Complement to the Rayleigh-Ritz principle for the energy spectrum of a system composed of identical particles

A theorem is proved which states that the energies $E_i$ of a general translation-invariant system of $N$ identical particles interacting by pair potentials and obeying nonrelativistic quantum mechanics are bounded below one by one by the energies $E_i^L$ of a related model system consisting of ($N-1\mathrm{}$) noninteracting particles of the same symmetry type bound to a fixed center. The theorem is applied to a three-fermion harmonic oscillator test system whose exact eigenvalues are found. The ratios $\frac{E_i^L}{E_i}$ for this problem satisfy $\frac{E_i^L}{E_i}\ge \frac{E_0^L}{E_0}=\frac{\sqrt{3}}{4}$ for at least the first twenty three-body states.