Energy Decay of the Schrödinger Equation Under Internal Fractional Integral Damping

In this article, we investigate the stabilization and energy decay properties of a Schrödinger equation with an internal fractional integral damping term. By introducing an appropriate diffusive representation, the original problem is reformulated as an augmented system without memory. The well-posedness of the resulting system is established using semigroup theory, ensuring the existence and uniqueness of solutions. We then prove the strong stability of the associated semigroup by means of the Arendt–Batty theorem. Furthermore, by combining multiplier techniques with a frequency domain approach, we derive an explicit polynomial decay rate for the energy of solutions. More precisely, the energy decays at a rate of order t−2/1−α, which is shown to be optimal for general initial data. These results highlight the effectiveness of fractional integral damping mechanisms in enhancing the stabilization of Schrödinger-type equations.