Zero mode in the fermion-vacuum sector ($f=C{e}^{\chi }$)

For the gradient superpotential Hamiltonian in supersymmetric quantum mechanics, $H_0={\sum }_a{A}_a^{†}{A}_a$. A zero-energy normalizable wavefunction therefore satisfies $A_{a}f=0$ for every $a$, because its energy is ${\sum }_a\Vert A_{a}f{\Vert }^2$. Equivalently ${\partial }_a(e^{-\chi }f)=0$, so on a connected configuration space $f=C{e}^{\chi }$. It exists in the Hilbert space precisely when $e^{\chi }$ is square-integrable. In three dimensions with radial $\chi$, this requires $4\pi {\int }_0^{\infty }{r}^2{e}^{2\chi (r)}dr<\infty$. The sign depends on the convention $A=\partial -\partial \chi$; the filled sector instead has the possible zero mode $e^{-\chi }$.