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Journal of Operator Theory

Volume 86, Issue 1, Summer 2021  pp. 61-91.

High energy bounds on wave operators

Authors:  Henning Bostelmann (1), Daniela Cadamuro (2), Gandalf Lechner (3)
Author institution: (1) Department of Mathematics, University of York, York YO10 5DD, U.K.
(2) Institut fuer Theoretische Physik, Universitaet Leipzig, Bruederstrasse 16, 04103 Leipzig, Germany
(3) School of Mathematics, Cardiff University, Senghennydd Road, CF24 4AG Cardiff, U.K.

Summary: The wave operators $W_\pm(H_1,H_0)$ of two selfadjoint operators $H_0$ and $H_1$ are analyzed at asymptotic spectral values. Sufficient conditions for $\|(W_\pm(H_1,H_0)-P_{1}^\mathrm{ac}P_{0}^\mathrm{ac})f(H_0)\| <\infty$ are given, where $P_{j}^\mathrm{ac}$ projects onto the subspace of absolutely continuous spectrum of $H_j$ and $f$ is an unbounded function ($f$-boundedness), both in the case of trace-class perturbations and in terms of the high-energy behaviour of the boundary values of the resolvent of $H_0$ (smooth method). Examples include $f$-boundedness for the perturbed polyharmonic operator and for Schr\"odinger operators with matrix-valued potentials. We discuss an application to the problem of quantum backflow.

Keywords: scattering theory, wave operator, high energy behaviour, Schr\"odinger operator

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