High energy bounds on wave operators

Abstract

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 $\Vert (W_{\pm }(H_1,H_0)-P_1^{\mathrm{ac}}{P}_0^{\mathrm{ac}})f(H_0)\Vert <\infty$ 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 Schrödinger operators with matrix-valued potentials. We discuss an application to the problem of quantum backflow.

as published in J. Operator Theory; 23 pages

References in corpus (2)

• Multi-Component NLS Models on Symmetric Spaces: Spectral Properties versus Representations Theory
• Quantum backflow and scattering