The Green's-function (GF) method identifies single-particle resonant states in the continuum directly from the density of states and the poles of the single-particle Green's function, treating bound, resonant, and non-resonant continuum states on the same footing. It was extended to quadrupole-deformed Dirac equations (coupled-channel form, fixed Woods-Saxon potentials) by Sun, Qian, Chen, Ring, and Li in 2020, who left open the transfer of this machinery into fully self-consistent calculations: implementing the deformed GF method within deformed relativistic Hartree-Bogoliubov (RHB) theory so that deformation, pairing, and the continuum are treated self-consistently, and then applying it to the deformed quasiparticle random-phase approximation (QRPA) for excitation modes coupled to the continuum. Spherical self-consistent GF-continuum calculations with pairing exist in both Skyrme and relativistic form, and self-consistent deformed continuum covariant density-functional theory exists by other continuum treatments (Dirac-Woods-Saxon-basis DRHBc, complex momentum representation, complex scaling), but no self-consistent deformed GF implementation and no GF-based deformed QRPA are known in the literature. The problem is therefore open as a method-development program with a recognizable completion boundary: formalism, validated implementation, cross-benchmarks, and controlled convergence.

Background

Near the neutron and proton drip lines (the limits of nuclear existence set by zero binding energy), nuclei become weakly bound and the unbound single-particle states above the threshold matter: among them are single-particle resonances, quasi-bound states with a decay width Gamma embedded in the continuum. In covariant density functional theory (CDFT), nucleons move in scalar and vector fields generated self-consistently by a relativistic mean-field (RMF) density functional, and the single-particle motion obeys a Dirac equation. Because most nuclei are deformed, the interplay between deformation and the orbital structure near threshold is central for exotic nuclei; in an axially quadrupole-deformed potential the Dirac equation couples the spherical partial waves, giving a coupled-channel problem whose channels are labeled by the Nilsson quantum numbers Omega[N n_z Lambda] (projection of total angular momentum on the symmetry axis, and the associated oscillator labels).

For resonances, several techniques exist: the analytical continuation of the coupling constant (ACCC), the complex scaling method (CSM), the complex momentum representation (CMR), scattering phase shift (SPS) analysis, and the Green's-function (GF) method. In the GF method one constructs the single-particle Green's function in coordinate space; the density of states (DOS) is its energy-integrated imaginary part, and resonance energies and widths follow either from the DOS peaks or, as proposed in the source paper, from the extrema of the GF/DOS in the complex-energy plane, which locate the resonance poles E−iΓ/2E - i\Gamma/2 exactly. A distinctive advantage is that bound and resonant wave functions, with proper asymptotic behavior, are obtained on the same footing, so the method combines naturally with many-body models.

The source paper (T.-T. Sun, L. Qian, C. Chen, P. Ring, Z. P. Li, Phys. Rev. C 101, 014321 (2020)) applied the GF method for the first time to the coupled-channel Dirac equation with quadrupole-deformed Woods-Saxon potentials, verified it against spherical GF, CSM/CMR, ACCC, and SPS results, and used it to compute Nilsson levels of bound and resonant orbitals in the halo candidate 37^{37}Mg. It closed with an explicit outlook: apply the GF method to the deformed relativistic Hartree-Bogoliubov theory (RHB, the CDFT framework in which mean field and pairing correlations are treated self-consistently via Bogoliubov quasiparticles, the particle-hole mixed excitations of the paired system) to describe deformed halos, and to the deformed quasiparticle random-phase approximation (QRPA, the small-amplitude collective-excitation theory built on the Hartree-Bogoliubov ground state) to describe excitation modes coupled to the continuum. These two goals are the open problem; the accuracy for very broad, strongly mixed deformed resonances and the convergence in the partial-wave truncation NN, box size RboxR_{\mathrm{box}}, mesh, and energy step dεd\varepsilon form part of the completion boundary.

The surrounding landscape: self-consistent continuum calculations with pairing via the GF method exist only for spherical systems, in Skyrme Hartree-Fock-Bogoliubov (HFB) form (giant-halo and quasiparticle-resonance studies) and in spherical relativistic continuum Hartree-Bogoliubov form (canonical states, i.e., the single-particle states that diagonalize the Hartree-Bogoliubov density matrix, in Zr giant-halo nuclei). Conversely, self-consistent deformed continuum covariant DFT with pairing has been achieved by other continuum treatments: the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc, built on a Dirac Woods-Saxon basis) underlies large mass-table, fission-barrier, and halo studies, and deformed RMF plus BCS exists in the complex momentum representation. The GF route into the deformed self-consistent regime, and a GF-based deformed QRPA, are the missing steps.

