Iso-bispectral potentials for Sturm-Liouville-type operators with small delay
Abstract
In recent years, there appeared a considerable interest in the inverse spectral theory for functional-differential operators with constant delay. In particular, it is well known that, for each fixed ∈\0,1\, the spectra of two operators generated by one and the expression -y''(x)+q(x)y(x-a) and the boundary conditions y()(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∈[π/2,π). For a<π/2, the main equation of the corresponding inverse problem is nonlinear, and it actually became the basic question of the inverse spectral theory for Sturm-Liouville operators with constant delay whether the uniqueness holds also in this nonlinear case. A few years ago, a positive answer was obtained for a∈[2π/5,π/2). Recently, the authors gave, however, a negative answer for a∈[π/3,2π/5) by constructing infinite families of iso-bispectral potentials. Meanwhile, the question remained open for the most difficult nonlinear case a∈(0,π/3), allowing the parameter a to approach the classical situation a=0, in which the uniqueness is well known. In the present paper, we address this gap and give a negative answer in this remarkable case by constructing appropriate iso-bispectral potentials.