Green's function method for the single-particle resonances in a deformed Dirac equation

Single-particle resonances are crucial for exotic nuclei near and beyond the drip lines. Since the majority of nuclei are deformed, the interplay between deformation and orbital structure near threshold becomes very important and can lead to an improved description of exotic nuclei. In this work, the Green's function (GF) method is applied to solve the coupled-channel Dirac equation with quadrupole-deformed Woods-Saxon potentials for the first time. The detailed formalism for the partial-wave expansion of the Green's function is presented. A new approach getting exact values for energies and widths of resonant states by the GF method is proposed. Numerical checks are carried out by comparing with our previous implementation of the spherical GF method and the results from the deformed complex momentum representation~(CMR), the analytical continuation of the coupling constant (ACCC), and the scattering phase shift (SPS) methods, and it is proved that the GF method is very effective and reliably for describing resonance states, no matter they are narrow or broad, spherical or deformed. Finally, Nilsson levels for bound and resonant orbitals in the halo candidate nucleus $^{37}$Mg are calculated from the deformed GF method over a wide range of deformations and some decisive hints of $p$-wave halo formation are shown in this nucleus, namely, the crossing between the configurations $1/2[321]$ and $5/2[312]$ at deformation parameter $\beta>0.5$ may enhance the probability to occupy the $1/2[321]$ orbital that is originated from the $2p_{3/2}$ shell.