Inverse spectral problem for integro-differential operators

Abstract

In this paper, we study the inverse spectral problem on a finite interval for the integro-differential operator ℓ which is the perturbation of the Sturm-Liouville operator by the Volterra integral operator. The potential q belongs to L 2[0, π] 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 ℓ and by two pairs of boundary conditions coinciding at one of the finite points.