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 $-y^{\prime \prime }(x)+q(x)y(x-a)$ 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.

6 pages

References in corpus (1)

• An inverse spectral problem for second-order functional-differential pencils with two delays