ANTI Sturm-Liouville Spectral Problem With The Transfer Conditions On The Half Line Operator

In this paper, we main discuss the inverse spectral problem of the Sturm-Liouville op-erators on the half line with the discontinuous conditions and boundary conditions. The Jost solution is important in solving the inverse problems of Sturm-Liouville operators on the half-line, and the Weyl function is defined by the Jost solution which is used in the proof of the uniqueness theorem. First, we solve the equation and get the fundamental systems of the usual situation. Second, together with the discontinuous condition, we can obtain the basic solutions of the problem. Third, we solve the Jost solution by the basic solutions. Thus, we can define the Weyl solution and the Weyl function. When we obtain the Weyl function, the uniqueness theorem can be proved. We use the Weyl function defined before to deduce the integral equation which the basic solutions of problem L and L[9] satisfied, while it is de-duced through the eigenvalues and the normalized coefficients of the eigenvectors in Yurko’s result, thus the method we used is easier in calculation. At the end, an algorithm is proposed to calculate the potential and the parameters of the boundary conditions and discontinuous conditions. Under the conditions that the problem L and the Weyl function of problem L are known, we obtain the basic solution of problem L by solving the integral equation with numerical methods, thus we can get the potential.