  A classic no-go theorem in one-dimensional quantum mechanics can be evaded
when the potentials are unbounded below, thus allowing for novel parity-paired
degenerate energy bound states. We numerically determine the spectrum of one
such potential and study the parametric variation of the transition wavelength
between a bound state lying inside the valley of the potential and another, von
Neumann-Wigner-like state, appearing above the potential maximum. We then
construct a modified potential which is bounded below except when a parameter
is tuned to vanish. We show how the spacing between certain energy levels
gradually decreases as we tune the parameter to approach the value for which
unboundedness arises, thus quantitatively linking the closeness of degeneracy
to the steepness of the potential. Our results are generic to a large class of
such potentials. Apart from their conceptual interest, such potentials might be
realisable in mesoscopic systems thus allowing for the experimental study of
the novel states. The numerical spectrum in this study is determined using the
asymptotic iteration method which we briefly review.
