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Longtime decay estimates for the Schrödinger equation on manifolds. (English) Zbl 1133.35022

Bourgain, Jean (ed.) et al., Mathematical aspects of nonlinear dispersive equations. Lectures of the CMI/IAS workshop on mathematical aspects of nonlinear PDEs, Princeton, NJ, USA, 2004. Princeton, NJ: Princeton University Press (ISBN 978-0-691-12955-6/pbk; 978-0-691-12860-3/hbk). Annals of Mathematics Studies 163, 223-253 (2007).
Let \((M,g)=({\mathbb R}^3,g)\) be a compact perturbation of Euclidean space \({\mathbb R}^3\), i.e. manifold \(M\) is \({\mathbb R}^3\) endowed with a smooth metric \(g\) outside of a ball \(\{x\in{\mathbb R}^3:| x| <R \}\) for some \(R\). It is assumed that \(M\) obeys the non-trapping condition. The authors consider smooth solutions to the Schrödinger equation \(u_t=-iHu, u(0)=u_0\), where \(H=-{1\over 2}\Delta_M\), \(\Delta_M\) is the Laplace-Beltrami operator on a manifold \(M\). The main result is the following global-in-time smoothing estimate
\[ \begin{split}\int_{\mathbb R}\left(\left\| (1+| x| ^2)^{-1/2}\nabla e^{-itH}u_0\right\| ^2_{L^2(M)}\right)\\ + \left\| (1+| x| ^2)^{-3/2-\sigma} e^{-itH}u_0 \right\| ^2_{L^2(M)} dt\leq C_{\sigma,M}\| u_0\| ^2_{H^{1/2}(M)}\end{split} \]
for any \(\sigma>0\). The proof is based on the decomposition of the evolution into low-energy, medium-energy, and high-energy components. The most difficult part is the treatment of the medium-energy component. For this purpose the quantitative version of V. Enss’ method [Commun. Math. Phys. 61, 285–291 (1978; Zbl 0389.47005)] is used.
For the entire collection see [Zbl 1113.35005].

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35J10 Schrödinger operator, Schrödinger equation
58J99 Partial differential equations on manifolds; differential operators

Citations:

Zbl 0389.47005
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