High energy bounds on wave operatorsDecember 23rd, 2019
In a general setting of scattering theory, we consider two self-adjoint operators $H_0$ and $H_1$ and investigate the behaviour of their wave operators $W_\pm(H_1,H_0)$ at asymptotic spectral values of $H_0$ and $H_1$. Specifically, we analyse when $\|(W_\pm(H_1,H_0)-P^{\rm ac}_1P^{\rm ac}_0)f(H_0)\| <\infty$, where $P^{\rm ac}_j$ is the projector onto the subspace of absolutely continuous spectrum of $H_j$, and $f$ is an unbounded function ($f$-boundedness). We provide sufficient criteria both in the case of trace-class perturbations $V= H_1 – H_0$ and within the general setting of the smooth method of scattering theory, where the high-energy behaviour of the boundary values of the resolvent of $H_0$ plays a major role. In particular, we establish $f$-boundedness for the perturbed polyharmonic operator and for Schr\”odinger operators with matrix-valued potentials. Applications of these results include the problem of quantum backflow.
