  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 $Ly\equiv-y''(x)+q(x)y(a),$ $y^{(\nu)}(0)=\gamma y^{(\nu)}(1),$
$\nu=0,1,$ 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 $\gamma\ne\pm1.$
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 $\gamma=\pm1.$ 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.
