Exactness of the supersymmetric WKB approximation scheme

Exactness of the lowest-order supersymmetric WKB (SWKB) quantization condition integral(x1)(x2)root E-omega(2)(x)dx=nHBAR pi, for certain potentials, is examined, using the complex integration technique. Comparison of the above scheme with a similar, but exact quantization condition, closed integral(c)p(x,E)dx=2 pi nHBAR, originating from the quantum Hamilton-Jacobi formalism, reveals that the locations and the residues of the poles that contribute to these integrals match identically for both of these cases. As these poles completely determine the eigenvalues in these two cases, the exactness of the SWKB for these potentials is accounted for. Three nonexact cases are also analyzed; the origin of this nonexactness is shown to be due to the presence of additional singularities in root E-omega(2)(x), like branch cuts in the complex x plane.