  A general solution to the "shutter" problem is presented. The propagation of
an arbitrary initially bounded wavefunction is investigated, and the general
solution for any such function is formulated. It is shown that the exact
solution can be written as an expression that depends only on the values of the
function (and its derivatives) at the boundaries. In particular, it is shown
that at short times ($t\ll2mx^2/\hbar$, where $x$ is the distance to the
boundaries) the wavefunction propagation depends only on the wavefunction's
values (or its derivatives) at the boundaries of the region. Finally, we
generalize these findings to a non-singular wavefunction (i.e., for wavepackets
with finite-width boundaries) and suggest an experimental verification.
