Numerical algorithms for recovering of one dimensional Schrödinger operators from the Weyl–Titchmarsh <i>m</i>-function

The purpose of this paper is to investigate a numerical algorithm for recovering the potential function of one-dimensional Schrödinger operator Ky(x)≔−y″(x)+q(x)y(x)=λy(x) with boundary condition y(0)cos α − y′(0)sin α = 0 and y(π)cos β + y′(π)sin β = 0 on (0, π) from the Weyl–Titchmarsh m-function given on a countable set of points. The primary concept involves employing the Fourier–Legendre series expansion of the integral kernel of the transmutation operator. It is demonstrated that the problem can be transformed into an infinite system of linear equations, which possesses a unique solution if the original problem does. The solution of this linear system allows one to reconstruct the characteristic determinant and hence to obtain the eigenvalues as its zeros and to compute the corresponding norming constants, transforming the origin problem into an inverse problem with a given spectral density function. The reconstruction approach delineated in this paper offers a robust numerical framework for addressing a variety of inverse spectral problems, while simultaneously minimizing the reconstruction error associated with the recovered potential to the greatest extent feasible.