No Zero Energy States for the Supersymmetric x2y2 Potential

We show that the positive supersymmetric matrix-valued differential operator H = px2 + py2 + x2y2 + xσ3 + yσ1 has no zero modes, i.e., Hψ = 0 implies ψ = 0. The result depends on a virial type argument for the corresponding supercharge. The model may be regarded as a simple relative of dimensional reductions of supersymmetric Yang–Mills theories.