An inverse spectral problem for second-order functional-differential pencils with two delays

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.