The Hartree-Fock-Bogolyubov equations

Writing the fields in matrix form

$$\mathcal{M}=\left(\begin{matrix}M& \tilde M \\ \tilde M & - M \\ ... ...{matrix}U-\lambda& \tilde U \\ \tilde U & - U+\lambda \\ \end{matrix}\right)\,,$$ (30)

$$\mathcal{U}_{\mathrm{so}}=\left(\begin{matrix}B & \tilde B \\ \tilde B & - B \\ \end{matrix}\right) \frac{j(j+1)-\ell(\ell+1)-\frac{3}{4}}{2r} \,.$$ (31)

The HFB equations read

$$\left[-\frac{d}{dr}\mathcal{M}\frac{d}{dr}+\mathcal{U}+\mathcal{M... ...\ \end{matrix}\right) =E \left(\begin{matrix}u_1\\ u_2\\ \end{matrix}\right)\,.$$ (32)

An $r$-dependant mixing of components and scaling described in Ref. [5] allows us to write this equation as an equation with no differential operator in the coupling terms and no first order derivative

$$\begin{array}{rcl} -M^*\frac{d^2}{dr^2}f_1+Vf_1+Wf_2 &=& Ef_1 ,\\ M^*\frac{d^2}{dr^2}f_2-Vf_2+Wf_1 &=& Ef_2 . \end{array}$$ (33)

This last form with no first order derivative of the functions is particularly suitable for the numerical integration by the Numerov algorithm briefly discussed in section 5.2.