strengths $v^{\rm even}_2$ and $v^{\rm even, \sigma}_2$ are adjusted so as to reproduce the $\Lambda$ binding energies of the $0^+$ and $1^+$ states of $^4_\Lambda$H with the use of the $N+N+N+\Lambda$ four-body model. Furthermore, strengths $v^{\rm odd}_2$ and $v^{\rm odd, \sigma}_2$ are adjusted within the framework of $\alpha+ \Lambda+ n+p$ four-body model so as to reproduce the observed binding energies of the ground-state spin doublet, $1/2^+$ and $3/2^+$ of $^7_{\Lambda}$Li. Our resulting parameters in Eq.(3.5) are listed in Table III. We further found that the energy of the ground state of $^6_\Lambda$He ($^6_\Lambda$Li) measured from the $^5_\Lambda{\rm He}-N$ threshold can be well reproduced with our $\Lambda N$ interaction in the $\alpha+N+\Lambda$ three-body calculation; for $^6_\Lambda$He ($^6_\Lambda$Li), the calculated energy is $-0.17$ MeV (0.57 MeV), while the observed one is $-0.17$ MeV (0.59 MeV).

$\Lambda \, \Lambda$ interactions
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In the present model, since the $\Lambda \Lambda$ relative motion is solved rigorously including the short-range correlations, it is not adequate to use the $\Lambda \Lambda$ $G$-matrix interaction given in Ref. [@Yama94]. However, our $\Lambda \Lambda$ interaction to be used in the present calculation should be still considered as an effective interaction, since the couplings to $\Xi N$ and $\Sigma \Sigma$ channels 