Effective limiting absorption principles, and applications

Details Effective limiting absorption principles, and applications Effective limiting absorption principles, and applications

2011-05-04

arxiv.org/abs/1105.0873v2

Mathematics

Analysis of PDEs

91 pages, no figures, submitted, CMP. Some references fixed, other minor corrections

Scientific paper

We investigate quantitative (or effective) versions of the limiting absorption principle, for the Schr\"odinger operator on asymptotically conic manifolds with short-range potentials, and in particular consider estimates of the form $$ \| R(\lambda+i\eps) f \|_{H^{0,-1/2-\sigma}} \leq C(\lambda, H) \| f \|_{H^{0,1/2+\sigma}}.$$ We are particularly interested in the exact nature of the dependence of the constants $C(\lambda,H)$ on both $\lambda$ and $H$. It turns out that the answer to this question is quite subtle, with distinctions being made between low energies $\lambda \ll 1$, medium energies $\lambda \sim 1$, and large energies $\lambda \gg 1$, and there is also a non-trivial distinction between "qualitative" estimates on a single operator $H$ (possibly obeying some spectral condition such as non-resonance, or a geometric condition such as non-trapping), and "quantitative" estimates (which hold uniformly for all operators $H$ in a certain class). Using elementary methods (integration by parts and ODE techniques), we give some sharp answers to these questions. As applications of these estimates, we present a global-in-time local smoothing estimate and pointwise decay estimates for the associated time-dependent Schr\"odinger equation, as well as an integrated local energy decay estimate and pointwise decay estimates for solutions of the corresponding wave equation, under some additional assumptions on the operator $H$.

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