Long-time decay estimates for the Schrödinger equation on manifolds
| dc.creator | Rodnianski, Igor | |
| dc.creator | Tao, Terence | |
| dc.date | 2004-12-20 | |
| dc.date | 2004-12-22 | |
| dc.date.accessioned | 2026-07-07T05:15:31Z | |
| dc.date.available | 2026-07-07T05:15:31Z | |
| dc.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). | |
| dc.description | 29 pages, no figures, submitted, IAS conference proceedings. Minor typos fixed, some references added | |
| dc.identifier | https://arxiv.org/abs/math/0412416 | |
| dc.identifier | http://arxiv.org/abs/math/0412416 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/73656 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 35J10 | |
| dc.title | Long-time decay estimates for the Schrödinger equation on manifolds | |
| dc.type | text |