Operator form of the three-nucleon scattering amplitude

To extend the applications of the so-called “three-dimensional” formalism to the description of three-nucleon scattering within the Faddeev formalism, we develop a general form of the three-nucleon scattering amplitude. This form significantly decreases the numerical complexity of the “three-dimensional” calculations by reducing the scattering amplitude to a linear combination of momentum-dependent spin operators and scalar functions of momenta. The number and structure of the spin operators is fixed and the scalar functions can be represented numerically using standard methods such as multidimensional arrays. In this paper, we show that all orders of the iterated Faddeev equation can be written in this general form. We argue that calculations utilizing the three-nucleon force will also conform to the same general form. Additionally, we show how the general form of the scattering amplitude can be used to transform the Faddeev equation to make it suitable for numerical calculations using iterative methods.

Figure 1

Distribution of nonzero elements in the $A_{\mathrm{rw}}$ matrix. The nonzero elements of this sparse matrix are marked using black squares. Each row and column of squares corresponds to the appropriate row and column of $A_{\mathrm{rw}}.$