Inverse spectral problems for Hill-type operators with frozen argument

The paper deals with nonlocal differential operators possessing a term frozen (fixed) argument appearing, in particular, modelling various physical systems feedback. presence of feedback means that the external affect on system depends its current state. If this state is taken into account only at some fixed point, then mathematically corresponds to an operator argument. In present paper, we consider $Ly\equiv-y''(x)+q(x)y(a),$ $y^{(\nu)}(0)=\gamma y^{(\nu)}(1),$ $\nu=0,1,$ where $\gamma\in{\mathbb C}\setminus\{0\}.$ $L$ analog classical Hill describing processes cyclic or periodic media. We study two inverse problems recovering complex-valued square-integrable potential $q(x)$ from spectral information about $L.$ first problem involves single spectrum as input data. obtain complete characterization and prove specification determines uniquely if $\gamma\ne\pm1.$ For rest (periodic antiperiodic) cases, describe classes iso-spectral potentials provide restrictions under which uniqueness holds. second spectra related $\gamma=\pm1.$ necessary sufficient conditions for solvability establish holds $a=0,1.$ $a\in(0,1),$ iso-bispectral give resumes. Algorithms solving both are provided. appendix, Riesz-basisness auxiliary two-sided sequence sines.