The symplectic geometry characterization of second order singular J-symmetric differential operators was investigated. By constructing different quotient spaces, self-adjoint extensions of second order J-symmetric differential operators were studied through using the method of symplectic geometry. Therefore the classification and description of complete J-Lagrangian submanifold corresponding with self-adjoint domains of second order differential operators were obtained.

Interior singular points were mainly studied in this paper,which means the characterization of self-adjoint domains for symmetric differential operators in the direct sum spaces. There exist the different deficiency indices at (n,n)singular points.Therefore by constructing different quotient spaces and using the method of symplectic geometry, it is possible to study self-adjoint extensions of symmetric differential operators in the direct sum spaces.The classification and description of complete Lagrangian submanifold that corresponds with self-adjoint domains of second order differential operators were also produced .

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 (2n,2n)singular points were studied. The classification and description of complete Lagrangian submanifold that correspond with self-adjoint domains of second order differential operators were given.