Problem Statement

Carry the deformed coupled-channel Green's-function continuum method from fixed Woods-Saxon potentials into fully self-consistent covariant density-functional theory, in the two linked stages left open by Sun, Qian, Chen, Ring, and Li (Phys. Rev. C 101, 014321 (2020)):

(i) Ground-state stage — implement the GF method within deformed relativistic Hartree-Bogoliubov theory, so that deformation, pairing correlations, and the continuum are treated self-consistently and on the same footing: construct the coupled-channel quasiparticle Green's function of the deformed RHB equations (including its normalization and completeness relations), build the densities and pairing fields from the full continuum (resonant and non-resonant contributions), iterate to self-consistency for axially deformed weakly bound nuclei (the target application being deformed halos), and extract quasiparticle resonance energies and widths directly from the Green's-function density of states, including broad and strongly mixed deformed resonances.

(ii) Response stage — apply the same Green's-function machinery to the deformed quasiparticle random-phase approximation, so that collective excitation modes of deformed weakly bound nuclei are computed with the exact continuum rather than a discretized box spectrum.

The two stages form one program with a common objective — exact-continuum Green's-function treatment inside self-consistent deformed covariant density-functional theory — with the response stage building directly on the ground-state machinery. A complete resolution requires both linked stages — stage (i) alone is not sufficient — and all four components of the completion boundary named above (formalism, validated implementation, cross-benchmarks, and controlled convergence) are required, not advisory.

