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.
@article{1512,
title = {Semi-Classical Dispersive Estimates for the Wave and Schr"odinger Equations with a Potential in Dimensions n >= 4},
journal = {CUBO, A Mathematical Journal},
volume = {10},
year = {2008},
language = {en},
url = {http://dml.mathdoc.fr/item/1512}
}
Cardoso, F.; Vodev, G. Semi-Classical Dispersive Estimates for the Wave and Schr¨odinger Equations with a Potential in Dimensions 𝓃 ≥ 4. CUBO, A Mathematical Journal, Tome 10 (2008) . http://gdmltest.u-ga.fr/item/1512/