# High energy bounds on wave operators

December 23, 2019

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:

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