  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 $\mathscr{W}_{\pm}(D_h, D_g, I_{g,h})$ and $\mathscr{W}_{\pm}(D_h^2,
D^2_g, I_{g,h})$, 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.
