小时滞Sturm-Liouville型算子的等双谱势
Iso-bispectral potentials for Sturm-Liouville-type operators with small delay
摘要：
近年来，常时滞泛函微分算子的逆谱理论引起了人们极大的兴趣。特别地，众所周知，对于每个固定的$ν\in\{0,1\},$，由一个算子生成的两个算子的谱和表达式$-y''(x)+q(x)y(x-a)$以及边界条件$y^{(ν)}(0)=y^{(j)}(π)=0,$ $j=0,1,$唯一地确定了复值平方可积势$q(x)$在$(0,a)$上消失，只要$a\in[π/2,π).$对于$a<π/2,$，对应的逆问题的主方程是非线性的，而在这种非线性情形下，是否唯一性也成立实际上成为常时滞Sturm-Liouville算子逆谱理论的基本问题.几年前，人们对$a\in[2π/5,π/2).$给出了肯定的回答。最近，作者通过构造等双谱势的无限族，对$a\in[π/3,2π/5)$给出了否定的回答。与此同时，对于最困难的非线性情况$a\in(0,π/3),$，这个问题仍然悬而未决，允许参数$a$接近经典情况$a=0,$，其中唯一性是众所周知的。在本文中，我们解决这个差距，并给出了一个否定的答案，在这个显着的情况下，通过构建适当的等双谱电位。
英文摘要：
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 $ν\in\{0,1\},$ the spectra of two operators generated by one and the expression $-y''(x)+q(x)y(x-a)$ and the boundary conditions $y^{(ν)}(0)=y^{(j)}(π)=0,$ $j=0,1,$ uniquely determine the complex-valued square-integrable potential $q(x)$ vanishing on $(0,a)$ as soon as $a\in[π/2,π).$ 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 $a\in[2π/5,π/2).$ Recently, the authors gave, however, a negative answer for $a\in[π/3,2π/5)$ by constructing infinite families of iso-bispectral potentials. Meanwhile, the question remained open for the most difficult nonlinear case $a\in(0,π/3),$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
