Schrödinger equations for the square root density of an eigenmixture and the square root of an eigendensity spin matrix

We generalize a “one eigenstate” theorem of Levy, Perdew and Sahni (LPS) [1] to the case of densities coming from eigenmixture density operators. The generalization is of a special interest for the radial density functional theory (RDFT) for nuclei [2], a consequence of the rotational invariance of the nuclear Hamiltonian; when nuclear ground states (GSs) have a finite spin, the RDFT uses eigenmixture density operators to simplify predictions of GS energies into one-dimensional, radial calculations. We also study Schrödinger equations governing spin eigendensity matrices. The theorem of Levy, Perdew and Sahni [1] may be described as follows: i) let HA be a Hamiltonian for A identical particles, with individual mass m, HA = A ∑ i=1 [−h̄2∆~ri/(2m) + u(~ri)] + A ∑ i>j=1 v(~ri, ~rj), (1) ii) consider a GS eigenfunction of HA, ψ(~r1, σ1, ~r2, σ2, ..., ~rA, σA), where σi denotes the spin state of the particle with space coordinates ~ri, iii) use a trace of |ψ〉〈ψ| upon all space coordinates but the last one, and upon all spins, to define the density, ρ(~r) = A ∑