Regularized index theory for non-Fredholm Dirac-Schrödinger operators

We present a new trace formula which connects heat semi-groups of a Dirac-Schrödinger operator with its operator potential. The trace formula induces a non-Fredholm extension of the classical Callias index theorem to the Witten index. We conclude with an invariant description of both sides of the generalized index formula in terms of (higher order) spectral shift functions.