We give a derivation of the main equation for Sturm-Liouville operator with Coulomb potential and prove its unique solvability. Using the solution of the main equation, we get an algorithm for the solution of the inverse problem.

In this study, we consider the discontinuous Dirac equations system with eigenparameter dependent boundary and finite number of transmission conditions. First, the space that corresponds to problem is introduced, the norm on this space is defined and the operator model that corresponds to the given problem is constructed on this space. Then the integral equations and asymptotics of eigenfunctions of the problem are obtained. The characteristic function is defined and the asymptotic formula of the characteristic function is given by using obtained asymptotics of eigenfunctions. After the Weyl solution and the Weyl function of the problem are formed. Finally, some uniqueness theorems are proved by using Weyl function and some spectral data.

In this paper, half-inverse problem is considered for Dirac equations with boundary and finite number of transmission conditions depending polynomially on the spectral parameter, if the potential is given over the half of the considered interval and if one spectrum is known then, potential function Ω(x) on the whole interval and the other coefficients of the considered problem can be determined uniquely.

In the present study, theorems related to the uniqueness of the solution of inverse problems for Dirac equations system are proved by applying spectral mapping method. With the help of this method, the inverse problem is reduced to the so-called main equation, which corresponds to the problem of existence and uniqueness of the solution of the system of linear equations in the Banach space.