  The paper study a possibility to recover solutions of Schr\"odinger equations
from its time-averages in the setting where the values at the initial time are
unknown. This problem can be reformulated as a new boundary value problem where
a Cauchy condition is replaced by a prescribed time-average of the solution. It
is shown that this new problem is well-posed in certain classes of solutions.
The paper establishes existence, uniqueness, and a regularity of the solution
for this new problem.
