中文摘要：

运用Legendre谱方法研究Schrodinger方程与具有黏性阻尼波方程耦合组成的系统稳定性问题．首先通过分析得到耦合系统能量不增长，再利用Legendre谱方法对该系统特征方程进行数值化，并利用MATLAB计算得到该耦合系统的谱分布，从而根据系统特征根全部分布在复平面的左半平面并距离虚轴一定距离得到该耦合系统达到了强稳定．最后将Legendre谱方法应用到热方程与Schrodinger耦合系统的稳定性．

英文摘要：

An Legendre spectral method is used to study the stability of a SchrSdinger equation coupled with a damped wave equation which acts a dynamic controller for the SchrSdinger equation. The graphic numerical solutions are presented. Remarkably, the spectrum generated by the closed-loop system is distributed into the left half-plane of the complex plane, which notes that the system is dissipative. Combining the energy of the system is not increasing, we can see the coupled system achieves the strong stability. At last, the Legendre spectral method is also used to verify the stability of a SchrSdinger system coupled with a heat equation.