2021/02/16 by Nebojs̆a Djurić, Djurić, Nebojša, Sergey Buterin +1
Mathematics · Computer Science · #Spectral Theory in Mathematical Physics #Numerical methods in inverse problems #Advanced Mathematical Modeling in Engineering
paper · pdf · doi:10.48550/arxiv.2102.08149
In recent years, there appeared a considerable interest in the inverse\nspectral theory for functional-differential operators with constant delay. In\nparticular, it is well known that, for each fixed \ν\∈ 0,1 , the spectra\nof two operators generated by one and the expression -y''(x)+q(x)y(x-a) and\nthe boundary conditions y(\ν)(0)=y(j)(\π)=0, j=0,1, uniquely\ndetermine the complex-valued square-integrable potential q(x) vanishing on\n(0,a) as soon as a\∈[\π/2,\π). For a<\π/2, the main equation of the\ncorresponding inverse problem is nonlinear, and it actually became the basic\nquestion of the inverse spectral theory for Sturm-Liouville operators with\nconstant delay whether the uniqueness holds also in this nonlinear case. A few\nyears ago, a positive answer was obtained for a\∈[2\π/5,\π/2). Recently,\nthe authors gave, however, a negative answer for a\∈[\π/3,2\π/5) by\nconstructing infinite families of iso-bispectral potentials. Meanwhile, the\nquestion remained open for the most difficult nonlinear case a\∈(0,\π/3),\nallowing the parameter a to approach the classical situation a=0, in which\nthe uniqueness is well known. In the present paper, we address this gap and\ngive a negative answer in this remarkable case by constructing appropriate\niso-bispectral potentials.\n