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arXiv · 2006.15860

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

Haruya Mizutani
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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.

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Haruya Mizutani. 2020-09-10. Scattering theory in homogeneous Sobolev spaces for Schrödinger and wave equations with rough potentials. https://arxiv.org/abs/2006.15860

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