Application of the inverse Hamiltonian method to Hartree-Fock-Bogoliubov calculations

We solve the Hartree-Fock-Bogoliubov (HFB) equations for a spherical mean field and a pairing potential with the inverse Hamiltonian method, which we have developed for the solution of the Dirac equation. This method is based on the variational principle for the inverse Hamiltonian and is applicable to Hamiltonians that are not bound from above or below. We demonstrate that the method works well not only for the Dirac, but also for the HFB equations.

Figure 1

Spectra of (a) a quasiparticle Hamiltonian $H$ itself and (b) the inverse of the Hamiltonian $1/(H-W)$. $\lambda$ is the chemical potential. The bound states of positive and negative energies are indicated by solid and open circles, respectively. The continuum states are represented by the thick solid lines. The energy shift $W$ is taken between the positive and negative spectra. The eigenvalues are labeled by an integer $k$ such that $E_{-k}=-E_k$.

Figure 2

Covergence properties of (a) the energy expectation value $\langle H\rangle$ and (b) the expectation value of the inverse of Hamiltonian $\langle {(H-W)}^{-1}\rangle$ for the lowest $s_{1/2}$ quasiparticle state. The energy shift and the step size of $T$ are taken to be $W=0.1$ MeV and $\Delta T=10$ MeV, respectively.

Figure 3

Comparisons of wave functions for the three lowest $s_{1/2}$ states. Figures 3a, 3c, and 3e show the upper components [$U_k\left(r\right)$] while Figs. 3b, 3d, and 3f show the lower components [$V_k\left(r\right)$] of the quasiparticle wave function, respectively. The exact wave functions are shown with the dashed lines and the ones obtained by the inverse Hamiltonian method are drawn with the solid lines.