A systematic treatment of the procedure used to construct and solve the Faddeev-Yakubovsky integral equations for three and four different particles with nonzero spin is introduced. The four-body T-matrices are written in terms of solutions of the three-body Faddeev integral equations in total angular momenta representation. General spin-angular structure of the four-body Faddeev-Yakubovsky equations is written down and reduced to a spin-angular structure of the three-body system by elimination of one body. Partial-wave expansion performed for the system of four-body integral equations gives an infinite system of one-dimensional coupled equations with integral kernels containing four types of partial components of the three-body Faddeev integral equation solutions. The given method is tested on and systems with two-body potential written down in only separable form. Both separable NN potential and ΛN potential rewritten in separable form for s-wave are used to determine the Λ-hyperon binding energies in and systems. Calculation also includes the hyperon conversion in process. Λ-hyperon separation energies were calculated from binding energies and turned out to be 0.147 MeV in system and 2.04 MeV in system, respectively, that is in close coincidence with experimental ones.
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