A Dirac-Schrödinger operator is a Dirac operator over a non-compact manifold M perturbed by an operator valued potential A, acting pointwise in a Hilbert space H. The potential A admits decay of its derivative in some sense, and it is usually required that A is invertible outside some compact region, allowing the operator D to be Fredholm. The Fredholm index problem of D is well studied, and there exists an abundance of associated index theorems, the most famous of which might be one of its earliest incarnations, the ”index=spectral flow”-theorem.
In this talk, the invertibility condition on A is suspended, and therefore D will not be Fredholm. However, there exists a notion of regularized index via trace limits (the so-called Witten index), which we will apply here. We will see that a general trace formula leads to a new index formula, which we will investigate in the example of the (d + 1)-massless Dirac-Schrödinger operators in Rd+1. In particular, we show that its Witten index may attain any real number, in stark contrast to the Fredholm index.