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

### Alan Carey

Australian National University, Canberra; and University of Wollongong, Australia### Fritz Gesztesy

Baylor University, Waco, USA### Galina Levitina

Australian National University, Canberra, Australia### Roger Nichols

The University of Tennessee at Chattanooga, USA### Fedor Sukochev

University of New South Wales, Sydney, Australia### Dmitriy Zanin

University of New South Wales, Sydney, Australia

This book is published *open access.*

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

in $[L_{2}(R_{n})]_{N}$, $n∈N$, $n≥2$, $N=2_{⌊(n+1)/2⌋}$. 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 $O(∣x∣_{−1−ε})$ as $∣x∣→∞$ for some $ε>0$. 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 $R\{0}$ and the eigenvalue $−1$ of the (normal boundary values of the) Birman–Schwinger-type operator

Upon expressing $ξ(⋅;H,H_{0})$ 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 $O(∣x∣_{−n−1−ε})$ as $∣x∣→∞$), the associated spectral shift function satisfies $ξ(⋅;H,H_{0})∈C((−∞,0)∪(0,∞))$, and that the left and right limits at zero, $ξ(0_{±};H,H_{0})=lim_{ε↓0}ξ(±ε;H,H_{0})$, exist.

This fact is then used to express the resolvent regularized Witten index of the non-Fredholm operator $D_{A}$ in $L_{2}(R;[L_{2}(R_{n})]_{N})$ given by

where

in terms of $ξ(⋅;H,H_{0})$. Here $A$, $A_{−}$, $A_{+}$, $B$, and $B_{+}$ in $L_{2}(R;[L_{2}(R_{n})]_{N})$ are generated with the help of the Dirac-type operators $H,H_{0}$ and potential matrices $V$, via

in $[L_{2}(R_{n})]_{N}$, assuming

In particular, $A_{±}$ are the asymptotes of the family $A(t)$, $t∈R$, as $t→±∞$ in the norm resolvent sense. (Here $L_{2}(R;H)=∫_{R}dtH$ and $T=∫_{R}dtT(t)$ represent direct integrals of Hilbert spaces and operators.)

Introducing the nonnegative, self-adjoint operators

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

Here the notation $ξ_{L}(0_{+};H_{2},H_{1})$ indicates that $0$ is a right Lebesgue point for $ξ(⋅;H_{2},H_{1})$, 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 $K$, defined by

whenever the limit exists for some $k∈N$.