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

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