2021/01/21 by Nebojs̆a Djurić, Djurić, Nebojša, Sergey Buterin +1
Computer Science · Mathematics · #34A55 34K29 #Advanced Mathematical Modeling in Engineering #Differential Equations and Boundary Problems #FOS: Mathematics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2101.08557
openalex publication_date 2021/01/21 · openalex created_date 2021/02/01 · openalex updated_date 2026/07/28
As is known, for each fixed ν∈\0,1\, the spectra of two operators generated by -y''(x)+q(x)y(x-a) and the boundary conditions y(ν)(0)=y(j)(π)=0, j=0,1, uniquely determine the complex-valued square-integrable potential q(x) vanishing on (0,a) as soon as a∈[2π/5,π). 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∈[π/3,2π/5) in the case ν=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 ν=1. Here we construct a counterexample giving a negative answer for ν=1, which is an important subcase of the Robin boundary condition at zero. We also refine the former counterexample for ν=0 to W21-potentials.