ground state of $^8_\Lambda$Li within the $\alpha+t+\Lambda$ model; the calculated energy is 6.80 MeV while the observed one is 6.80 $\pm$ 0.03 MeV.

$\Lambda N$ interaction in $_\Lambda^{ }$$_\Lambda^7$He ($_\Lambda^{ }$$_\Lambda^7$Li) 
---------------------------------------------------------------------------------------

In the study of $_\Lambda^{ }$$_\Lambda^7$He ($_\Lambda^{ }$$_\Lambda^7$Li) with the $\alpha+N+\Lambda+ \Lambda$ model and of the subsystem $^6_\Lambda{\rm He}\:(^6_\Lambda{\rm Li})$ with the $\alpha+N+\Lambda$ model, it is inadequate to use the G-matrix type $\Lambda N$ interaction because $\Lambda N$ correlations are fully taken into account in our model space. Here, we employ a simple free-space $\Lambda N$ interaction with a three-range Gaussian form, which simulates the Nijmegen model F (NF) $\Lambda N$ interaction. Here, the $\Sigma N$ channel coupling contribution is renormalized into the $\Lambda N$ single channel with the closure approximation. The even- and odd-state parts of our $\Lambda N$ interaction are represented as follows: $$\begin{aligned}
             &V\!\!\!\!&_{\Lambda N}(r)  =  \sum_{i=1}^{3}
             \Big[(v^{\rm even}_i + 
            v^{\rm even, \sigma}_i
              \mbox{\boldmath $\sigma$}_{\Lambda}\cdot
               \mbox{\boldmath $\sigma$}_N )
               \frac{1+P_r}{2}
                 \nonumber \\
             & &+\, (v^{\rm odd}_i
                + v^{\rm odd, \sigma}_i
                \mbox{\boldmath $\sigma$}_{\Lambda}\cdot
               \mbox{\boldmath $\sigma$}_N )
               \frac{1-P_r}{2} \Big] 
             \: e^{-\mu_i r^2} \; . %\nonumber \\
\label{inta}\end{aligned}$$ First, the parameters are dertemined so as to simulate the $\Lambda N$ scattering phase shifts calculated with NF. Next, the second-range 