In this paper, two classical inverse spectral problems are investigated, namely, the inverse second-order Sturm-Liouville problem and the inverse fourth-order Sturm-Liouville problem. Based on Lidskii’s theorem, we derive trace formulas showing relations between the unknown coefficients and eigenvalues explicitly for both problems. According to those trace formulas, two efficient algorithms are proposed to recover the symmetric potential from one spectrum for second-order Sturm-Liouville problem and two coefficients simultaneously from three spectra for fourth-order Sturm-Liouville problem, respectively. Numerical results are presented to illustrate the effectiveness of the proposed reconstruction algorithms.

本文研究了两个经典的逆谱问题，即二阶逆Sturm-Liouville问题和逆四阶Sturm-Liouville问题。基于 Lidskii 定理，我们推导出迹公式，明确显示两个问题的未知系数和特征值之间的关系。根据这些迹公式，分别针对二阶Sturm-Liouville问题提出了两种有效的算法来从一个谱中恢复对称势，而对于四阶Sturm-Liouville问题则分别从三个谱中恢复两个系数。给出了数值结果来说明所提出的重建算法的有效性。