Semi-Classical Dispersive Estimates for the Wave and Schr¨odinger Equations with a Potential in Dimensions 𝓃 ≥ 4

Abstract

We expand the operators  and , 0 < h ≪ 1, modulo operators whose L¹ → L^∞ norm is O_N(h^N), ∀ N ≥ 1, where ðœ‘, 𜓠∈  and V ∈ L^∞(ð“¡^ð“ƒ), 𓃠≥ 4, is a real-valued potential satisfying V(x) = O (〈x〉^-ð›¿), 𛿠> (𓃠+ 1)/2 in the case of the wave equation and 𛿠> (𓃠+ 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 ≤ (𓃠− 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 ≤ ð“ƒ ≤ 7 in the case of the wave group, and when ð“ƒ = 4, 5 in the case of the Schr¨odinger group.