  A non-Hermitian PT-symmetric system can have full real spectrum but does not
ensure probability preserving time evolution, in contrast to that of a
Hermitian system. We present a non-Hermitian two-band model, which is comprised
of dimerized hopping terms and staggered imaginary on-site potentials, and
study the dynamics in the exact PT-symmetric phase based on the exact solution.
It is shown that an initial state, which does not involve two
equal-momentum-vector eigenstates in different bands, obeys perfectly
probability-preserving time evolution in terms of the Dirac inner product.
Beyond this constriction, the quasi-Hermitian dynamical behaviors, such as
non-spreading propagation and fractional revival of a Gaussian wave packet, are
also observed.
