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, inverse two-spectra 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.

翻译：本文提出了一种用于求解Sturm-Liouville方程 -y''+q(x)y={\lambda}y（其中势函数q(x)取复数值）中多种反系数问题的方法。该方法基于解的Bessel函数级数表示的Neumann级数展开。借助这些级数表示，原问题被转化为关于表示系数的线性代数方程组。势函数可通过前两个系数的算术组合重构得出。所考虑问题的特例包括：基于Weyl函数的势函数重构、双谱逆Sturm-Liouville问题，以及有限区间上的逆散射问题。该方法为求解系数反问题提供了高效的数值算法，并通过若干算例验证了其数值效率。