  We study in this paper the well-posedness and stability for two linear
Schr\"odinger equations in $d$-dimensional open bounded domain under Dirichlet
boundary conditions with an infinite memory. First, we establish the
well-posedness in the sens of semigroup theory. Then, a decay estimate
depending on the smoothness of initial data and the arbitrarily growth at
infinity of the relaxation function is established for each equation with the
help of multipliers method.
