resolvent sense. (Here $L^2(\mathbb{R};\mathcal{H})={\int }_{\mathbb{R}}^{\oplus }dt\mathcal{H}$ and $\mathbf{T}={\int }_{\mathbb{R}}^{\oplus }dtT(t)$ represent direct integrals of Hilbert spaces and operators.)

Introducing the nonnegative, self-adjoint operators

in $L^2(\mathbb{R};[L^2({\mathbb{R}}^n){]}^N)$, one of the principal results proved in this manuscript expresses the resolvent regularized Witten index $W_{k,r}({\mathbf{D}}_{\mathbf{A}}^{})$ of ${\mathbf{D}}_{\mathbf{A}}^{}$ in terms of spectral shift functions via

Here the notation ${\xi }_L(0_{+};{\mathbf{H}}_2,{\mathbf{H}}_1)$ indicates that $0$ is a right Lebesgue point for $\xi (\cdot ;{\mathbf{H}}_2,{\mathbf{H}}_1)$, and $W_{k,r}(T)$ represents the $k$th resolvent regularized Witten index of the densely defined, closed operator $T$ in the complex, separable Hilbert space $\mathcal{K}$, defined by

whenever the limit exists for some $k\in \mathbb{N}$.