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

Abstract

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 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 Schrödinger 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.

22 pages, revised version, to appear in J. Math. Phys

References in corpus (2)

• Asymptotic stability of small solitons for 2D Nonlinear Schrödinger equations with potential
• On the asymptotic behavior of high order moments for a family of Schroedinger equations

Cited by in corpus (1)

• The $W^{s,p}$-boundedness of stationary wave operators for the Schrödinger operator with inverse-square potential