固定角散射：奇异点的恢复及其局限性
Fixed angle scattering: recovery of singularities and its limitations
摘要：
证明了在$n \ge 2$维上，具有某种先验正则性的复位势$q$的主奇点包含在由定角散射数据构造的玻恩近似$q_θ$中.此外，在Sobolev标度中，${q-q_θ}$可以比$q$多一个导数。事实上，这个结果是最优的，我们构造了一族紧支势和径向势，对于它们不可能有多于一个的导数增益。此外，这些函数表明，对于$n>3$，最大导数增益可以是非常小的索伯列夫规模的潜力，不具有一定的先验水平的规律性，随着尺寸的增长。
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 $q_θ$ constructed from fixed angle scattering data. Moreover, ${q-q_θ}$ can be up to one derivative more regular than $q$ in the Sobolev scale. In fact, this result is optimal, we construct a family of compactly supported and radial potentials for which it is not possible to have more than one derivative gain. Also, these functions show that for $n>3$, the maximum derivative gain can be very small for potentials in the Sobolev scale not having a certain a priori level of regularity which grows with the dimension.
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