arXiv 2009.026362020-09-06math.SP

关于恢复Sturm-Liouville型时滞算子的一个公开问题

On an open question in recovering Sturm-Liouville-type operators with delay

Nebojša Djurić, Sergey Buterin

备注：: 6 pages
DOI：: 10.1016/j.cnsns.2021.105900

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，在边界条件$y(0)=y^{(j)}(π)=0$下由同一个泛函微分表达式$-y''(x)+q(x)y(x-a)$生成的两个算子$\ell_j,$ $j=0,1,$的谱的规范唯一地确定了复值平方可积势$q(x)$在$a\in[π/2,π).$之后立即在$(0,a)$上消失多年来，最近，对于$a\in(0,π/2).$的情况获得了肯定的答案。然而，在本文中，我们给出了，通过构造等双谱势的无限族，对$a\in[π/3,2π/5)$给出了这个问题的{ \it否定}答案。对其他类型的边界条件构造类似的反例的可能性进行了一些讨论，并提出了新的未决问题。

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 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.

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摘要：

英文摘要：

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