Order reduction-based uniform approximation of exponential stability for one-dimensional Schrödinger equation

Abstract

This paper considers the uniform exponential stability approximation of a one-dimensional Schrödinger system with boundary damping. The continuous system is known to be exponentially stable. Firstly, the order reduction method is adopted to transform the original system into an equivalent one. Two second-order semi-discretized finite difference schemes are derived for both the transformed system and the original system, which are shown to be equivalent to each other. Secondly, the Lyapunov function method is used to prove the uniform exponential stability of the semi-discretized transformed system, which is parallel to the proof of the continuous transformed system. Finally, the solution of the semi-discretized system converges weakly to the solution of the original system and the discrete energy converges to the continuous energy.