On wave operators for Hardy operators with and without perturbations

Abstract

We prove asymptotic completeness of the wave operators in time-dependent scattering theory which intertwine the ordinary or fractional Laplacian and the corresponding Hardy operator, i.e., the fractional Laplacian with an added subcritical or critical Hardy potential, whenever the Hardy potential is short-range. Moreover, we prove asymptotic completeness of the wave operators intertwining Hardy operators with and without external short-range perturbations. We also show the nonexistence of the considered wave operators when the Hardy or external potentials are long-range. We arrive at these results using three different approaches. The first approach uses an abstract semigroup framework for asymptotic completeness of wave operators for Fourier multipliers with short-range potentials. The second approach uses Kato smoothness theory. In the third approach, we develop and apply new propagation estimates for Hardy operators with and without perturbations.