Constructing quantum observables and self-adjoint extensions of symmetric operators. II. Differential operators

Abstract

We discuss a problem of constructing self-adjoint ordinary differential operators starting from self-adjoint differential expressions based on the general theory of self-adjoint extensions of symmetric operators outlined in [1]. We describe one of the possible ways of constructing in terms of the closure of an initial symmetric operator associated with a given differential expression and deficient spaces. Particular attention is focused on the features peculiar to differential operators, among them on the notion of natural domain and the representation of asymmetry forms generated by adjoint operators in terms of boundary forms. Main assertions are illustrated in detail by simple examples of quantum-mechanical operators like the momentum or Hamiltonian.