2004/12/20 by Igor Rodnianski, Terence Tao, Rodnianski, Igor +1
Mathematics · #35J10 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #math.AP #msc:35J10
paper · pdf · doi:10.48550/arxiv.math/0412416
29 pages, no figures, submitted, IAS conference proceedings. Minor typos fixed, some references added
openalex publication_date 2004/12/20 · arxiv created 2004/12/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we develop a quantitative version of Enss' method to establish global-in-time decay estimates for solutions to Schrödinger equations on manifolds. To simplify the exposition we shall only consider Hamiltonians of the form H := - 1/2 ΔM, where ΔM is the Laplace-Beltrami operator on a manifold M which is a smooth compact perturbation of three-dimensional Euclidean space \R3 which obeys the non-trapping condition. We establish a global-in-time local smoothing estimate for the Schrödinger equation ut = -iHu. The main novelty here is the global-in-time aspect of the estimates, which forces a more detailed analysis on the low and medium frequencies of the evolution than in the local-in-time theory. In particular, to handle the medium frequencies we require the RAGE theorem (which reflects the fact that H has no embedded eigenvalues), together with a quantitative version of Enss' method decomposing the solution asymptotically into incoming and outgoing components, while to handle the low frequencies we need a Poincare-type inequality (which reflects the fact that H has no eigenfunctions or resonances at zero).