Does Huygens' principle hold for small perturbations of the wave equation?

The perturbed wave equation □u + q(x)u = 0 in $\text{R}$³ × $\text{R}$ with C∞ ($\text{R}$³) compactly supported initial data at t = 0 is considered. It is proven that the Huygens' principle does not hold for this equation if the potential is (essentially) non-negative, well-behaved at infinity and small in a suitable sense. The treatment is elementary and based on energy estimates and the positivity of the Riemann function for the wave equation in three space dimensions. The result still holds if the solution u is “small” over some space-time propagation cone. In the ease in which q has compact support, stronger results of this type for the above equation are obtained.