Iso-bispectral potentials for Sturm–Liouville-type operators with small delay. (February 2022) - Djurić, Nebojša; Buterin, Sergey
Abstract: The paper addresses nonlinear inverse Sturm–Liouville-type problems with constant delay. Since many processes in the real world possess nonlocal nature, operators with delay as well as other classes of nonlocal operators are continuously finding numerous applications in the natural sciences and engineering. However, in spite of a large number of works devoted to inverse problems for operators with delay, the existing results do not give a comprehensive picture for all values of the delay parameter. Namely, for small delays, even such a basic question as the unique solvability of the inverse problem has been remaining open for many years. Since the problems with delay approximate the classical Sturm–Liouville problems as soon as the delay parameter tends to zero, many researchers expected the unique solvability as in the classical case. Here we give, however, a negative answer to this long-term basic question by constructing infinite families of iso-bispectral potentials. For this purpose, we develop a unified general approach that simultaneously covers various types of boundary conditions and allows one to significantly shorten the related proofs.