In this paper, an efficient algorithm for recovering a density of a fourth-order Sturm–Liouville operator from two given spectra is investigated. Based on Lidskii’s theorem and Mercer’s theorem, we build a sequence of trace formulae which bridge explicitly the density and eigenvalues in terms of nonlinear Fredholm integral equations. Due to intrinsic difficulties on ill-posedness of an inverse spectral problem, a truncated Fourier series regularization method is utilized for reconstructing the unknown density. Moreover, shifted Legendre polynomials are carried to balance the different order of trace formulae. Numerical results are presented to illustrate the effectiveness of the proposed reconstruction algorithm.

在本文中，研究了一种从两个给定光谱中恢复四阶Sturm-Liouville算子密度的有效算法。基于Lidskii定理和Mercer定理，我们建立了一系列跟踪公式，它们根据非线性Fredholm积分方程明确地将密度和特征值联系在一起。由于反谱问题不适定性的固有困难，因此采用了一种截断的傅里叶级数正则化方法来重建未知密度。而且，移位的Legendre多项式可以平衡迹线公式的不同顺序。数值结果表明了所提出的重建算法的有效性。