**1. Introduction**

Exotic nuclei have been studied for many years using a new generation of accelerators, which are now able to reach nuclear drip lines [1–4]. Contrary to well-bound nuclei, which are closed quantum systems, drip line nuclei can be seen as open quantum systems, as they are either weakly bound or unbound with respect to the particle emission threshold [5]. Many interesting phenomena appear at drip lines, such as a halo structure [1,6,7] and particle emission in resonance states [4,8]. Continuum coupling plays an important role in these loosely bound and unbound nuclear systems [5]. The proper description of nuclei at drip lines is one of the main challenges of nuclear theory, which was mostly developed to account for the structure of well-bound nuclei [5,6].

A clear consequence of the strong intertwinings of the continuum degrees of freedom and internucleon correlations at drip lines consists of the odd–even staggering found in the helium chain [9,10]. Indeed, odd helium isotopes (except 3He) are all resonances and bear widths of several hundreds of keV [10–12]. Conversely, the even–even helium isotopes 4,6,8He are bound, with 6,8He both exhibiting halo properties [13–15]. To accurately reproduce nuclear halos, many-body wave functions in asymptotic regions must be treated properly, which demands to take into account continuum coupling [1,6,7,16–19]. Adding to that, these weakly bound and unbound drip line nuclei also provide good laboratories to understand the single-particle structure, continuum coupling, internucleon correlations, and nucleon-nucleon (NN) interactions, which are not well understood in these regions.

Most present nuclear models, such as the no-core shell model (NCSM) [20], selfconsistent Green's function [21], coupled-cluster (CC) [22,23], in-medium similarity renormalization group (IM-SRG) [24], and standard shell model (SM) [25,26] have been developed for the study of well-bound nuclei, whereby continuum coupling is absent. Only few

**Citation:** Li, J.; Ma, Y.; Michel, N.; Hu, B.; Sun, Z.; Zuo, W.; Xu, F. Recent Progress in Gamow Shell Model Calculations of Drip Line Nuclei. *Physics* **2021**, *3*, 977–997. https:// doi.org/10.3390/physics3040062

Received: 17 August 2021 Accepted: 25 October 2021 Published: 8 November 2021

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models explicitly include continuum coupling. The main models including both internucleon correlations and continuum coupling in a unified picture are the no-core Gamow shell model (NCGSM) [27–29], the no-core shell model with continuum (NCSMC) [30], the complex CC [31,32], the Gamow IMSRG (Gamow-IMSRG) [33], continuum shell model (CSM) [34,35], and the Gamow shell model (GSM) [19,36–39], which are extensions of the NCSM, CC, IM-SRG, and SM, respectively. However, due to their huge model space dimensions, the NCGSM and NCSMC can only be used to describe light nuclei [27,28,30]. Furthermore, only nuclei in the vicinity of closed-shell nuclear systems can be investigated by the complex-CC and Gamow-IMSRG methods [31–33]. CSM [34,35] takes into account the continuum effect by projecting the model space onto the subspaces of bound and scattering states in a real-energy basis, in which, resonance states are not included. Within GSM, continuum coupling is treated at basis level by way of the Berggren basis [36–38]. The latter comprises bound, resonance, and continuum scattering states, with all of these states treated on an equal footing within the Berggren ensemble [40]. Internucleon correlations in GSM are induced by configuration mixing, similarly to conventional SM. GSM has been seen to successfully reproduce many situations of physical interest [5,38]; for example, the resonances of oxygen drip line nuclei [38,41,42] and the neutron halo structure of 31F [18].

The GSM was introduced in nuclear physics in 2002 [36,37], where only simple phenomenological nuclear potentials were used, while calculations were limited to only two valence neutrons outside of the inner core. After that, the GSM was extended to many valence particle systems, such as 8He [17] and *psd*-shell nuclei [43]. The realistic Gamow shell model was proposed in Refs. [44,45], with which, two- or three-particle systems could be investigated. An effective Hamiltonian based on realistic interactions was constructed by using the degenerate *Q*ˆ-box approach; however, folded diagrams are neglected [45]. A folded diagram sums up the subset of diagrams to infinite order so as to include high-order effects. In 2017, we developed the realistic GSM method with the full *Q*ˆ-box foldeddiagram method using the nondegenerate Berggren basis [38]. We applied it to the case of the neutron-rich oxygen isotopes up to the neutron drip line. After that, many extensions of the realistic GSM were developed, such as performing the realistic GSM in the Gamow Hartree-Fock basis (GHF) [41].

In the present review, the framework of the two types of GSM (realistic GSM and phenomenological GSM) is first introduced in Section 2. Then, we review our recent applications of GSM, including the calculations of neutron-rich oxygen and fluorine isotopes [38,39,41,42,46], neutron-rich calcium isotopes [47], and proton decays in 16Ne and 18Mg [48]. Finally, a short summary of the review and the future challenges of the next GSM calculations are given.

