Thanks for Answer.
The question is concerning the sparcity of the matrix representation of the propagator in a improper basis set. for practical matters, a way to perform practical computation using propagators is represent it in a conveniently chosen basis set and work with a matrix...
Considering the real-time quantum propagator originated by the green's function of the time-dependent Schrödinger equation:
i\hbar \frac{d\psi}{dt} =H\psi
namely,
\psi(t) =e^{-i\hbar H t}\psi(0)
it is straightforward to see that using as basis set the eigenvector of
H\phi_k...