English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

As is known, for each fixed $\nu\in\{0,1\},$ the spectra of two operatorsgenerated 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-valuedsquare-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 theinverse spectral theory for Sturm-Liouville operators with constant delaywhether the uniqueness holds also for smaller values of $a.$ Recently, anegative 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 familyof iso-bispectral potentials. Moreover, an essential and dramatic reason wasestablished why this strategy, generally speaking, fails in the remarkable casewhen $\nu=1.$ 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.

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