  To describe a quantum system whose potential is divergent at one point, one
must provide proper connection conditions for the wave functions at the
singularity. Generalizing the scheme used for point interactions in one
dimension, we present a set of connection conditions which are well-defined
even if the wave functions and/or their derivatives are divergent at the
singularity. Our generalized scheme covers the entire U(2) family of
quantizations (self-adjoint Hamiltonians) admitted for the singular system. We
use this scheme to examine the spectra of the Coulomb potential $V(x) = - e^2 /
| x |$ and the harmonic oscillator with square inverse potential $V(x) = (m
\omega^2 / 2) x^2 + g/x^2$, and thereby provide a general perspective for these
models which have previously been treated with restrictive connection
conditions resulting in conflicting spectra. We further show that, for any
parity invariant singular potentials $V(-x) = V(x)$, the spectrum is determined
solely by the eigenvalues of the characteristic matrix $U \in U(2)$.
