We expand the operators  and , 0 < h ≪ 1, modulo operators whose L1 → L∞ norm is ON(hN), ∀ N ≥ 1, where 𝜑, 𝜓 ∈  and V ∈ L∞(𝓡𝓃), 𝓃 ≥ 4, is a real-valued potential satisfying V(x) = O (⟨x⟩-𝛿), 𝛿 > (𝓃 + 1)/2 in the case of the wave equation and 𝛿 > (𝓃 + 2)/2 in the case of the Schr¨odinger equation. As a consequence, we give sufficent conditions in order that the wave and the Schr¨odinger groups satisfy dispersive estimates with a loss of ν derivatives, 0 ≤ ν ≤ (𝓃 − 3)/2. Roughly speaking, we reduce this problem to estimating the L1 → L∞ norms of a finite number of operators with almost explicit kernels. These kernels are completely explicit when 4 ≤ 𝓃 ≤ 7 in the case of the wave group, and when 𝓃 = 4, 5 in the case of the Schr¨odinger group.