Scattering theory in homogeneous Sobolev spaces for Schr\"odinger and wave equations with rough potentials

TL;DR

This paper develops scattering theory for Schrödinger and wave equations with rough potentials in homogeneous Sobolev spaces, covering inverse-square and strongly singular potentials, and describes asymptotic behaviors of solutions.

Contribution

It introduces new results on the existence of wave operators in homogeneous Sobolev spaces for critical and subcritical rough potentials, including inverse-square and strongly singular cases.

Findings

Existence of wave operators in homogeneous Sobolev spaces for inverse-square potentials.

Asymptotic decomposition of solutions in the critical case.

A criterion linking standard and homogeneous Sobolev space wave operators.

Abstract

We study the scattering theory for the Schr\"odinger and wave equations with rough potentials in a scale of homogeneous Sobolev spaces. The first half of the paper concerns 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 Schr\"odinger and wave equations. In the critical case, it is shown that the solution is asymptotically a sum of a $n$-dimensional free wave and a rescaled two-dimensional free wave. The second half of the paper is concerned with a generalization to a class of strongly singular decaying potentials. We provides 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.