  Considerable attention has been recently focused on quantum-mechanical
systems with boundaries and/or singular potentials for which the construction
of physical observables as self-adjoint (s.a.) operators is a nontrivial
problem. We present a comparative review of various methods of specifying
ordinary s.a. differential operators generated by formally s.a. differential
expressions based on the general theory of s.a. extensions of symmetric
operators. The exposition is untraditional and is based on the concept of
asymmetry forms generated by adjoint operators. The main attention is given to
a specification of s.a. extensions by s.a. 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 s.a. differential operators by explicit s.a.
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.
