  Investigations for decay of unstable D-brane and rolling of accelerated
D-brane dynamics have revealed that various proposed prescriptions give
different result for spectral amplitudes and observables. Here, we study them
with particular attention to unitarity and open-closed channel duality. From
"ab initio" derivation in the open string channel, both in Euclidean and
Lorentzian worldsheet approaches, we find heretofore overlooked contribution to
the spectral amplitudes and obervables. The contribution is fortuitously absent
for decay of unstable D-brane, but is present for rolling of accelerated
D-brane. We finally show that the contribution is imperative for ensuring
unitarity and optical theorem at each order in string loop expansion.
