SUSY QM - working out energy spectrum and wavefunctions from a given superpotential

Question

I'm currently self-studying F. Cooper and al.'s Supersymmetry in Quantum Mechanics, and I need help working out a particular case on shape-invariance.

From a given superpotential of the form $W(x)=ax^3$, where $a>1$, I can derive the two partner potentials: $$V_{2,1}=W(x)^2\pm W(x)'=a^2x^6\pm 3ax^2$$ I can also say that $V_1(x)$ and $V_2(x)$ are shape-invariant potentials and write: $$V_2(x;a_1)=V_1(x;a_2)+R(a_1)$$ where $a_1$ is a set of parameters, $a_2$ is a function of $a_1$ and $R(a_1)$ is independent of $x$.

From there, I am a bit lost. How do I work out the energy spectrum for the first few partner Hamiltonians?

This post imported from StackExchange Physics at 2014-10-23 21:54 (UTC), posted by SE-user Demosthene

Comments

$V_2-V_1$ depends on $x$ for all choices of $a_1$ and $a_2$. Why then do you think that you have shape invariance?