We study the scattering theory for the Schrodinger 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 Schrodinger 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.

中文翻译：

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薛定谔和具有粗糙势的波动方程的齐次 Sobolev 空间中的散射理论

我们在齐次 Sobolev 空间的尺度上研究了薛定谔和具有粗糙势的波动方程的散射理论。本文的前半部分涉及亚临界和临界常数情况下的平方反比势，这是标度临界奇异扰动的特殊模型。在亚临界情况下，获得了在齐次 Sobolev 空间范围内定义的波算符和逆波算符的存在性。特别是，对于薛定谔方程和波动方程，我们都有均匀能量空间中自由解的散射。在临界情况下，它表明解是一个 n 维自由波和一个重新缩放的二维自由波的渐近和。本文的后半部分涉及对一类强奇异衰减势的推广。我们在抽象框架中提供了一个简单的标准，以从基 Hilbert 空间上定义的标准算子的存在推断出在齐次 Sobolev 空间上定义的波算子的存在。