Abstract—In this work, we will define two new types of exponential asymptotic stability for the Euler-Maruyama numerical scheme in the stochastic controlled Schrödinger equation dependent on time, corresponding to a stochastic quantum two-level system, namely, almost sure exponential robust stability and asymptotic stability. These types of stability are variants of Mao and Mora definitions for the stability of numerical schemes, in the control case. Through the techniques used by Tsoi, we demonstrate the almost sure exponential robust stability and the asymptotic stability of the Euler-Maruyama numerical scheme for the stochastic controlled Schrödinger equation.

Index Terms—Asymptotic stability, Euler-Maruyama numerical scheme, exponential robust stability, time-dependent stochastic controlled Schrödinger equation.

Cutberto Romero-Meléndez is with Department of Basic Sciences, Autonomous Metropolitan University, Mexico. David Castillo-Fernández is with Engineering Department, La Salle University, México.

Cite:Cutberto Romero-Meléndez1, David Castillo-Fernández, "Numerical Exponential Stability of a Time-Dependent Stochastic Schrödinger Equation," International Journal of Applied Physics and Mathematics vol. 11, no. 1, pp. 1-8, 2021.

Copyright © 2021 by the authors. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).