An algebraic form of the Marchenko inversion. Partial waves with orbital momentum $l\ge 0$

Abstract

We present a generalization of the algebraic method for solving the Marchenko equation (fixed-$l$ inversion) for any values of the orbital angular momentum $l$. We expand the Marchenko equation kernel in a separable form using a triangular wave set. The separable kernel allows a reduction of the equation to a system of linear equations. We obtained a linear expression of the kernel expansion coefficients in terms of the Fourier series coefficients of $q(1-S(q))$ function ($S(q)$ is the scattering matrix) depending on the momentum $q$. The linear expression is valid for any orbital angular momentum $l$. The kernel expansion coefficients are determined by the scattering data in the finite range $0\leq q\leq\pi/h$. In turn, the thus defined Marchenko kernel of the equation allows one to find the potential function of the radial Schr\"odinger equation with $h$-step accuracy.