Splitting of degenerate states in one-dimensional quantum mechanics

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.