Separability of Solutions to a Schrödinger Equation

We discuss separability of solutions to a Schrödinger equation that describes a composite quantum system and give some kinds of Hamiltonians H(t) such that the solution to Schrödinger equation induced by H(t) is separable at any time provided that it is separable at t = 0. For example, we prove that if the Hamiltonian H is time-independent and equals to the product P_A ⊗ P_B of two projections on the subsystems K_A and K_B, respectively, then the state |ψ(t)〉 of the composite system starting from a separable initial |ψ(0)〉 = |ψ_A〉 ⊗ |ψ_B〉 is separable for all t ∈ [0, T] if and only if either |ψ_A〉 is an eigenstate of P_A, or |ψ_B〉 is an eigenstate of P_B.