An approach for solving a variety of inverse coefficient problems for the Sturm-Liouville equation -y''+q(x)y={\lambda}y 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, the inverse two-spectra Sturm-Liouville problem as well as the recovery of the potential from the output boundary values of an incident wave interacting with the potential. The approach leads to efficient numerical algorithms for solving coefficient inverse problems. Numerical efficiency is illustrated by several examples.

翻译：针对具有复值势函数q(x)的Sturm-Liouville方程-y''+q(x)y=λy，提出了一种求解多种反系数问题的方法。该方法基于解的Bessel函数级数Neumann级数表示形式，借助该表示将问题转化为关于表示系数的线性代数方程组，势函数则通过前两个系数的算术组合重构。所考虑问题的特例包括：利用Weyl函数恢复势函数、双谱逆Sturm-Liouville问题，以及根据入射波与势函数相互作用的输出边界值恢复势函数。该方法为求解系数反问题提供了高效数值算法，并通过多个算例验证了其数值效能。