We consider an inverse scattering problem for the Schrödinger operator in two dimensions. The aim of this work is to discuss some first numerical results on Saito’s formula. Saito’s formula is an explicit integral formula, which at the high-frequency limit gives a uniqueness result for the inverse scattering problem. The numeric approach is quite straight-forward: we take a large enough fixed wave number and evaluate the integrals in Saito’s formula numerically. The potential function can then be recovered from the blurry measurements by using the fast Fourier transform and a high-pass filter. We also discuss in detail how the synthetic data is generated via a matrix-based approach. Several numerical examples are shown to demonstrate the results.

中文翻译：

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反散射理论中斋藤唯一性定理的数值结果

我们考虑了二维Schrödinger算子的逆散射问题。这项工作的目的是讨论有关Saito公式的一些最初的数值结果。Saito公式是一个显式积分公式，在高频极限下，它可以解决逆散射问题。数值方法非常简单：我们采用足够大的固定波数，并以数字方式评估Saito公式中的积分。然后可以通过使用快速傅立叶变换和高通滤波器从模糊测量中恢复势函数。我们还将详细讨论如何通过基于矩阵的方法生成综合数据。显示了几个数值示例来证明结果。