Numerical algorithms for inverse Sturm-Liouville problems - Jiang, Xiaoying; Li, Xiaowen; Xu, Xiang (Numerical algorithms, 2022)
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.