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 \(Ly\equiv -y^{\prime \prime }(x)+q(x)y(a),\) \(y^{(\nu )}(0)=\gamma y^{(\nu )}(1),\) \(\nu =0,1,\) where \(\gamma \in {\mathbb C}{\setminus }\{0\}.\) 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 \(\gamma \ne \pm 1.\) 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 \(\gamma =\pm 1.\) 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.

本文研究的是非局部微分算子，该算子具有带有冻结（固定）参数的项，尤其是在对带有反馈的各种物理系统进行建模时。反馈的存在意味着对系统的外部影响取决于其当前状态。如果仅在某个固定物理点考虑此状态，则在数学上，这对应于具有冻结参数的运算符。在本文中，我们考虑算子\（Ly \ equiv -y ^ {\ prime \ prime}（x）+ q（x）y（a），\） \（y ^ {（\ nu）}（0 ）= \ gamma y ^ {（\ nu}}（1），\） \（\ nu = 0,1，\）其中\（\ gamma \ in {\ mathbb C} {\ setminus} \ {0 \} 。\）运算符L是经典Hill算子的非本地类似物，描述了循环或周期性介质中的各种过程。我们研究了从有关L的某些光谱信息中恢复复数值平方可积电势q（x）的两个反问题。第一个问题仅涉及单个光谱作为输入数据。我们获得了频谱的完整表征，并证明了当且仅当\（\ gamma \ ne \ pm 1. \）时，它的规范才能唯一确定q（x）。对于其余（周期性和反周期性）情况，我们描述了等频谱势，并提供了保持唯一性的限制。第二个反问题涉及恢复q（X从两个光谱有关）\（\伽马= \下午1 \）我们获得用于其有解的充分必要条件，并建立该唯一性成立当且仅当\（α= 0,1。\）对于\ （a \ in（0,1），\）我们描述了等双谱势，并给出了恢复唯一性的限制。提供了解决两个反问题的算法。在附录中，我们证明了一个辅助的双向正弦序列的Riesz基性。