Green's function relativistic mean field theory for $\Lambda$ hypernuclei

The relativistic mean-field theory with Green's function method is extended to study $\Lambda$ hypernuclei. Taking hypernucleus $^{61}_{\Lambda}$Ca as an example, the single-particle resonant states for $\Lambda$ hyperons are investigated by analyzing density of states and the corresponding energies and widths are given. Different behaviors are observed for the resonant states, i.e., the distributions of the very narrow $1f_{5/2}$ and $1f_{7/2}$ states are very similar as bound states while that of the wide $1g_{7/2}$ and $1g_{9/2}$ states are like scattering states. Besides, the impurity effect of $\Lambda$ hyperons on the single-neutron resonant states are investigated. For most of the resonant states, both the energies and widths decrease with adding more $\Lambda$ hyperons due to the attractive $\Lambda N$ interaction. Finally, the energy level structure of $\Lambda$ hyperons in the Ca hypernucleus isotopes with mass number $A=53-73$ are studied, obvious shell structure and small spin-orbit splitting are found for the single-$\Lambda$ spectrum.