Algebraic model for single-particle energies of $\Lambda$ hypernuclei

A model is proposed for the spectrum of $\Lambda$ hypernuclei based on the $u(3)\times u(2)$ Lie algebra, in which the internal degrees of freedom of the spin-1/2 $\Lambda$ particle are treated in the Fermionic $u(2)$ scheme, while the motion of the hyperon inside a nucleus is described in the Bosonic $u(3)$ harmonic oscillator scheme. Within this model, a simple formula for single-particle energies of the $\Lambda$ particle is obtained from the natural dynamical symmetry. The formula is applied to the experimental data on the reaction spectroscopy for the $^{89}_\Lambda$Y and $^{51}_\Lambda$V hypernuclei, providing a clear theoretical interpretation of the observed structures.