Abstract: The characterization of self-adjoint domains for symmetric differential operators with interior singular points in the direct sum spaces was investigated. By constructing different quotient spaces, using the method of symplectic geometry, self-adjoint extensions of symmetric differential operators in the direct sum spaces for the different deficiency indices at (2, 2)and(1,1)singular points was studied. The classification and description of complete Lagrangian submanifold that correspond with self-adjoint domains of second order differential operators were given.

Key words: Differential operators , Symplectic spaces , Lagrangian submanifold , Singular points , Direct sum spaces

摘要： 研究了具有内部奇异点,即直和空间上的二阶对称微分算子辛几何刻画问题。由于对称微分算子在
端点处的亏指数取值情况不同,当微分算子在端点处分别取(2,2)和(1,1)时,通过构造商空间,应用辛几何的方法
讨论了直和空间的对称微分算子的自共轭扩张问题。给出了与二阶微分算子自共轭域相对应的完全Lagrangian子
流型的分类与描述。

关键词: 微分算子 , 辛空间 ,  Lagrangian子流型 , 奇异点 , 直和空间