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 ${\ell }_j,$ $j=0,1,$ generated by one and the same functional-differential expression $-y^{\prime \prime }(x)+q(x)y(x-a)$ under the boundary conditions $y(0)=y^{(j)}(\pi )=0$ uniquely determines the complex-valued square-integrable potential $q(x)$ vanishing on $(0,a)$ as soon as $a\in [\pi /2,\pi ).$ For many years, it has been a challenging {\it open question} whether this uniqueness result would remain true also when $a\in (0,\pi /2).$ Recently, a positive answer was obtained for the case $a\in [2\pi /5,\pi /2).$ In this paper, we give, however, a {\it negative} answer to this question for $a\in [\pi /3,2\pi /5)$ 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.

6 pages