Deeply quasi-bound state in single- and double-(K)over-bar nuclear clusters

New calculations of the quasi-bound state positions in K-K-pp kaonic nuclear cluster are performed using non-relativistic four-body Faddeev-type equations in AGS form. The corresponding separable approximation for the integral kernels in the three-and four-body kaonic clusters is obtained by using the Hilbert-Schmidt expansion procedure. Different phenomenological models of (K) over bar N-pi Sigma potentials with oneand two-pole structure of Lambda(1405) resonance and separable potential models for (K) over bar -(K) over bar and nucleon-nucleon interactions, are used. The dependence of the resulting four-body binding energy on models of (K) over bar N-pi Sigma interaction is investigated. We obtained the binding energy of the K-K- pp quasi-bound state similar to 80-94 MeV with the phenomenological (K) over bar N potentials. The width is about similar to 5-8MeV for the two-pole models of the interaction, while the one-pole potentials give similar to 24-31 MeV width.