Semi-Classical Dispersive Estimates for the Wave and Schr¨odinger Equations with a Potential in Dimensions ð“ƒ â‰¥ 4
We expand the operators and , 0 < h â‰ª 1, modulo operators whose L1 â†’ Lâˆž norm is ON(hN), âˆ€ N â‰¥ 1, where ðœ‘, ðœ“ âˆˆ and V âˆˆ Lâˆž(ð“¡ð“ƒ), ð“ƒ â‰¥ 4, is a real-valued potential satisfying V(x) = O (âŒ©xâŒª-ð›¿), ð›¿ > (ð“ƒ + 1)/2 in the case of the wave equation and ð›¿ > (ð“ƒ + 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 Î½ derivatives, 0 â‰¤ Î½ â‰¤ (ð“ƒ âˆ’ 3)/2. Roughly speaking, we reduce this problem to estimating the L1 â†’ Lâˆž norms of a finite number of operators with almost explicit kernels. These kernels are completely explicit when 4 â‰¤ ð“ƒ â‰¤ 7 in the case of the wave group, and when ð“ƒ = 4, 5 in the case of the Schr¨odinger group.