#### **2. Method**

GSM is built within a configuration interaction framework based on the one-body Berggren basis [5,36–38]. The Berggren basis [40,49] is generated by a finite-range potential, which can be written as the solutions of the one-body Schröndinger equation in the complex momentum space, which reads

$$\frac{d^2u(k,r)}{dr^2} = \left[\frac{l(l+1)}{r^2} + \frac{2m}{\hbar^2}lI(r) - k^2\right]u(k,r),\tag{1}$$

where *l* is the orbital angular momentum of the nucleon motion, *m* is the mass of the nucleon, *r* stands for the radius, and *h*¯ is the reduced Planck constant. The momentum *k* and wave function *u*(*k*,*r*) can be complex. *U*(*r*) is the finite-range potential, which is, in practice, a Woods–Saxon (WS) [50] or GHF potential [33,44]. When considering protons, the Coulomb potential must be included in *U*(*r*). Bound, resonance, and scattering states can then be generated. The eigenenergy of single-particle states in the above equation is complex in general, and reads *<sup>e</sup>*˜*<sup>n</sup>* = *<sup>k</sup>*2/2*<sup>m</sup>* = *en* − *<sup>i</sup>γn*/2, where *<sup>n</sup>* denotes the state [40,49]. *en* stands for the energy, whereas *γ<sup>n</sup>* represents the particle decay width, so that *γ<sup>n</sup>* = 0 for bound states and *γ<sup>n</sup>* > 0 for resonance states. A schematic Berggren basis set of states in the complex *k*-plane is illustrated in Figure 1. The wave function of a resonance state

is not square-integrable, as its exponential increase in modulus implies that the wave function of a resonance state cannot be normalized with conventional techniques [40,49]. Consequently, one has to rely on the complex scaling method, which has been seen to properly account for the normalization of resonance states [51].

The completeness relation borne by Berggren basis states [40,49] reads

$$\sum\_{n} |n\rangle\langle n| + \int\_{L\_{+}} |k\rangle\langle k| dk = \mathbf{1},\tag{2}$$

where |*n* states are bound states and resonance states inside the *L*<sup>+</sup> contour of Figure 1. These states are called pole states, as they are the *S*-matrix poles of the finite-range potential. |*k* states are scattering states and follow the *L*<sup>+</sup> contours in the complex *k*-plane, starting from *k* = 0 and going to *k* → +∞, as shown in Figure 1. Scattering states initially form a continuum. Hence, in order to be used in numerical applications, the scattering states along the *L*+ contour must be discretized with a Gauss–Legendre quadrature [5,49]. It has been checked that 10–45 states per contour are necessary to have converged results [5,38]. Once discretized, the Berggren basis is, in effect, the same as that of the harmonic-oscillator (HO) states within the standard SM [5,49]. Concerning resonance states, only narrow resonance states contribute to the physical states, and thus are included in the real calculations, whereas broad resonance states are not included, as they lie below the *L*+ contour.

**Figure 1.** Depiction of the Berggren basis in the complex-momentum-*k* plane for a fixed partial wave. Typical complex momenta of bound, narrow, and broad-resonance states, i.e., *S*-matrix pole, are provided. The *L*+ contour of scattering states encompasses the *S*-matrix poles of interest.

In fact, the Berggren basis is the complex extension of the real-energy completeness relation of Newton [52], which consists of bound states and of a continuum of real-energy scattering states. Contrary to the Newton completeness relation [52], with which, only localized states can be expanded, the Berggren basis can expand unbound resonance states [40,49]. The many-body completeness relation is obtained by constructing Slater determinants from the one-body Berggren basis, which contains bound, resonance, and scattering states [5,49]. In the GSM, the Hamiltonian is represented by a complex symmetric matrix when using the one-body Berggren basis, which has to be diagonalized [5,49]. This process can be handled efficiently by using the complex symmetric extension of the Jacobi-Davidson method [49,53], where one can take advantage of the relatively small

coupling to continuum states in order to have a fast convergence of calculations [5]. The full configuration space is extremely large due to the many scattering states within the model space. In practical calculations, however, we often truncate basis model spaces so that only two particles can occupy scattering states. It has been checked that this is sufficient to obtain converged results for both the energy and decay width of many-body states [5,38,54].

In GSM calculations, an effective Hamiltonian must be constructed. There are two main methods to build the effective Hamiltonian in GSM calculations. One is to construct an effective Hamiltonian based on realistic nuclear force [38,39,41], and hence in the frame of the realistic GSM, whereas the other one consists of using an effective phenomenological nuclear potential [5,36,37,49], in which, the parameters of the potential are optimized to reproduce experimental data. In the following, we give details about these two versions of GSM.
