  For a class of singular potentials, including the Coulomb potential (in three
and less dimensions) and $V(x) = g/x^2$ with the coefficient $g$ in a certain
range ($x$ being a space coordinate in one or more dimensions), the
corresponding Schr\"odinger operator is not automatically self-adjoint on its
natural domain. Such operators admit more than one self-adjoint domain, and the
spectrum and all physical consequences depend seriously on the self-adjoint
version chosen. The article discusses how the self-adjoint domains can be
identified in terms of a boundary condition for the asymptotic behaviour of the
wave functions around the singularity, and what physical differences emerge for
different self-adjoint versions of the Hamiltonian. The paper reviews and
interprets known results, with the intention to provide a practical guide for
all those interested in how to approach these ambiguous situations.

Quantum Physics