Theorems on ground-state phase transitions in Kohn-Sham models given by the Coulomb density functional

Some theorems on derivatives of the Coulomb density functional with respect to the coupling constant $\lambda$ are given. Consider an electron density $n_{GS}({\bf r})$ given by a ground state. A model Fermion system with the reduced coupling constant, $\lambda<1$, is defined to reproduce $n_{GS}({\bf r})$ and the ground state energy. Fixing the charge density, possible phase transitions as level crossings detected in a value of the reduced density functional happen only at discrete points along the $\lambda$ axis. If the density is $v$-representable also for $\lambda<1$, accumulation of phase transition points is forbidden when $\lambda\rightarrow 1$. Relevance of the theorems for the multi-reference density functional theory is discussed.