Reduced density matrices of energy eigenstates

The following question is considered: What special properties are possessed by those reduced density matrices which come from energy eigenstates? Using the fact that 〈φ[H, A] |φ〉 = 0, where A is any operator and |φ〉 an energy eigenstate, it is shown that the elements of the two-particle density matrix are severely restricted by homogeneous linear relations. Their full content is expressed in terms of an auxiliary one-particle density which possesses additional positivity properties in the ground state.