Iterative solution of a Dirac equation with an inverse Hamiltonian method

We solve a singe-particle Dirac equation with Woods-Saxon potentials using an iterative method in the coordinate space representation. By maximizing the expectation value of the inverse of the Dirac Hamiltonian, this method avoids the variational collapse in which an iterative solution dives into the Dirac sea. We demonstrate that this method works efficiently, reproducing the exact solutions of the Dirac equation.

Figure 1

(a) A schematic spectrum of a Dirac Hamiltonian $H$. $V(r)$ and $S(r)$ are vector and scalar potentials, respectively, plotted as functions of $r$. $E_0$ is an arbitrary constant located between the lowest energy state in the Fermi sea and the highest energy state in the Dirac sea. The hatched regions indicate the continuum spectra. (b) The spectrum of $1/(H-E_0)$ corresponding to (a). The filled circles indicate the bound states in the Fermi sea, while the open circles indicate the bound states in the Dirac sea. The continuum region is denoted by the thick solid line.

Figure 2

The expectation value of the Hamiltonian $H$ (a) and that of the inverse of the Hamiltonian, $1/(H-E_0)$ (b), at each iteration for the neutron 1$p$${}_{1/2}$ state in ${}^{16}\mathrm{O}$. $E_0$ is set to be the depth of the Woods-Saxon potential $U(r)=V(r)+S(r)$, that is, $E_0=-71.28$ MeV, and the step $\Delta T$ is taken to be 10 MeV. The open circles in (a) are obtained by improving the energy shift $E_0$ at each iteration as described in the text.

Figure 3

Comparison of the wave function for the neutron 1$p$${}_{1/2}$ state in ${}^{16}\mathrm{O}$ obtained with the inverse Hamiltonian method (the dashed lines) to the exact solutions (the solid lines).

Figure 4

Same as Fig. 3, but for the neutron $2s_{1/2}$ state in ${}^{16}\mathrm{O}$.