On an open question in recovering Sturm–Liouville-type operators with delay

Abstract

In recent years, there appeared a considerable interest in the inverse spectral theory for functional-differential operators with constant delay. In particular, it is well known that specification of the spectra of two operators ℓj,j=0,1, generated by one and the same functional-differential expression −y′′(x)+q(x)y(x−a) under the boundary conditions y(0)=y(j)(π)=0 uniquely determines the complex-valued square-integrable potential q(x) vanishing on (0,a) as soon as a∈[π∕2,π). For many years, it has been a challenging open question whether this uniqueness result would remain true also when a∈(0,π∕2). Recently, a positive answer was obtained for the case a∈[2π∕5,π∕2). In this paper, we give, however, a negative answer to this question for a∈[π∕3,2π∕5) by constructing an infinite family of iso-bispectral potentials. Some discussion on a possibility of constructing a similar counterexample for other types of boundary conditions is provided, and new open questions are outlined.