A short proof of an index theorem (Q2750872)

This article gives a \(KK\)-theoretic proof of the index theorem for Callias-type operators (also known as perturbed Dirac operators and Dirac-Schrödinger operators) on a class of complete, noncompact, odd-dimensional manifolds with warped-product ends. The index formula is expressed as the integral of a differential form on the cross-section of the end. The proof relies on the associativity of the Kasparov product of \(KK\) cycles representing the perturbation, restriction to the cross-section (an analogue of restriction to a boundary), and the Dirac operator. The result is a new proof of the generalization by \textit{N. Anghel} [Commun. Math. Phys. 128, No. 1, 77-97 (1990; Zbl 0697.58048)] of the theorem of \textit{C. Callias} [Commun. Math. Phys. 62, 213-234 (1978; Zbl 0416.58024)].