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

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