We consider two main inverse Sturm-Liouville problems: the problem of recovery of the potential and the boundary conditions from two spectra or from a spectral density function. A simple method for practical solution of such problems is developed, based on the transmutation operator approach, new Neumann series of Bessel functions representations for solutions and the Gelfand-Levitan equation. The method allows one to reduce the inverse Sturm-Liouville problem directly to a system of linear algebraic equations, such that the potential is recovered from the first element of the solution vector. We prove the stability of the method and show its numerical efficiency with several numerical examples.

中文翻译：

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求解逆 Sturm-Liouville 问题的直接方法

我们考虑两个主要的逆 Sturm-Liouville 问题：从两个谱或谱密度函数恢复势和边界条件的问题。基于嬗变算子方法、新的 Neumann 级数 Bessel 函数表示和 Gelfand-Levitan 方程，开发了一种用于实际解决此类问题的简单方法。该方法允许将逆 Sturm-Liouville 问题直接简化为线性代数方程组，以便从解向量的第一个元素恢复势能。我们证明了该方法的稳定性，并通过几个数值例子证明了其数值效率。