  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.
