On non-uniqueness of recovering Sturm-Liouville operators with delay and the Neumann boundary condition at zero

Abstract: As is known, for each fixed $\nu\in{0,1},$ the spectra of two operators generated by $-y''(x)+q(x)y(x-a)$ and the boundary conditions $y^{(\nu)}(0)=y^{(j)}(\pi)=0,$ $j=0,1,$ uniquely determine the complex-valued square-integrable potential $q(x)$ vanishing on $(0,a)$ as soon as $a\in[2\pi/5,\pi).$ Meanwhile, it actually became the main question of the inverse spectral theory for Sturm-Liouville operators with constant delay whether the uniqueness holds also for smaller values of $a.$ Recently, a negative answer was given by the authors [Appl. Math. Lett. 113 (2021) 106862] for $a\in[\pi/3,2\pi/5)$ in the case $\nu=0$ by constructing an infinite family of iso-bispectral potentials. Moreover, an essential and dramatic reason was established why this strategy, generally speaking, fails in the remarkable case when $\nu=1.$ Here we construct a counterexample giving a negative answer for $\nu=1,$ which is an important subcase of the Robin boundary condition at zero. We also refine the former counterexample for $\nu=0$ to $W_2^1$-potentials.