Inverse spectral problems for Hill-type operators with frozen argument

Abstract

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.