Inverse Spectral Problem for Integro-Differential Sturm–Liouville Operators with Discontinuity Conditions

Abstract

We consider the Sturm–Liouville operator perturbed by a convolution integral operator on a finite interval with Dirichlet boundary conditions and discontinuity conditions in the middle of the interval. We study the inverse problem of recovering the convolution term from the spectrum. The problem is reduced to solving the so-called main nonlinear integral equation with a singularity. To derive and study this equation, we do detailed analysis of kernels of transformation operators for the integro-differential expression considered. We prove global solvability of the main equation, which enables us to prove uniqueness of solution of the inverse problem and to obtain necessary and sufficient conditions for its solvability in terms of asymptotics of the spectrum. The proof is constructive and gives an algorithm for solving the inverse problem.