Well-posedness and stability for Schr\"odinger equations with infinite memory

TL;DR

This paper investigates the well-posedness and stability of linear Schrödinger equations with infinite memory in bounded domains, establishing conditions for existence, uniqueness, and decay of solutions based on initial data and relaxation functions.

Contribution

It provides a rigorous analysis of well-posedness and decay estimates for Schrödinger equations with infinite memory, extending understanding of their stability properties.

Findings

Well-posedness established via semigroup theory

Decay estimates depend on initial data smoothness

Stability influenced by relaxation function growth

Abstract

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.