= Gradient superpotential Hamiltonian in supersymmetric quantum mechanics
{title2=$H=\{q,q^\dagger\}$}
For a real smooth function $\chi$ on <Euclidean space>, put $A_a=\partial_a-\partial_a\chi$ and $q=\sum_a\psi_a^\dagger A_a$. The <adjoint operator> for the usual $L^2$ <inner product> is $A_a^\dagger=-\partial_a-\partial_a\chi$. Since $[A_a,A_b]=0$, the <canonical anticommutation relations> give $q^2=0$. Expanding $H=\{q,q^\dagger\}$ and using $[A_a,A_b^\dagger]=-2\partial_a\partial_b\chi$ gives
$$
H=(-\Delta+|\nabla\chi|^2+\Delta\chi)I-2\sum_{a,b}(\partial_a\partial_b\chi)\psi_a^\dagger\psi_b.
$$
In the empty <fermionic Fock space> sector the last term vanishes; in the filled sector $\psi_a^\dagger\psi_b=\delta_{ab}$. 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 $\langle v,Hv\rangle=\|qv\|^2+\|q^\dagger v\|^2$.
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