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Iso-bispectral potentials for Sturm–Liouville-type operators with small delay
Nonlinear Analysis: Real World Applications ( IF 2 ) Pub Date : 2021-07-22 , DOI: 10.1016/j.nonrwa.2021.103390
Nebojša Djurić 1 , Sergey Buterin 2
Affiliation  

The paper addresses nonlinear inverse Sturm–Liouville-type problems with constant delay. Since many processes in the real world possess nonlocal nature, operators with delay as well as other classes of nonlocal operators are continuously finding numerous applications in the natural sciences and engineering. However, in spite of a large number of works devoted to inverse problems for operators with delay, the existing results do not give a comprehensive picture for all values of the delay parameter. Namely, for small delays, even such a basic question as the unique solvability of the inverse problem has been remaining open for many years. Since the problems with delay approximate the classical Sturm–Liouville problems as soon as the delay parameter tends to zero, many researchers expected the unique solvability as in the classical case. Here we give, however, a negative answer to this long-term basic question by constructing infinite families of iso-bispectral potentials. For this purpose, we develop a unified general approach that simultaneously covers various types of boundary conditions and allows one to significantly shorten the related proofs.



中文翻译:

具有小延迟的 Sturm-Liouville 型算子的等双谱势

该论文解决了具有恒定延迟的非线性逆 Sturm-Liouville 型问题。由于现实世界中的许多过程都具有非本地性质,因此延迟算子以及其他类别的非本地算子在自然科学和工程中不断得到大量应用。然而,尽管有大量工作致力于解决具有延迟的算子的反问题,但现有的结果并没有给出延迟参数的所有值的全面图景。也就是说,对于小的延迟,即使是这样一个基本问题因为逆问题的独特可解性多年来一直保持开放。由于延迟问题一旦延迟参数趋于零就接近经典的 Sturm-Liouville 问题,因此许多研究人员期望在经典情况下具有独特的可解性。然而,在这里我们通过构建无限的等双谱势族来对这个长期的基本问题给出否定的答案。为此,我们开发了一种统一的通用方法,同时涵盖各种类型的边界条件,并允许显着缩短相关证明。

更新日期:2021-07-23
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