Inverse Spectral Problem for Integro-Differential Operators

In this paper, we study the inverse spectral problem on a finite interval for the integro-differential operator $\ell$ which is the perturbation of the Sturm–Liouville operator by the Volterra integral operator. The potential $q$ belongs to $L_2[0,\pi]$ and the kernel of the integral perturbation is integrable in its domain of definition. We obtain a local solution of the inverse reconstruction problem for the potential $q$, given the kernel of the integral perturbation, and prove the stability of this solution. For the spectral data we take the spectra of two operators given by the expression for $\ell$ and by two pairs of boundary conditions coinciding at one of the finite points.