= Zero mode in the fermion-vacuum sector
{title2=$f=Ce^\chi$}
For the <gradient superpotential Hamiltonian in supersymmetric quantum mechanics>, $H_0=\sum_aA_a^\dagger A_a$. A zero-energy <normalizable wavefunction> therefore satisfies $A_af=0$ for every $a$, because its energy is $\sum_a\|A_af\|^2$. Equivalently $\partial_a(e^{-\chi}f)=0$, so on a connected configuration space $f=Ce^\chi$. It exists in the <Hilbert space> precisely when $e^\chi$ is <square-integrable>. In three dimensions with radial $\chi$, this requires $4\pi\int_0^\infty r^2e^{2\chi(r)}\,dr<\infty$. The sign depends on the convention $A=\partial-\partial\chi$; the filled sector instead has the possible zero mode $e^{-\chi}$.
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