Analysis of the stability of finite subspaces in density functional theory

We study the problem of the stability of finite subspaces with respect to the external potential in the formulation of the Hohenberg-Kohn theorem in density functional theory. We provide general procedures to construct potentials that make any finite dimensional subspace unstable, i.e., we construct potentials that acting over functions that belong to the subspace, generate functions that do not belong to that subspace. Explicit calculations of these instability generating potentials are carried out for the particle-in-a-box problem and for the hydrogen atom. We also discuss the consequences of these instabilities on the Kohn-Sham equations, as well as conditions for stability and the relation between instability and nonuniqueness of potentials.