Batu Güneysu

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.

We prove a completely new integral criterion for the existence and completeness of the wave operators $W_{\pm }(-{\Delta }_h,-{\Delta }_g,I_{g,h})$ corresponding to the (unique self-adjoint realizations of) the Laplace-Beltrami operators $-{\Delta }_j$, $j=1,2$, that are induced by two quasi-isometric complete Riemannian metrics $g$ and $h$ on an open manifold $M$. In particular, this result provides a criterion for the absolutely continuous spectra of $-{\Delta }_g$ and $-{\Delta }_h$ to coincide. Our proof relies on estimates that are obtained using a probabilistic Bismut type formula for the gradient of a heat semigroup. Unlike all previous results, our integral criterion only requires some lower control on the Ricci curvatures and some upper control on the heat kernels, but no control at all on the injectivity radii. As a consequence, we obtain a stability result for the absolutely continuous spectrum under a Ricci flow.