Self-Adjoint Operators. SchrÖdinger Operators

Abstract

In Section 2.1 we define symmetric and self-adjoint operators and give criteria for a symmetric operator to be self-adjoint. In Section 2.2 we study simple spectral properties of self-adjoint operators. A particular class of self-adjoint operators, the socalled multiplication operators, are introduced in Section 2.3, and the results are applied to proving the essential self-adjointness of the Laplacian. In Section 2.4 we give a criterion for the invariance of self-adjointness under perturbations and apply it to Schrödinger operators with non-singular potentials. Finally, in Section 2.5, we give a characterization of the domain of Schrödinger operators with strongly singular potentials. The importance of self-adjointness will be discussed in Section 4.1.