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We consider the operatorℓ  y  :  =  -   y  ′′    (  x  )   +  q   (  x  )   y   (  a  ){\ell y\mathrel{\mathop{:}}=-y^{\prime\prime}(x)+q(x)y(a)}   ,0  &lt;  x  &lt;  π{0&lt;x&lt;\pi}   ,y  ⁢   (  0  )    =   y  ⁢   (  π  )    =  0{y(0)=y(\pi)=0}   , whereq  ⁢   (  x  )    ∈    L  2   ⁢   (  0  ,  π  ){q(x)\in L_{2}(0,\pi)}    is a complex-valued function anda  /  π   ∈   [  0  ,  1  ]{a/\pi\in[0,1]}    is a rational number. The inverse problem of recovering the potentialq  ⁢   (  x  ){q(x)}    from the spectrum ofℓ{\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 .

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