Applicability of the Richardson method in a complex-energy basis: Fermionic case

We are testing the applicability of the algebraic procedure developed by Richardson for bound states to solve the pairing-force problem in presence of states with complex energies. This scenario, in which nucleons occupy single-particle states with complex energies and interact via the pairing force, is closely related to the microscopic description of nuclei with a large neutron (or proton) excess in basis which includes single-particle resonances. It is shown that the method gives results which are in good agreement with exact solutions. This finding is in coincidence with previously published works, by other authors, based on different methods to describe resonances.

Figure 1

Exact eigenvalues for a two-level system with ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=\mathrm{\Omega }=10$, and energies ${\epsilon }_1=\frac{\epsilon }{2}$ and ${\epsilon }_2=-\frac{\epsilon }{2}$ and number of fermion pairs, $n_p=3$. Both levels have real energy. The energies are given in arbitrary units.

Figure 2

Richardson pair energies ($x_{\alpha }$) for the system of Fig. 1. In the insets of the figure, from bottom to top, we show the real (left panel) and imaginary (right panel) components of the solutions for the ground state, and first-, second-, and third-excited states, respectively. The energies are given in arbitrary units.

Figure 3

Exact eigenvalues, for three fermion pairs moving in two levels of degeneracy $\mathrm{\Omega }$. The lower level has a real energy ${\epsilon }_1=-\epsilon$ and the upper one has complex energy with real part ${\epsilon }_2=\epsilon$ and imaginary part $\mathrm{\Gamma }=-0.05$. All values are given in arbitrary units of energy.

Figure 4

Richardson pair energies ($x_{\alpha }$) for the system of Fig. 3. From bottom to top the insets show the results for the real (left panel) and imaginary (right panel) part of the solutions for the ground state and for the excited states, respectively.

Figure 5

Exact solutions for the two-level system with $\mathrm{\Gamma }=-2.59$, and for the same other parameters given in the caption to Fig. 1.

Figure 6

Real and imaginary parts of the solutions $x_i$, for each configuration, for the same case as that of Fig. 5.

Figure 7

Results of the calculations, applying Richardson's method, after adding up the values of $x_i$, both real and imaginary, for each eigenvalue. The results shown in this figure are to be compared with the results shown in Fig. 5.

Figure 8

Richardson method for a two-level system with ${\mathrm{\Omega }}_i=10, n_p=10$. Parameters for the ground state, for the case with $\mathrm{\Gamma }=0.0$.

Figure 9

Richardson method for a two-level system with ${\mathrm{\Omega }}_i=10$ and $n_p=10$. Parameters $x_{\alpha }$ for the first-excited state, for the case with $\mathrm{\Gamma }=0.0$.

Figure 10

Richardson method for a two-level system with ${\mathrm{\Omega }}_i=10$, and $n_p=10$. Parameters for the ground state, for the case with $\mathrm{\Gamma }=-0.1$.

Figure 11

Richardson method for a two-level system with ${\mathrm{\Omega }}_i=10$, and $n_p=10$. The curves show the value of the parameters $x_{\alpha }$ of the first-excited state for the case $\mathrm{\Gamma }=-0.1$.

Figure 12

Real part of the energy difference between the first-excited state and the ground state, $W=E_{\text{1st}\text{exc.}}-E_{\text{g.s.}}$, obtained by the exact diagonalization and by applying Richardson method, for a two-level system with ${\mathrm{\Omega }}_i=10$ and $n_p=10$. The results correspond to $\mathrm{\Gamma }=0.0$ (continuous line) and $\mathrm{\Gamma }=-0.1$ (dashed line).