Scattering Theory and Spectral Stability under a Ricci Flow for Dirac Operators

Abstract

Given a noncompact spin manifold $M$ with a fixed topological spin structure and two complete Riemannian metrics $g$ and $h$ on $M$ with bounded sectional curvatures, we prove a criterion for the existence and completeness of the wave operators ${\mathcal{W}}_{\pm }(D_h,D_g,I_{g,h})$ and ${\mathcal{W}}_{\pm }(D_h^2,D_g^2,I_{g,h})$, where $I_{g,h}$ is the canonically given unitary map between the underlying $L^2$-spaces of spinors. This criterion does not involve any injectivity radius assumptions and leads to a criterion for the stability of the absolutely continuous spectrum of a Dirac operator and its square under a Ricci flow.