The Symplectic Geometry Characterization Of Seif-Adjoint Domains Of Symmetric Differential Operators In Direct Sum Spaces

In this paper we investigate the characterization of self-adjoint domains of symmetric differential operators with interior singular points in the direct sum spaces. Through constructing different quotient spaces, using the method of symmetric spaces, we study self-adjoint extensions of symmetric differential operators in the direct sum spaces for the different deficiency indices at singular points. We give the classification and description of self-adjoint domains by complete Lagrangian submanifold, and necessary and sufficient conditions for K-grade self-adjoint domain of differential operators are obtained.