Gradient superpotential Hamiltonian in supersymmetric quantum mechanics ($H=\{q,q^{†}\}$)

For a real smooth function $\chi$ on Euclidean space, put $A_a={\partial }_a-{\partial }_a\chi$ and $q={\sum }_a{\psi }_a^{†}{A}_a$. The adjoint operator for the usual $L^2$ inner product is $A_a^{†}=-{\partial }_a-{\partial }_a\chi$. Since $[A_a,A_b]=0$, the canonical anticommutation relations give $q^2=0$. Expanding $H=\{q,q^{†}\}$ and using $[A_a,A_b^{†}]=-2{\partial }_a{\partial }_b\chi$ givesIn the empty fermionic Fock space sector the last term vanishes; in the filled sector ${\psi }_a^{†}{\psi }_b={\delta }_{ab}$. Consequently the two scalar Hamiltonian operators are $H_0=-\Delta +|\nabla \chi {|}^2+\Delta \chi$ and $H_d=-\Delta +|\nabla \chi {|}^2-\Delta \chi$. On the compatible self-adjoint operator domain the Hamiltonian operator is nonnegative because $⟨v,Hv⟩=\Vert qv{\Vert }^2+\Vert q^{†}v{\Vert }^2$.

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 $e^{-\chi }$.