2017/09/05 by Batu Güneysu, Güneysu, Batu, Anton Thalmaier +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Physics (math-ph) #Nonlinear Partial Differential Equations #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.1709.01612
openalex publication_date 2017/09/05 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We prove a completely new integral criterion for the existence and\ncompleteness of the wave operators W\±(-\Δh,-\Δg, Ig,h)\ncorresponding to the (unique self-adjoint realizations of) the Laplace-Beltrami\noperators -\Δj, j=1,2, that are induced by two quasi-isometric\ncomplete Riemannian metrics g and h on an open manifold M. In particular,\nthis result provides a criterion for the absolutely continuous spectra of\n-\Δg and -\Δh to coincide. Our proof relies on estimates that are\nobtained using a probabilistic Bismut type formula for the gradient of a heat\nsemigroup. Unlike all previous results, our integral criterion only requires\nsome lower control on the Ricci curvatures and some upper control on the heat\nkernels, but no control at all on the injectivity radii. As a consequence, we\nobtain a stability result for the absolutely continuous spectrum under a Ricci\nflow.\n