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Scattering theory without injectivity radius assumptions and spectral stability for the Ricci flow

by Batu Güneysu and Anton Thalmaier

**Abstract**

We prove a completely new integral criterion for the existence and completeness of the wave
operators W_{±}(−Δ_{h},−Δ_{g},I_{g,h}) corresponding to the (unique self-adjoint realizations of) the
Laplace-Beltrami operators −Δ_{j}, j = g, h, 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 −Δ_{g} and −Δ_{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.

The paper is available here:

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