On The Self-Adjoint Extensions Of Differential Operator

There are two ways to extend a symmetric operator to a self-adjoint one, Von Neumann's and Calkin's. In this paper, we discuss the relation of the two methods. Then we realize the domain of a abstract Von Neumann's self-adjoint extension as the boundary condition description for the ordinary differential operator. These results are helpful when we discuss the spectra of self-adjoint operators getting by Von. Neumann's extension. On the other hand, we also realize the domain of a self-adjointness of boundary condition form as the Von Neumann description.