  The boundary state associated with the rolling tachyon solution on an
unstable D-brane contains a part that decays exponentially in the asymptotic
past and the asymptotic future, but it also contains other parts which either
remain constant or grow exponentially in the past or future. We argue that the
time dependence of the latter parts is completely determined by the requirement
of BRST invariance of the boundary state, and hence they contain information
about certain conserved charges in the system. We also examine this in the
context of the unstable D0-brane in two dimensional string theory where these
conserved charges produce closed string background associated with the discrete
states, and show that these charges are in one to one correspondence with the
symmetry generators in the matrix model description of this theory.
