# Interpreting Argyres' spectrum of spontaneously broken SUSY QM

I can't understand the spectrum in the figure on page 19 from Argyres' lecture notes on supersymmetry: http://www.physics.uc.edu/~argyres/661/susy1996.pdf Argyres is considering a supersymmetric quantum mechanical system of an anharmonic oscillator, with a superpotential $W\sim x^3$. The plots of $W$ and $V$ make perfect sense. What doesn't make sense is the spectrum on the right.

Why are there both x's and o's over each Hamiltonian $H_1$ and $H_2$?? I thought $H_1$ is exclusively the spin-up Hamiltonian and $H_2$ is exclusively the spin-down Hamiltonian, so therefore the spectrum consist of a column of just x's over $H_1$ and a column just of o's over $H_2$.

Additional request: Would someone please write down the form of $H_1$ and $H_2$ in their answer, so to make sure we're on the same page? A graph of the the respective potentials $V_1$ and $V_2$ would be even better.

Take a look at the much more sensible figure on page 7. This is something that I can comprehend. It turns out that this is only true classically. Non-perturbative tunneling-effects (instantons) destroy this equivalence: the spectrum in which the vacuum is a spin-down state is no longer degenerate with the spin-up vacuum. This "lifting" of the energy levels is shown in the figure on page 19. $H_1$ and $H_2$ correspond to the aforementioned classically degenerate, but quantum mechanically non-degenerate ground states.
Despite the use of the letter $H$, $H_1$ and $H_2$ are not two different hamiltonians, but refer to the hamiltonian $H$ acting on the ground states $|1\rangle$ and $|2\rangle$. Furthermore, there is only one potential, i.e. the one in the plot you have included in your question.
• @QuantumDot: There is only one Hamiltonian, $H_1$ and $H_2$ refer to two different ground states. May 4 '14 at 11:45
• @QuantumDot: You get the respective energies by acting with the Hamiltonian $H$ on the ground states $|1\rangle$ and $|2\rangle$. May 4 '14 at 15:07