Wave discrete inverse problems: From boundary control to Gelfand–Levitan–Krein equations

This paper develops a unified discrete framework for dynamic, spectral, and scattering inverse problems associated with the one-dimensional wave equation with a potential on a semi-axis. Using a second-order explicit finite-difference scheme, we reduce the continuous initial-boundary value problem to a discrete dynamical system governed by a tridiagonal Jacobi matrix. We establish explicit Fourier-type integral transforms connecting the dynamic response operator, the Weyl function, and the spectral measure. Our main theoretical result is that, in the discrete one-dimensional setting, the Boundary control method and the Krein integral equation approach are strictly equivalent: their corresponding systems of linear algebraic equations coincide. We construct a discrete analogue of the Gelfand–Levitan equation and prove necessary and sufficient conditions for the existence and uniqueness of the solution to the discrete coefficient inverse problem, namely the unique solvability of an auxiliary linear system for each depth level. Additionally, we analyze the relationship between discrete scattering data (Jost solutions and norming constants) and spectral measures for Jacobi operators. The proposed framework provides a rigorous foundation for layer-stripping numerical inversion algorithms applicable to acoustic flaw detection, seismic tomography, and diagnostic wave imaging.