Spectral analysis of the indefinite non-self-adjoint Sturm–Liouville operator

The study investigates the inverse scattering problem for the Schrodinger operator with complex potentials, considering indefinite discontinuous coefficients on the axis. Using the integral representation of the Jost solutions on the real and imaginary axes, solved the direct scattering problem. An additional study of the operator’s spectrum was conducted, scattering data was introduced, and the eigenfunction expansion was obtained. Integral equations derived play a crucial role in solving the inverse problem and finally prove the uniqueness theorem for the solution.