On justification of the Gelfand–Levitan–Krein method for a two-dimensional inverse problem

For a hyperbolic equation of the second order, we consider the inverse problem of recovering the coefficient $q(x,y)$ in this equation. We discuss the scheme of solution of the problem which was proposed by Kabanikhin about 30 years ago. This scheme generalizes the Gelfand–Levitan–Krein method for the solution of the inverse spectral problem to the multidimensional case and reduces the solution of the inverse problem to some infinite system of linear integral equations. No mathematical justification for this scheme has been obtained yet. But numerical experiments based on the $N$-approximation produced good results. In this article, we justify some elements of the scheme related to the construction of the infinite system of integral equations in the case when the coefficient $q(x,y)$ is analytic in $x$. In particular, we prove the convergence of the series in these equations and find the conditions for the $N$-approximation of the system. We also establish that the infinite system of integral equations is not Fredholm. The question of the solvability of the systems remains open.