论文标题
具有粗糙电势的Schrödinger和波动方程的均匀Sobolev空间中的散射理论
Scattering theory in homogeneous Sobolev spaces for Schrödinger and wave equations with rough potentials
论文作者
论文摘要
我们研究了schrödinger和波动方程的散射理论，具有均匀的Sobolev空间的粗糙电位。本文的前半部分涉及在亚临界和临界恒定病例中具有反向的潜力，这是缩放至关重要的奇异扰动的特定模型。在亚临界情况下，获得了在一系列均匀的Sobolev空间上定义的波浪和反波算子的存在。特别是，对于施罗丁和波方程的同质能量空间中，我们将散射到一个自由溶液中。在关键情况下，结果表明该解决方案是渐近的$ n $维自由波和重新续订的二维自由波的总和。本文的后半部分涉及对一类强烈奇异衰减电位的概括。我们在抽象框架中提供了一个简单的标准，以推断出在基本希尔伯特空间上定义的标准标准空间的存在上定义在均匀Sobolev空间上的波动算子的存在。
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.
