*2.1. Realistic Gamow Shell Model Calculations*

Within realistic GSM, one starts from the intrinsic Hamiltonian of an *A*-body system, which reads

$$H = \sum\_{i=1}^{A} \frac{\mathbf{p}\_i^2}{2m} + \sum\_{i$$

where *<sup>p</sup><sup>i</sup>* is the nucleon momentum in laboratory frame, *<sup>P</sup>* <sup>=</sup> <sup>∑</sup>*<sup>A</sup> <sup>i</sup>*=<sup>1</sup> *<sup>p</sup><sup>i</sup>* is the center-ofmass (CoM) momentum of the system, and *<sup>V</sup>*(*ij*) NN is the realistic NN interaction, such as CD-Bonn [55] and N3LO [56] interaction. In the above Hamiltonian, the CoM energy is removed. In order to construct the effective Hamiltonian to be used in GSM calculations, an auxiliary potential is usually introduced [38,57,58], so that the Hamiltonian can be rewritten as,

$$\begin{aligned} H &= \sum\_{i=1}^{A} (\frac{\mathbf{p}\_i^2}{2m} + \mathsf{U}) + \sum\_{i$$

where *H*<sup>0</sup> = ∑*<sup>A</sup> <sup>i</sup>*=1( *<sup>p</sup>*<sup>2</sup> *i* <sup>2</sup>*<sup>m</sup>* <sup>+</sup> *<sup>U</sup>*) has a one-body form, and *<sup>H</sup>*<sup>1</sup> <sup>=</sup> <sup>∑</sup>*<sup>A</sup> i*<*j* (*V*(*ij*) NN <sup>−</sup> *<sup>U</sup>* <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *i* <sup>2</sup>*Am* <sup>−</sup> *<sup>p</sup>i*·*p<sup>j</sup> Am* ) is the residual two-body interaction, including the correction issued from the CoM motion. For the auxiliary potential *U*, we usually take a WS finite-range potential. To speed up the convergence of many-body calculations, the bare force is often softened by a similarity renormalization group method [59], such as the similarity renormalization group (SRG) and *V*low-*k*,in order to remove the strong short-range repulsive core.

The realistic NN interaction is firstly defined in a relative momentum space. However, the many-body problem is usually solved in the laboratory frame (with, e.g., the HO basis), so that a transformation from relative and CoM coordinates to the laboratory frame is necessary. This procedure can be conveniently carried out in the HO basis via Brody– Moshinsky brackets [60]. In the HO basis, the two-body completeness relation reads

$$\sum\_{\alpha \le \beta} |\alpha \beta\rangle \langle \alpha \beta| = \mathbf{1},\tag{5}$$

where |*αβ* is the two-particle state of the HO basis. The two-body interaction in the HO basis is given by

$$V\_{\rm HO} = \sum\_{a \le \beta}^{N\_{\rm shell}} \sum\_{\gamma \le \delta}^{N\_{\rm shell}} |a\beta\rangle \langle a\beta| V\_{\rm low-k} |\gamma \delta\rangle \langle \gamma \delta|,\tag{6}$$

where *N*shell = 2*n* + *l*, indicates that a finite summation is performed up to *N*shell. The GSM calculations are carried out in the Berggren basis, so that the transformation of the interaction matrix elements from the HO basis to the Berggren basis needs to be carried out. This is achieved, in practice, by computing the overlaps between the Berggren and HO basis wave functions,

$$
\langle ab|V|cd\rangle \approx \sum\_{a \le \beta}^{N\_{\text{shell}}} \sum\_{\gamma \le \delta}^{N\_{\text{shell}}} \langle ab|a\beta\rangle \langle a\beta|V\_{\text{low}-k}|\gamma\delta\rangle \langle \gamma\delta|cd\rangle,\tag{7}
$$

where |*ab* (|*cd*) is a two-particle state of the Berggren basis. For identical particles (proton– proton or neutron–neutron), the overlap of the two-body state reads

$$
\langle ab|a\beta\rangle = \frac{\langle a|a\rangle\langle b|\beta\rangle - (-1)^{f-j\_{\alpha}-j\_{\beta}}\langle a|\beta\rangle\langle b|a\rangle}{\sqrt{(1+\delta\_{ab})(1+\delta\_{a\beta})}},\tag{8}
$$

where *J* is the total angular momentum of the two-particle state, and *j* is the angular momentum of a single-particle basis state. The *a*|*α*(*b*|*β*) is the overlap of the one-body basis state, and *δαβ* is the Kronecker delta. For the proton–neutron coupling, the overlap of the two-body state is simply given by

$$
\langle ab|a\beta\rangle = \langle a|a\rangle \langle b|\beta\rangle. \tag{9}
$$

