The inverse Kohn-Sham problem asks: given an electron density, find the local effective potential whose noninteracting ground state reproduces that density. This is the reverse of the usual Kohn-Sham calculation (given a potential, find the density) and is essential for constructing and testing exchange-correlation functionals.
The paper on a unified variational framework (arXiv: 2603.23452) identifies the Kohn-Sham potential as the variational dual object of the fixed-density constrained search.
The insight is structural: the principal inversion formulations that have been proposed — Wu-Yang, Zhao-Morrison-Parr, PDE-constrained approaches — are all realizations of the same underlying variational structure, classified by how they treat the state equations and density-reproduction conditions (as objectives, constraints, penalties, or feasibility relations). They are not different methods; they are different parametrizations of the same dual problem.
The framework also clarifies long-standing technical issues: the additive-constant ambiguity of the potential, the asymptotic normalization condition, the non-smooth variational structure at degenerate densities, and the weak-gap instability that makes some methods numerically fragile.
The through-claim: the Kohn-Sham potential is a Lagrange multiplier. The density is the primal variable; the potential is the dual. Every inversion method is a different way of solving the same dual problem. The apparent proliferation of methods reflects choices in numerical strategy, not differences in mathematical structure.
2603.23452. Quantum chemistry / density functional theory / inverse problems / Kohn-Sham / variational duality.