Abstract: We show that the positive supersymmetric matrix-valued differential operator $H = p_x^2 + p_y^2 + x^2y^2 + x \sigma_3 + y \sigma_1$ has no zero modes, i.e. $H \psi = 0$ implies $\psi = 0$. 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. In addition we discuss other related models, i.e. special cases of the Witten Laplacian with continuous spectrum down to zero, where the existence or the absence of zero modes can be shown.