How good is the supersymmetry-inspired WKB quantization condition?

Abstract

I show that unlike the standard lowest order WKB, the supersymmetry-inspired WKB quantization condition gives correct n-dependence for all the energy eigenvalues (except ground state) of the potential V(x) = A²x⁶ + 3Ax². I also display a number of other potentials for which SWKB is then shown to give exact eigenvalues. Finally, I conjecture that for the class of models given by V(x) = A²x^4d+2 + (2d + 1) Ax^2d (d = 0,1,2,..), the exact energy eigenvalues obey the SWKB-predicted energy dependence E_n ∞ (n + 1)^(2d+1)/(d+1).