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 ≤ q ≤ π/h. In turn, the thus defined Marchenko kernel of the equation allows one to find the potential function of the radial Schrödinger equation with h-step accuracy.