An approach for solving a variety of inverse coefficient problems for the Sturm–Liouville equation −y″ + q(x)y = ρ2y with a complex valued potential q(x) is presented. It is based on Neumann series of Bessel functions representations for solutions. With their aid the problem is reduced to a system of linear algebraic equations for the coefficients of the representations. The potential is recovered from an arithmetic combination of the first two coefficients. Special cases of the considered problems include the recovery of the potential from a Weyl function, inverse two-spectrum Sturm–Liouville problems, as well as the inverse scattering problem on a finite interval. The approach leads to efficient numerical algorithms for solving coefficient inverse problems. Numerical efficiency is illustrated by several examples.

中文翻译：

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复势重建技术

提出了一种解决具有复值势 q(x) 的 Sturm–Liouville 方程 −y″ + q(x)y = ρ2y 的各种反系数问题的方法。它基于 Neumann 级数的 Bessel 函数表示形式来求解。在他们的帮助下，问题被简化为表示系数的线性代数方程组。势能通过前两个系数的算术组合来恢复。所考虑问题的特殊情况包括从 Weyl 函数恢复势能、逆双谱 Sturm-Liouville 问题以及有限区间上的逆散射问题。该方法产生了解决系数逆问题的有效数值算法。通过几个例子说明了数值效率。