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 on the real line away from zero for the n-dimensional free massless Dirac operator H₀, and then use this to demonstrate the absence of singular continuous spectrum of interacting massless Dirac operators H = H₀ + V, where the entries of the (essentially bounded) matrix-valued potential V decay appropriately at spatial infinity. This includes the special case of electromagnetic potentials. In addition, we derive a one-to-one correspondence between embedded eigenvalues of H (away from zero) and the eigenvalue −1 of an underlying Birman–Schwinger-type operator.
In addition, expressing the spectral shift function for the pair (H, H₀) as normal boundary values of associated regularized Fredholm determinants to the real axis, we then prove that under appropriate additional decay hypotheses on V, the spectral shift function for (H, H₀) is continuous away from zero and that its left and right limits at zero exist.
This fact is then used to express the resolvent regularized Witten index of the non-Fredholm operator DA = (d/dt) + A, where A represents the direct integral over a family operators A(t) that has asymptotes A₊ and A₋ as t tends to + infinity and − infinity (in the norm resolvent sense), respectively, in terms of the spectral shift function for the pair (H, H₀).