论文标题
关于延迟恢复Sturm-Liouville型操作员的公开问题
On an open question in recovering Sturm-Liouville-type operators with delay
论文作者
论文摘要
近年来，对于持续延迟的功能分化算子的反光谱理论似乎引起了极大的兴趣。 In particular, it is well known that specification of the spectra of two operators $\ell_j,$ $j=0,1,$ generated by one and the same functional-differential expression $-y''(x)+q(x)y(x-a)$ under the boundary conditions $y(0)=y^{(j)}(π)=0$ uniquely determines the complex-valued square-integrable potential $q(x)$在[π/2，π）中以$ a \ $ a \ $ a \ in [π/2，π）的消失而消失。$多年来，当$ a \ in（0，π/2）$ a \ in（最近，π/2）$ a a \ in [2）$ a \ in [2π/5，we 2π/5）时，这是一个充满挑战的{\ it oble ate Ission}。负面}通过构建无限的ISO-biseptral潜力家族来回答$ a \ in [π/3,2π/5）$的问题。提供了一些关于为其他类型的边界条件构建类似反例的可能性的讨论，并概述了新的开放问题。
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 specification of the spectra of two operators $\ell_j,$ $j=0,1,$ generated by one and the same functional-differential expression $-y''(x)+q(x)y(x-a)$ under the boundary conditions $y(0)=y^{(j)}(π)=0$ uniquely determines the complex-valued square-integrable potential $q(x)$ vanishing on $(0,a)$ as soon as $a\in[π/2,π).$ For many years, it has been a challenging {\it open question} whether this uniqueness result would remain true also when $a\in(0,π/2).$ Recently, a positive answer was obtained for the case $a\in[2π/5,π/2).$ In this paper, we give, however, a {\it negative} answer to this question for $a\in[π/3,2π/5)$ by constructing an infinite family of iso-bispectral potentials. Some discussion on a possibility of constructing a similar counterexample for other types of boundary conditions is provided, and new open questions are outlined.
