High frequency dispersive estimates for the Schrödinger equation in high dimensions

Article type: Research Article

Authors: Cardoso, Fernando | Cuevas, Claudio | Vodev, Georgi^;

Affiliations: Departamento de Matemática, Universidade Federal de Pernambuco, CEP. 50540-740 Recife-Pe, Brazil. E-mails: {fernando, cch}@dmat.ufpe.br | Département de Mathématiques, Université de Nantes, UMR 6629 du CNRS, 2, rue de la Houssinière, BP 92208, 44332 Nantes Cedex 03, France. E-mail: vodev@math.univ-nantes.fr

Note: [] Corresponding author.

Abstract: We prove optimal dispersive estimates at high frequency for the Schrödinger group for a class of real-valued potentials V(x)=O(〈x〉−δ), δ>n−1, and V∈Ck(Rn), k>kn, where n≥4 and (n−3)/2≤kn<n/2. We also give a sufficient condition in terms of L1→L∞ bounds for the formal iterations of Duhamel's formula, which might be satisfied for potentials of less regularity.

Keywords: dispersive estimates, high frequency

DOI: 10.3233/ASY-2010-1020

Journal: Asymptotic Analysis, vol. 71, no. 4, pp. 207-225, 2011