  We re-adapt a spectral renormalization method, introduced in nonlinear
optics, to solve the Kohn-Sham (KS) equations of density functional theory
(DFT), with a focus on functionals based on the strictly-correlated electrons
(SCE) regime, which are particularly challenging to converge. Important aspects
of the method are: (i) the eigenvalues and the density are computed
simultaneously; (ii) it converges using randomized initial guesses; (iii) easy
to implement. Using this method we could converge for the first time the
Kohn-Sham equations with functionals that include the next leading term in the
strong-interaction limit of density functional theory, the so-called zero-point
energy (ZPE) functional as well as with an interaction-strength-interpolation
(ISI) functional that includes both the exact SCE and ZPE terms. This work is
the first building block for future studies on quantum systems confined in low
dimensions with different statistics and long-range repulsions, such as
localization properties of fermions and bosons with strong long-range repulsive
interactions in the presence of a random external potential.
