从一般维的背散射数据恢复势函数的奇异性
Recovery of the singularities of a potential from backscattering data in general dimension
摘要：
证明了在$n\ge 2$维上，具有先验正则性的复位势$q$的主要奇点包含在由后向散射数据构造的玻恩近似$q_{B}$中.这是存档使用一个新的显式公式的多重色散算子在傅里叶变换侧。我们还表明，${q-q_{B}}$可以比$q$在Sobolev尺度的一个衍生物更经常。另一方面，我们构造反例表明，在一般情况下，它是不可能有一个以上的衍生增益，有时甚至严格少，这取决于先验的规律性$q$。
英文摘要：
We prove that in dimension $n\ge 2$ the main singularities of a complex potential $q$ having a certain a priori regularity are contained in the Born approximation $q_{B}$ constructed from backscattering data. This is archived using a new explicit formula for the multiple dispersion operators in the Fourier transform side. We also show that ${q-q_{B}}$ can be up to one derivative more regular than $q$ in the Sobolev scale. On the other hand, we construct counterexamples showing that in general it is not possible to have more than one derivative gain, sometimes even strictly less, depending on the a priori regularity of $q$.
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