Long-time decay estimates for the Schrödinger equation on manifolds
Abstract
Description
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 $H := - {1/2} Δ_M$, where $Δ_M$ is the Laplace-Beltrami operator on a manifold $M$ which is a smooth compact perturbation of three-dimensional Euclidean space $\R^3$ which obeys the non-trapping condition. We establish a global-in-time local smoothing estimate for the Schrödinger equation $u_t = -iHu$. 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).
29 pages, no figures, submitted, IAS conference proceedings. Minor typos fixed, some references added
29 pages, no figures, submitted, IAS conference proceedings. Minor typos fixed, some references added