We consider a one-dimensional model where the confining potential is parametrized/obtained from finite element calculations. The bare Coulomb interaction is divergent in 1D (REF) and it is customary to use a smoothed Coulomb interaction given by
$$ \begin{align} u(x_1, x_2) = \frac{\alpha}{\sqrt{(x_1 - x_2)^2 + a^2}}, \tag{6} \end{align} $$where \( \alpha \) adjusts the strength of the interaction and \( a \) removes the singularity at \( x_1 = x_2 \).
The single-particle functions are chosen as the eigenfunctions of the single-particle Hamiltonian
$$ \begin{equation} \left( -\frac{d^2}{dx^2}+v(x) \right) \psi_p(x) = \epsilon_p \psi_p(x). \tag{7} \end{equation} $$