High energy bounds on wave operators

TL;DR

This paper establishes conditions under which wave operators remain bounded at high energies, with applications to quantum backflow and specific operators like polyharmonic and matrix Schrödinger operators.

Contribution

It provides new criteria for the boundedness of wave operators at spectral asymptotes, extending previous results to unbounded functions and high-energy regimes.

Findings

Boundedness of wave operators at asymptotic spectral values established.

Conditions for $f$-boundedness in trace-class and high-energy cases derived.

Applications demonstrated for polyharmonic and matrix Schrödinger 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 $\|(W_\pm(H_1,H_0)-P_{1}^\mathrm{ac}P_{0}^\mathrm{ac})f(H_0)\| <\infty$ 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.