论文标题
山型操作员的逆频谱问题与冻结的争论
Inverse spectral problems for Hill-type operators with frozen argument
论文作者
论文摘要
本文介绍了具有一个术语的非局部差分运算符，尤其是在用反馈对各种物理系统建模时出现的冷冻（固定）参数。反馈的存在意味着对系统的外部影响取决于其当前状态。如果仅在某个固定的物理点考虑此状态，则在数学上，这对应于具有冷冻参数的操作员。在本文中，我们考虑运营商$ ly \ equiv-y'（x）+q（x）y（a），$ $ y^{（ν）}（0）=γy^{（ν）}（（ν）}（1），$ $ $ $ $ $ um $ $ n = 0,1，$ in {古典山丘操作员的非局部类似物描述了循环或周期性介质中的各种过程。我们研究了从某些有关$ L的一些频谱信息中恢复复杂值的正方形累积潜力$ q（x）$的两个反问题。$第一个问题仅涉及单个频谱作为输入数据。我们获得了频谱的完整表征，并证明其规格确定$ q（x）$唯一时，并且仅当$γ\ ne \ pm1。$ $ q. $ $（周期性和反碘）案例时，我们描述了iSo-Spectral潜力的类别，并提供限制性限制。第二个反问题涉及从与$γ= \ pm1相关的两个光谱中恢复$ q（x）$。$。$我们获得其可溶解度的必要条件，并确定仅当$ a = 0,1。$ a =0,1。$ a \ in（0,1）中的独特性才能保持不变，我们描述了$ bisectry的潜力和限制性的限制性。提供解决两个反问题的算法。在附录中，我们证明了辅助双侧罪恶序列的riesz-basisness。
The paper deals with nonlocal differential operators possessing a term with frozen (fixed) argument appearing, in particular, in modelling various physical systems with feedback. The presence of a feedback means that the external affect on the system depends on its current state. If this state is taken into account only at some fixed physical point, then mathematically this corresponds to an operator with frozen argument. In the present paper, we consider the operator $Ly\equiv-y''(x)+q(x)y(a),$ $y^{(ν)}(0)=γy^{(ν)}(1),$ $ν=0,1,$ where $γ\in{\mathbb C}\setminus\{0\}.$ The operator $L$ is a nonlocal analog of the classical Hill operator describing various processes in cyclic or periodic media. We study two inverse problems of recovering the complex-valued square-integrable potential $q(x)$ from some spectral information about $L.$ The first problem involves only single spectrum as the input data. We obtain complete characterization of the spectrum and prove that its specification determines $q(x)$ uniquely if and only if $γ\ne\pm1.$ For the rest (periodic and antiperiodic) cases, we describe classes of iso-spectral potentials and provide restrictions under which the uniqueness holds. The second inverse problem deals with recovering $q(x)$ from the two spectra related to $γ=\pm1.$ We obtain necessary and sufficient conditions for its solvability and establish that uniqueness holds if and only if $a=0,1.$ For $a\in(0,1),$ we describe classes of iso-bispectral potentials and give restrictions under which the uniqueness resumes. Algorithms for solving both inverse problems are provided. In the appendix, we prove Riesz-basisness of an auxiliary two-sided sequence of sines.
