Quality of the supersymmetric WKB quantization condition for non-shape-invariant potentials

The supersymmetric WKB (SWKB) approximation is known to give exact results for translationally shape-invariant potentials. To examine how this approximation fares for potentials which are not shape invariant, we investigate two such potentials: the double oscillator well and the finite square well. In the case of the double oscillator, where the energy levels are nearly degenerate, no analytic expression for the superpotential is possible and the SWKB approximation is far worse than the WKB approximation. But for the finite square well, where the energy spectrum is nondegenerate and an analytic expression for the superpotential is possible, the SWKB approximation is much better than the WKB approximation for almost all states. Thus the discreteness of the energy spectrum and the existence of an analytic expression for the superpotential are preconditions for applicability of the SWKB approximation.