Recently, there appeared a considerable interest in inverse Sturm–Liouville-type problems with constant delay. However, necessary and sufficient conditions for solvability of such problems were obtained only in one very particular situation. Here we address this gap by obtaining necessary and sufficient conditions in the case of functional-differential pencils possessing a more general form along with a nonlinear dependence on the spectral parameter. For this purpose, we develop the so-called transformation operator approach, which allows reducing the inverse problem to a nonlinear vectorial integral equation. In Appendix A, we obtain as a corollary the analogous result for Sturm–Liouville operators with delay. Remarkably, the present paper is the first work dealing with an inverse problem for functional-differential pencils in any form. Besides generality of the pencils under consideration, an important advantage of studying the inverse problem for them is the possibility of recovering both delayed terms, which is impossible for the Sturm–Liouville operators with two delays. The latter, in turn, is illustrated even for different values of these two delays by a counterexample in Appendix B. We also provide a brief survey on the contemporary state of the inverse spectral theory for operators with delay observing recently answered long-term open questions.

最近，人们对具有恒定延迟的逆 Sturm-Liouville 型问题产生了相当大的兴趣。然而，这些问题的可解性的充要条件只能在一种非常特殊的情况下获得。在这里，我们通过在具有更一般形式以及对光谱参数的非线性依赖性的功能微分铅笔的情况下获得必要和充分条件来解决这一差距。为此，我们开发了所谓的变换算子方法，它允许将逆问题简化为非线性矢量积分方程。在附录 A 中，我们得到了带有延迟的 Sturm-Liouville 算子的类似结果作为推论。值得注意的是，本文是第一部处理任何形式的功能差分铅笔的逆问题的工作。除了所考虑的铅笔的一般性之外，研究它们的反问题的一个重要优点是可以恢复两个延迟项，这对于具有两个延迟的 Sturm-Liouville 算子是不可能的。反过来，即使对于这两个延迟的不同值，附录 B 中的反例也说明了后者。 我们还为具有延迟观察的操作员提供了关于逆谱理论的当代状态的简要调查最近回答的长期开放问题.