Position-momentum uncertainty products

We point out two interesting features of position-momentum uncertainty product: $U=\mathrm{\Delta }x\mathrm{\Delta }p$. We show that two special (non-differentiable) eigenstates of the Schr{"o}dinger operator with the Dirac Delta potential $[V(x)=-V_0\delta (x)],V_0>0$, also satisfy the Heisenberg’s uncertainty principle by yielding $U>\frac{\mathrm{\hslash }}{2}$. One of these eigenstates is a zero-energy and zero-curvature bound state.