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

Abstract

Constructing physical observables as self-adjoint operators under quantum-mechanical description of systems with boundaries and/or singular potentials is a nontrivial problem. We present a comparative review of various methods for constructing ordinary self-adjoint differential operators associated with self-adjoint differential expressions based on the general theory of self-adjoint extensions of symmetric operators. The exposition is nontraditional and is based on the concept of asymmetry forms generated by adjoint operators. The main attention is given to a specification of self-adjoint extensions by self-adjoint boundary conditions. All the methods are illustrated by examples of quantum-mechanical observables like momentum and Hamiltonian. In addition to the conventional methods, we propose a possible alternative way of specifying self-adjoint differential operators by explicit self-adjoint boundary conditions that generally have an asymptotic form for singular boundaries. A comparative advantage of the method is that it allows avoiding an evaluation of deficient subspaces and deficiency indices. The effectiveness of the method is illustrated by a number of examples of quantum-mechanical observables.