This paper deals with the discrete system being the finite-difference approximation of the Sturm–Liouville problem with frozen argument. The inverse problem theory is developed for this discrete system. We describe the two principal cases: degenerate and non-degenerate. For these two cases, appropriate inverse problems statements are provided, uniqueness theorems are proved, and reconstruction algorithms are obtained. Moreover, the relationship between the eigenvalues of the continuous problem and its finite-difference approximation is investigated. We obtain the “correction terms” for approximation of the discrete problem eigenvalues by using the eigenvalues of the continuous problem. Relying on these results, we develop a numerical algorithm for recovering the potential of the Sturm–Liouville operator with frozen argument from a finite set of eigenvalues. The effectiveness of this algorithm is illustrated by numerical examples.

本文讨论离散系统是 Sturm-Liouville 问题的有限差分逼近，其具有冻结参数。逆问题理论是为这个离散系统开发的。我们描述了两种主要情况：退化和非退化。针对这两种情况，给出了适当的逆问题陈述，证明了唯一性定理，并得到了重构算法。此外，研究了连续问题的特征值与其有限差分近似之间的关系。我们通过使用连续问题的特征值来获得近似离散问题特征值的“修正项”。依靠这些结果，我们开发了一种数值算法，用于从有限的特征值集恢复带有冻结参数的 Sturm-Liouville 算子的潜力。数值例子说明了该算法的有效性。