Effective limiting absorption principles, and applications

Abstract

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 $H$.