Lower bounds for quasi-bound state energy of three-Body kaonic clusters

The kaonic clusters $K^{-}K^{-}p$ and $ppK^{-}$ are described based on the configuration space Faddeev equations for $AAB$ system. The $AB$ interaction is given by phenomenological isospin-dependent potentials. We show that the relation $\left\vert E_{3}(V_{AA}=0)\right\vert~<~2\left\vert E_{2}\right\vert$ is satisfied when $E_{2}$ is the binding energy of the $AB$ subsystem and $E_{3}(V_{AA}=0)$ is the three-body binding energy and $V_{AA}$ is the interaction between the identical particles. For the $NN{\bar K}$ system, taking into account weak attraction of $NN$ interaction the relation leads to the evaluation $|E_3|\le 2|E_2|$.