The overlaps of one-body basis states are directly obtained from an integration in *r*-space

$$
\langle a|a\rangle = \int dr r^2 u\_a(r) \mathcal{R}\_a(r) \delta\_{l\_a l\_a} \delta\_{j\_a j\_a} \delta\_{t\_a t\_a \nu} \tag{10}
$$

where *u*(*r*) and *R*(*r*) are the radial parts of the single-particle Berggren and HO basis wave functions, with *l*, *j*, and *t* being the orbital, total angular momentum, and isospin quantum number, respectively. The single-particle wave functions of resonance and continuum states are not localized and hence are not square-integrable. The transformation defined by Equation (7), in fact, utilizes the short-range nature of nuclear force. Indeed, the Gaussian fall-off of the HO wave function renders the overlaps integrable, even when one considers resonances or scattering states of complex energy. For long-range operators, such as the one-body kinetic energy and Coulomb potential, using Equation (7) is not suitable in practice. In this case, we use the exterior complex scaling technique [51] to treat the kinetic and Coulomb operator, i.e., terms proportional to *p*<sup>2</sup> and 1/*r*, respectively, with the Berggren basis.

The obtained interaction matrix elements in the Berggren basis are complex, and associated operators are non-Hermitian. The many-body perturbation theory (MBPT), named the full *Q*ˆ-box folded-diagram method [61], is employed to construct the realistic complex effective Hamiltonian in the defined model space for GSM calculations. The complex-*k* Berggren basis states in the model space are non-degenerate; therefore a nondegenerate *Q*ˆ-box folded-diagram perturbation, i.e., the extended Kuo–Krenciglowa (EKK) method [62], has been used. For this, we first calculate the *Q*ˆ-box using MBPT in the Berggren complex-*k* basis,

$$\begin{split} \hat{Q}(E) &= PPV + PVQ \frac{Q}{E - QHQ} QVP \\ &= PPV + PVQ \frac{Q}{E - QH\_0Q} QVP + \dots \end{split} \tag{11}$$

where *E* is starting energy, *P* and *Q* represent the model space and the excluded space, respectively, with *P* + *Q* = **1**. The *Q*ˆ-box is composed of irreducible valence-linked diagrams [57,58], which can be built order-by-order. *V* and *H* are the two-body interaction and two-body Hamiltonian, respectively, and *H*<sup>0</sup> is the unperturbed one-body Hamiltonian. The derivatives of the *Q*ˆ-box are defined as

$$\begin{split} \hat{Q}\_s(E) &= \frac{1}{s!} \frac{d^s \hat{Q}(E)}{dE^s} \\ &= (-1)^s P V Q \frac{Q}{(E - QHQ)^{s+1}} Q V P\_\prime \end{split} \tag{12}$$

where *s* denotes the *s*-th derivative.

The effective Hamiltonian *H*eff can then be constructed in operator form [63], written as

$$
\tilde{H}\_{\rm eff} = \tilde{H}\_{BH}(E) + \sum\_{k=1}^{\infty} \hat{\mathcal{Q}}\_k(E) \tilde{H}\_{\rm eff} \tag{13}
$$

where *H*eff stands for *H*eff = *H*eff − *E*, and *HBH*(*E*) = *HBH*(*E*) − *E* is the Block–Horowitz Hamiltonian shifted by an energy *E*, with

$$\begin{aligned} \left(H\_{BH}(E)\right) &= PH\_0P + \tilde{Q}(E) \\ &= PH\_0P + PVP + PVQ \frac{Q}{E - QHQ} QVP. \end{aligned} \tag{14}$$

*H*eff is obtained by performing iterations of Equation (13), which is equivalent to calculate folded-diagrams, where one considers high-order contributions by summing up the subsets of diagrams to finite order. When convergence is obtained, the effective Hamiltonian is given by *H*eff = *H*eff − *E*, and the effective interaction reads *V*eff = *H*eff − *PH*0*P*. The extended *Q*ˆ-box folded-diagram calculations provide a useful approach for including effects from the continuum coupling and core polarization [57,58,61,62].
