Complex geometric optics solutions to a system of d-bar equations appearing in the context of electrical impedance tomography and the scattering theory of the integrable Davey-Stewartson II equations are studied for large values of the spectral parameter k. For potentials $q\in {⟨\cdot ⟩}^{-2}{H}^s\left(ℂ\right)$ for some $s\in \left]1,2\right]$, it is shown that the solution converges as the geometric series in $1/{\left|k\right|}^{s-1}$. For potentials q being the characteristic function of a strictly convex open set with smooth boundary, this still holds with s = 3/2, i.e., with $1/\sqrt{\mid k\mid }$ instead of $1/{\left|k\right|}^{s-1}$. The leading-order contributions are computed explicitly. Numerical simulations show the applicability of the asymptotic formulae for the example of the characteristic function of the disk. © 2022 Courant Institute of Mathematics and Wiley Periodicals LLC.

中文翻译：

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d-bar 问题的复杂几何光学解决方案的大 ∣k∣ 行为

研究了电阻抗断层扫描中出现的 d-bar 方程组的复杂几何光学解以及可积 Davey-Stewartson II 方程的散射理论，以获取较大的光谱参数 k值。对于潜力$q\epsilon {⟨\cdot ⟩}^{-2}{H}^s\left(ℂ\right)$对于一些$s\epsilon \left]1,2\right]$，结果表明解收敛为几何级数$1/{\left|k\right|}^{s-1}$。对于势q是具有光滑边界的严格凸开集的特征函数，这在 s = 3/2 时仍然成立，即$1/\sqrt{\mid k\mid }$代替$1/{\left|k\right|}^{s-1}$。主序贡献是明确计算的。数值模拟显示了渐近公式对于圆盘特征函数示例的适用性。© 2022 Courant 数学研究所和 Wiley periodicals LLC。