Scattering theory in homogeneous Sobolev spaces for Schrödinger and wave equations with rough potentials

Journal of Mathematical Physics, Volume 61, Issue 9, September 2020. We study the scattering theory for the Schrödinger and wave equations with rough potentials in a scale of homogeneous Sobolev spaces. The first half of this paper is concerned with an inverse-square potential in both of subcritical and critical constant cases, which is a particular model of scaling-critical singular perturbations. In the subcritical case, the existence of the wave and inverse wave operators defined on a range of homogeneous Sobolev spaces is obtained. In particular, we have the scattering to a free solution in the homogeneous energy space for both of the Schrödinger and wave equations. In the critical case, it is shown that the solution is asymptotically a sum of an n-dimensional free wave and a rescaled two-dimensional free wave. The second half of this paper is concerned with a generalization to a class of strongly singular decaying potentials. We provide a simple criterion in an abstract framework to deduce the existence of wave operators defined on a homogeneous Sobolev space from the existence of the standard ones defined on a base Hilbert space.