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Algebraic model for single-particle energies of $Λ$ hypernuclei

A model is proposed for the spectrum of $Λ$ hypernuclei based on the $u(3)\times u(2)$ Lie algebra, in which the internal degrees of freedom of the spin-1/2 $Λ$ 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 $Λ$ particle is obtained from the natural dynamical symmetry. The formula is applied to the experimental data on the reaction spectroscopy for the $^{89}_Λ$Y and $^{51}_Λ$V hypernuclei, providing a clear theoretical interpretation of the observed structures.