  We point out that bound states, degenerate in energy but differing in parity,
may form in one dimensional quantum systems even if the potential is
non-singular in any finite domain. Such potentials are necessarily unbounded
from below at infinity and occur in several different contexts, such as in the
study of localised states in brane-world scenarios. We describe how to
construct large classes of such potentials and give explicit analytic
expressions for the degenerate bound states. Some of these bound states occur
above the potential maximum while some are below. Various unusual features of
the bound states are described and after highlighting those that are ansatz
independent, we suggest that it might be possible to observe such parity-paired
degenerate bound states in specific mesoscopic systems.

Quantum Physics;Mesoscale and Nanoscale Physics;High Energy Physics - Theory