Well-Posedness and Stability for Schrödinger Equations with Infinite Memory

We study in this paper the well-posedness and stability for two linear Schrödinger equations in d-dimensional open bounded domain under Dirichlet boundary conditions with an infinite memory. First, we establish the well-posedness in the sense 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 and some arguments devised in (Guesmia in J Math Anal Appl 382:748–760, 2011) and (Guesmia in Applicable Anal 94:184–217, 2015).