References

  1. T.-T. Sun, L. Qian, C. Chen, P. Ring, and Z. P. Li, Green's function method for the single-particle resonances in a deformed Dirac equation, Phys. Rev. C 101, 014321 (2020), DOI: 10.1103/PhysRevC.101.014321, arXiv:1710.02923, https://arxiv.org/abs/1710.02923
  2. T. T. Sun, S. Q. Zhang, Y. Zhang, J. N. Hu, and J. Meng, Green's function method for single-particle resonant states in relativistic mean field theory, Phys. Rev. C 90, 054321 (2014), DOI: 10.1103/PhysRevC.90.054321, arXiv:1409.8412, https://arxiv.org/abs/1409.8412 (spherical predecessor of the source method)
  3. Y. Zhang, M. Matsuo, and J. Meng, Pair correlation of giant halo nuclei in continuum Skyrme-Hartree-Fock-Bogoliubov theory, Phys. Rev. C 86, 054318 (2012), DOI: 10.1103/PhysRevC.86.054318, arXiv:1209.5263, https://arxiv.org/abs/1209.5263
  4. Cheng Chen, Zhipan Li, Yuxiao Li, and Tingting Sun, Single-particle resonant states with Green's function method, Chin. Phys. C 44, 084105 (2020), DOI: 10.1088/1674-1137/44/8/084105, arXiv:2004.11632, https://arxiv.org/abs/2004.11632
  5. Ya-Tian Wang and Ting-Ting Sun, Searching for single-particle resonances with the Green's function method, Nucl. Sci. Tech. 32, 46 (2021), DOI: 10.1007/s41365-021-00884-0, arXiv:2105.04071, https://arxiv.org/abs/2105.04071
  6. Ying Zhang and Xiao-Ying Qu, Effects of pairing correlation on the quasiparticle resonance in neutron-rich Ca isotopes, Phys. Rev. C 102, 054312 (2020), DOI: 10.1103/PhysRevC.102.054312, https://doi.org/10.1103/PhysRevC.102.054312
  7. Xiao-Ying Qu, Hui Tong, and S. Q. Zhang, Canonical states in relativistic continuum theory with the Green's function method: Neutrons in continuum of zirconium giant-halo nuclei, Phys. Rev. C 105, 014326 (2022), DOI: 10.1103/PhysRevC.105.014326, https://doi.org/10.1103/PhysRevC.105.014326
  8. En-Bo Huo, Ke-Ran Li, Xiao-Ying Qu, Ying Zhang, and Ting-Ting Sun, Continuum Skyrme Hartree-Fock-Bogoliubov theory with Green's function method for neutron-rich Ca, Ni, Zr, and Sn isotopes, Nucl. Sci. Tech. 34, 105 (2023), DOI: 10.1007/s41365-023-01261-9, https://doi.org/10.1007/s41365-023-01261-9
  9. Ting-Ting Sun, Bing-Xin Li, and Kun Liu, Probing spin and pseudospin symmetries in deformed nuclei by the Green's function method, Phys. Rev. C 109, 014323 (2024), DOI: 10.1103/PhysRevC.109.014323, arXiv:2310.17203, https://arxiv.org/abs/2310.17203
  10. Wei Gao, Ting-Ting Sun, and Wen-Hui Long, Single-particle resonant states in relativistic Hartree-Fock theory: A Green's function approach, Chin. Phys. C 50, 084109 (2026), DOI: 10.1088/1674-1137/ae6a85, arXiv:2603.04769, https://arxiv.org/abs/2603.04769
  11. Xiang-Xiang Sun and Shan-Gui Zhou, Angular momentum projection in the deformed relativistic Hartree-Bogoliubov theory in continuum, Phys. Rev. C 104, 064319 (2021), DOI: 10.1103/PhysRevC.104.064319, arXiv:2107.05925, https://arxiv.org/abs/2107.05925
  12. K. Y. Zhang, S. Q. Yang, J. L. An, S. S. Zhang, P. Papakonstantinou, M.-H. Mun, Y. Kim, and H. Yan, Missed prediction of the neutron halo in 37^{37}Mg, Phys. Lett. B 844, 138112 (2023), DOI: 10.1016/j.physletb.2023.138112, arXiv:2306.16011, https://arxiv.org/abs/2306.16011
  13. Wei Zhang, Jin-Ke Huang, Ting-Ting Sun, Jing Peng, and Shuang-Quan Zhang, Inner fission barriers of uranium isotopes in the deformed relativistic Hartree-Bogoliubov theory in continuum, Chin. Phys. C 48, 104105 (2024), DOI: 10.1088/1674-1137/ad62dd, arXiv:2403.19092, https://arxiv.org/abs/2403.19092
  14. Peng Guo, Xiaojie Cao, Kangmin Chen, Zhihui Chen, Myung-Ki Cheoun, Yong-Beom Choi, et al. (DRHBc Mass Table Collaboration), Nuclear mass table in deformed relativistic Hartree-Bogoliubov theory in continuum, II: Even-ZZ nuclei, At. Data Nucl. Data Tables 158, 101661 (2024), DOI: 10.1016/j.adt.2024.101661, https://doi.org/10.1016/j.adt.2024.101661
  15. Xue-Neng Cao, Xian-Xian Zhou, Ming Fu, and Xin-Xing Shi, Research on the influence of quadrupole deformation and continuum effects on the exotic properties of 15,17,19^{15,17,19}B with the complex momentum representation method, Nucl. Sci. Tech. 34, 25 (2023), DOI: 10.1007/s41365-023-01177-4, https://doi.org/10.1007/s41365-023-01177-4
  16. Min Shi, Xin-Xing Shi, Zhong-Ming Niu, Ting-Ting Sun, and Jian-You Guo, Relativistic extension of the complex scaled Green's function method for resonances in deformed nuclei, Eur. Phys. J. A 53, 40 (2017), DOI: 10.1140/epja/i2017-12241-6, https://doi.org/10.1140/epja/i2017-12241-6
  17. K. Hagino, Nguyen Van Giai, and H. Sagawa, Continuum QRPA response for deformed neutron-rich nuclei, Nucl. Phys. A 731, 264-272 (2004), DOI: 10.1016/j.nuclphysa.2003.11.037, https://doi.org/10.1016/j.nuclphysa.2003.11.037
  18. Kazuhito Mizuyama, Unified theoretical framework for Hartree-Fock-Bogoliubov resonant states: Integration of the complex-scaled Jost function and Autonne-Takagi normalization, Phys. Rev. C 114, 014619 (2026), DOI: 10.1103/zmjz-42kp, arXiv:2603.21420, https://arxiv.org/abs/2603.21420