On recovering a Sturm–Liouville-type operator with the frozen argument rationally proportioned to the interval length

We consider the operator {\ell y\mathrel{\mathop{:}}=-y^{\prime\prime}(x)+q(x)y(a)} , {0<x<\pi} , {y(0)=y(\pi)=0} , where {q(x)\in L_{2}(0,\pi)} is a complex-valued function and {a/\pi\in[0,1]} is a rational number. The inverse problem of recovering the potential {q(x)} from the spectrum of {\ell} is studied. We describe the sets of iso-spectral potentials and prove the uniqueness theorem in the class of potentials possessing some symmetry-type property. Moreover, we obtain a constructive procedure for solving this inverse problem along with necessary and sufficient conditions of its solvability, which in turn give the characterization of the spectrum. In parallel, we establish that the informativity of the spectrum is severely unstable with respect to the parameter a.