reproduce reasonably well the low-lying states and low-energy scattering phase shifts of the $\alpha \, x$ systems. The potentials are described in the following parity-dependent form with the central and spin-orbit terms: $$\begin{aligned}
&V&_{\!\!\!\!\!\alpha x}(r) =  \sum_{i=1}^{i_{\rm max}}
       V_i \: e^{-\beta_i r^2}
        + \sum_{i=1}^{i'_{\rm max}}
        (-)^lV_i^{\rm p} \: e^{-\beta_i^{\rm p} r^2} \nonumber \\  
         &  + & \big[\sum_{i=1}^{i''_{\rm max}}
           V_i^{ls} \: e^{-\gamma_i r^2}
           +  \sum_{i=1}^{i'''_{\rm max}}(-)^lV_i^{ls,{\rm p}}
        \: e^{-\gamma_i^{\rm p} r^2} \; \big] \;
        \mbox{\boldmath $l$}\cdot
               \mbox{\boldmath $s$}_x \;, \nonumber \\\end{aligned}$$ where is the relative angular momentum between $\alpha$ and $x$, and $_x$ is the spin of $x$. In the $\alpha \alpha$ system the spin-orbit term is missing and the odd wave is forbidden by the Pauli principle. The additional Coulomb potentials are constructed by folding the $pp$ Coulomb force into the proton densities of the $\alpha$ and $x$ clusters. The parameters in Eq.(3.2) are listed in Table I (we slightly modified the strength of the central force in $V_{\alpha d}$ and that of the spin-orbit force in $V_{\alpha t}$ to obtain better agreement with the energy levels of $^6$Li and $^7$Li, respectively).

$\Lambda \,x$ interactions
--------------------------

We derive the interaction between the $\Lambda$ particle and the $x$ cluster by folding 