Exact spectral decomposition of a time-dependent one-particle reduced density matrix

We determine the exact time-dependent non-idempotent one-particle reduced density matrix and its spectral decomposition for a harmonically confined two-particle correlated one-dimensional system when the interaction terms in the Schr\"odinger Hamiltonian are changed abruptly. Based on this matrix in coordinate space we derivea precise condition for the equivalence of the purity and the overlap-square of the correlated and non-correlated wave functions as the system evolves in time. This equivalence holds only if the interparticle interactions are affected, while the confinement terms are unaffected within the stability range of the system. Under this condition we also analyze various time-dependent measures of entanglement and demonstrate that, depending on the magnitude of the changes made in the Schr\"odinger Hamiltonian, periodic, logarithmically incresing or constant value behavior of the von Neumann entropy can occur.