Solving the relativistic Hartree-Bogoliubov equation with the finite-difference method

The Relativistic Hartree-Bogoliubov (RHB) theory is a powerful tool for describing the exotic nuclei near drip lines. The key technique is to solve the RHB equation in the coordinate space to obtain the quasi-particle states. In this paper, we solve the RHB equation with the Woods-Saxon type mean-field potential and Delta type pairing field potential by using the finite-difference method (FDM). We inevitably obtain the spurious states when using the common symmetric central difference formula (CDF) to construct the Hamiltonian matrix, which is similar to the problem when solving the Dirac equation with the same method. This problem is solved by using the asymmetric difference formula (ADF). In addition, we show that a large enough box is necessary to describe the continuum quasi-particle states. The canonical states obtained by diagonalizing the density matrix constructed by the quasi-particle states are not that sensitive to the box size. Part of the asymptotic wave functions can be improved due to the ADF in FDM compared to the shooting method with the same box boundary condition.