On non-uniqueness of recovering Sturm–Liouville operators with delay

Abstract

The inverse spectral theory was developed most widely for purely differential operators. Meanwhile, various their nonlocal perturbations are often more adequate for modelling various real-world processes frequently possessing a nonlocal nature. Thus, there recently appeared a growing interest in the nonlinear inverse problems for differential operators with constant delay. However, because of their peculiarity, such problems are still insufficiently studied. In particular, there has been a long-term basic open question about the unique solvability of such inverse problems for small values of the delay. In this paper, we address that question by giving a negative answer in an important and illustrative situation. Our study is based on numerical simulations, which have finally led to a deep theoretical result. Moreover, we develop a general approach for constructing relevant iso-bispectral potentials. The obtained results essentially change the future strategy of studying inverse problems for differential operators with delay.