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).