Relative index theory

AbstractWe develop the framework for the heat equation method in the relative index theory. We study pairs of Dirac operators on complete manifolds and give an analytic interpretation of the difference of the integrals over their local index densities. This yields a generalization of the relative index theorem of Gromov and Lawson. Under some assumptions on the curvature we show, that supersymmetric scattering theories of Borisov, Müller, and Schrader arise naturally. The scattering index is related to the relative topological index