A Method for Solving the Inverse Sturm–Liouville Problem

The inverse Sturm–Liouville problem is to find the coefficient (potential) in the stationary Schrödinger equation on a segment for the collection of eigenvalues. This paper considers a numerical solution of the inverse problem for a finite set of first eigenvalues of two Sturm–Liouville problems. The remaining eigenvalues are specified by classical asymptotics. The method for solving the inverse spectral problem is based on the one-to-one correspondence between the inverse spectral problem and the nonstationary inverse problem for the telegraph equation with a variable coefficient (potential). The reduction to the nonstationary problem is performed analytically using the inverse Laplace transform given by Mellin formula. An explicit formula is obtained for the reaction function in the inverse scattering problem. The inverse scattering problem for the telegraph equation is to find the unknown coefficient using the reaction function. This problem is solved numerically using the method of inversion of difference schemes. This paper solves a series of inverse Sturm–Liouville problems. In conclusion, it is noted that the number of given eigenvalues corresponds to the number of harmonics in the expansion of the sought potential.