研究方向：

几何数值方法，哈密尔顿系统，微分算子谱理论

主要成果：

1.   发展了保体积系统的生成函数理论, 给出无源系统保体积算法的一般性构造方法(其中部分成果与冯康合作)
2.   发现计算不变环面时的步长共振现象并给出步长远离共振的Diophantine条件，证明了Diophantine时间步长集合的大测度性质,证明了辛几何算法的KAM (Kolmogorov-Arnold-Moser)定理
3.   证明了高维小扭转辛映射不变环面的存在性（Moser小扭转定理的高维推广）, 给出辛映射情形KAM定理的完整证明以及有关重要估计
4.   给出奇异常微分算式J-自伴边界条件的完整解析描述（获1993年国家教委科技进步二等奖，排名第二）

发表论著：

1.   Shang Zaijiu: Stability of symplectic integrators, Preprint 2008.
2.   Shang Zaijiu, Song Lina: Exponentially small splittings of homoclinic orbits of Hamiltonian systems under symplectic discretizations, Preprint 2008.
3.   Shang Zaijiu: Volume-preserving maps, source-free systems and their local structures, Journal of Physics A: Mathematical and General, 39:19 (2006), 5601-561.
4.   Shang Zaijiu: Resonant and Diophantine step sizes in computing invariant tori of Hamiltonian systems. Nonlinearity 13 (2000), 299-308.
5.   Shang Zaijiu: A note on the KAM theorem for symplectic mappings. Journal of Dynamics and Differential Equations 12 (2000), 357-383.
6.   Shang Zaijiu: KAM theorem of symplectic algorithms for Hamiltonian systems. Numerische Mathematik 83 (1999), 477-496.
7.   Feng Kang and Shang Zai-jiu: Volume-preserving algorithms for source-free dynamical systems. Numerische Mathematik 71 (1995), 451-463.
8.   Shang Zaijiu: Generating functions for volume-preserving mappings and Hamilton-Jacobi equations for source-free systems. Science in China (Series A) 37 (1994), 1172-1188.
9.   Shang Zaijiu: On the construction of the volume-preserving difference schemes for source-free systems via generating functions. Journal of Computational Mathematics 12 (1994), 265-272.
10. Shang Zaijiu: On J-selfadjoint extensions of J-symmetric ordinary differential operators. Journal of Differential Equations 73 (1988), 153-177.