Precise comparison of the Gaussian expansion method and the Gamow shell model

We perform a detailed comparison of results of the Gamow shell model and the Gaussian expansion method supplemented by the complex scaling method for the same translationally invariant cluster-orbital shell model Hamiltonian. As a benchmark test, we calculate the ground state $0^{+}$ and the first excited state $2^{+}$ of mirror nuclei ${}^6$He and ${}^6$Be in the model space consisting of two valence nucleons in a $p$ shell outside of a ${}^4$He core. We find a good overall agreement of results obtained in these two different approaches, also for many-body resonances.

Figure 1

Coordinate system of the COSM approach.

Figure 2

Complex-scaled eigenstates of the three-body Hamiltonian for the Borromean system. Solid circles are bound and resonance states, and open circles are continuum states.

Figure 3

Deformed contour on the complex momentum plane (a), and discretized continuum states along the deformed contour (b). Solid circles are bound and resonant states, and open circles are discretized continuum states.

Figure 4

Convergence of the poles of the ground $0_1^{+}$ and the first excited $2_1^{+}$ states of ${}^6$He, which are calculated using the GEM+CS approach and the GSM for $1\le {\ell }_{\max }\le 5$. Open and solid circles denote GEM+CS and GSM results, respectively.

Figure 5

The density of valence neutrons in the COSM coordinate system for the ground $0_1^{+}$ state of ${}^6$He. The normalization of the density distribution is 1.

Figure 6

Poles of the ground and first excited states of ${}^6$Be calculated using the GEM+CS approach and the GSM from ${\ell }_{\max }=1$ to 5. Open and solid circles correspond to GEM+CS and GSM results, respectively.

Figure 7

Poles of ${}^6$He $\left(2^{+}\right)$ with ${\ell }_{\max }=1$. For the GEM+CS, we change the rotation angle $\theta$ from $5^{\circ }$ to ${25}^{\circ }$ in steps of $1^{\circ }$.