New and efficient method for solving the eigenvalue problem for the two-center shell model with finite-depth potentials

We propose a method to solve the eigenvalue problem with a two-center single-particle potential. This method combines the usual matrix diagonalization with the method of separable representation of a two-center potential; that is, an expansion of the two-center potential with a finite basis set. To this end, we expand the potential on a harmonic-oscillator basis, while single-particle wave functions on a combined basis with a harmonic oscillator and eigenfunctions of a one-dimensional two-center potential. To demonstrate its efficiency, we apply this method to a system with two ${}^{16}\mathrm{O}$ nuclei, in which the potential is given as a sum of two Woods–Saxon potentials.

Figure 1

A schematic view of a two-center potential given by Eq. (1).

Figure 2

(a) Neutron single-particle energies for the ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$ system as a function of the separation distance between the two ${}^{16}\mathrm{O}$ nuclei. The solid and the dashed lines indicate the energies for the positive- and the negative-parity states, respectively. (b) The dependence of the depth parameter $V_0$ in the Woods–Saxon potential on the separation distance as determined from the volume conservation condition given by Eq. (26). (c) The same as the panel (b), but for the radius parameter, $R$.

Figure 3

The single-particle energy for the second positive-parity state with $j_z^{\pi }=1/2^{+}$ as a function of the number of basis states in Eq. (19) for the $z$ direction. The upper and the lower panes show the result at the separation distance of $z=3$ fm and $z=10$ fm, respectively. The solid lines with filled circles indicate the results with the eigensolutions for the one-dimensional two-center potentials given by Eq. (22), while the dashed lines and the solid lines with filled triangles show the results with one-center and two-center harmonic-oscillator bases, respectively.

Figure 4

Same as Fig. 3, but for the fourth positive-parity state with $j_z^{\pi }=1/2^{+}$.