---
abstract: |
    Energy levels of the double-$\Lambda$ hypernuclei $_\Lambda^{ }$$_\Lambda^7$He, $_\Lambda^{ }$$_\Lambda^7$Li, $_\Lambda^{ }$$_\Lambda^8$Li, $_\Lambda^{ }$$_\Lambda^9$Li, $_\Lambda^{ }$$_\Lambda^9$Be and $_\Lambda^{ }$$_\Lambda^{\!\!\!10}$Be are predicted on the basis of the $\alpha+x+\Lambda
    +\Lambda$ four-body model with $x=n, p, d, t, ^3$He and $\alpha$, respectively. Interactions between the constituent particles are determined so as to reproduce reasonably the observed low-energy properties of the $\alpha +x$ nuclei $(^5{\rm He}, ~^5{\rm Li}, ~^6{\rm Li}, ~^7{\rm Li},
    ~^7{\rm Be}, ~^8{\rm Be})$ and the existing data of $\Lambda$-binding energies of the $x+\Lambda$ and $\alpha+x+\Lambda$ systems $(^3_{\Lambda}{\rm H},
    ~^4_{\Lambda}{\rm H}, ~^5_{\Lambda}{\rm He},
    ~^6_{\Lambda}{\rm He}, ~^6_{\Lambda}{\rm Li},  ~^7_{\Lambda}{\rm Li},
    ~^8_{\Lambda}{\rm Li},  ~^8_{\Lambda}{\rm Be},
    ~^9_{\Lambda}{\rm Be})$. Here, an effective $\Lambda \Lambda$ interaction is constructed so as to to reproduce, within the $\alpha+\Lambda+\Lambda$ model, the $B_{\Lambda \Lambda}$ of $_\Lambda^{ }$$_\Lambda^6$He which was discovered recently in the NAGARA event of the emulsion experiment. With no adjustable parameters for the $\alpha+x+\Lambda+\Lambda$ systems, $B_{\Lambda \Lambda}$ of the ground and bound excited states of the double-$\Lambda$ hypernuclei with $A=7 - 10$ are accurately calculated with the Gaussian-basis coupled-rearrangement-channel method. The [*Demachi-Yanagi*]{} event, observed recently for $_\Lambda^{ }$$_\Lambda^{\!\!\!10}$Be, is interpreted as observation of its $2^+$ excited state on the basis of the present calculation. Structure change 