wave function is given by $$\Psi_{JM}(\alpha+x) = \sum_{n,l}  C_{nl}  {\cal S}_\alpha 
   \Phi(\alpha) \big[\Phi_s(x)  \phi_{nl}({\bf r}) \big]_{JM} \; .
%(2.6)$$

Interactions
============

In the study of double-$\Lambda$ hypernuclei based on the $\alpha + x + \Lambda + \Lambda$ four-body model, it is absolutely necessary and important to examine, before the four-body calculation, that the model with the interactions adopted is able to reproduce reasonably well the following observed quantities: (i) energies of the low-lying states and scattering phase shifts of the $\alpha+x$ nuclear systems, (ii) $B_\Lambda$ of hypernuclei composed of $x+\Lambda$, $x$ being $d, t, ^3$He, $\alpha$, (iii) $B_\Lambda$ of hypernuclei composed of $\alpha+ x +\Lambda$, $x$ being $ n, p, d, t,^3$He, $\alpha$ and (iv) $B_{\Lambda \Lambda}$ of $_\Lambda^{ }$$_\Lambda^6$He $=\alpha+\Lambda+\Lambda$. We emphasize that this severe examination were successfully done in the present model as mentioned below. This encourages us to perform the four-body calculations, with no adjustable parameters at this stage, with high reliability of the results expected.

Pauli principle between $\alpha$ and $x$ clusters
-------------------------------------------------

The Pauli principle between nucleons belonging to $\alpha$ and $x$ clusters is taken into account by the orthogonality condition model (OCM)[@ocm]. The OCM projection operator $V_{\rm Pauli}$ is represented by $$V_{\rm Pauli}=\lim_{\lambda\to\infty}
        \lambda 