$\alpha+x+ \Lambda$ hypernuclei have been reproduced accurately enough to give reliable predictions for the double-$\Lambda$ hypernuclei with no adjustable parameters of the interactions in the four-body calculations. It offers the most reliable ground for our cluster model that the threshold energies for every partition into sub-cluster systems are assured to be reproduced.

Energy spectra
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In Figs. $2 - 7$, the calculated level structure of $\alpha+x$ core nuclei, $\alpha+x+ \Lambda$ hypernuclei and $\alpha+x+ \Lambda + \Lambda$ hypernuclei are illustrated side by side. There are shown all the ground and bound excited states of double-$\Lambda$ hypernuclei predicted in the present model. In these figures, one sees clearly that injection of one and two $\Lambda$ particles leads to stronger binding of the whole system and prediction of more bound states. But, there is no bound ’$p$-orbit’ of $\Lambda$ particle in single- and double-$\Lambda$ hypernuclei with $A \leq 10$. In the bound states of double-$\Lambda$ hypernuclei, two $\Lambda$ particles are coupled to $S=0$ and therefore the spins and parities are the same as those of its nuclear core.

Table V summarizes the calculated ground-state energies for the double-$\Lambda$ hypernuclei including the $2^+$ excited state of $_\Lambda^{ }$$_\Lambda^{\!\!\!10}$Be. The results are expressed in terms 