Energy-independent complex single $P$-waves $NN$ potential from Marchenko equation

We extend our previous results of solving the inverse problem of quantum scattering theory (Marchenko theory, fixed-$l$ inversion). In particular, we apply an isosceles triangular-pulse function set for the Marchenko equation input kernel expansion in a separable form. The separable form allows a reduction of the Marchenko equation to a system of linear equations for the output kernel expansion coefficients. We show that in the general case of a single partial wave, a linear expression of the input kernel is obtained in terms of the Fourier series coefficients of $q^{1-m}(1-S(q))$ functions in the finite range of the momentum $0\leq q\leq\pi/h$ [$S(q)$ is the scattering matrix, $l$ is the angular orbital momentum, $m=0,1,\dots,2l$]. Thus, we show that the partial $S$--matrix on the finite interval determines a potential function with $h$-step accuracy. The calculated partial potentials describe a partial $S$--matrix with the required accuracy. The partial $S$--matrix is unitary below the threshold of inelasticity and non--unitary (absorptive) above the threshold. We developed a procedure and applied it to partial-wave analysis (PWA) data of $NN$ elastic scattering up to 3 GeV. We show that energy-independent complex partial potentials describe these data for single $P$-waves.