## Analyzing our results, decomposing the Hamiltonian

The two-particle matrix elements are normalized and antisymmetrized. A similar expression applies to the medium-modified two-body interaction $$\tilde{v}$$ of Eq. (19) as well. The symbol $$\langle \alpha_p\alpha_q|LSJ \rangle$$ is a shorthand for the $$LS-jj$$ transformation coefficient, $$\begin{equation*} \langle \alpha_p\alpha_q|\lambda SJ \rangle = \sqrt{(2j_{p}+1)(2j_{q}+1)(2\lambda+1)(2S+1)} \left\{ \begin{array}{ccc} l_{p}&\frac{1}{2}&j_{p}\\ l_{q}&\frac{1}{2}&j_{q}\\ \lambda &S &J \end{array} \right\} \end{equation*}$$