  The quantum dynamics of a free particle on a circle with point interaction is
described by a U(2) family of self-adjoint Hamiltonians. We provide a
classification of the family by introducing a number of subfamilies and thereby
analyze the spectral structure in detail. We find that the spectrum depends on
a subset of U(2) parameters rather than the entire U(2) needed for the
Hamiltonians, and that in particular there exists a subfamily in U(2) where the
spectrum becomes parameter-independent. We also show that, in some specific
cases, the WKB semiclassical approximation becomes exact (modulo phases) for
the system.
