解决斯图姆-利奥维尔问题的直接方法 (A direct method for solving inverse Sturm-Liouville problems)

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.

翻译：我们考虑了两大反面的Sturm-Liouville问题:从两个光谱或光谱密度函数中恢复潜力和边界条件的问题;根据变异操作器方法、新的新Neumann系列贝塞尔功能表示式解决方案和Gelfand-Levitan等式,为实际解决这些问题制定了一种简单的方法;这种方法可以将反Sturm-Liouville问题直接降低为线性代数方程系统,从而从解决办法矢量的第一个要素中恢复潜力。我们用几个数字实例证明了这种方法的稳定性并展示了其数字效率。