Effective Limiting Absorption Principles, and Applications

The limiting absorption principle asserts that if H is a suitable Schrödinger operator, and f lives in a suitable weighted L² space (namely (Formula presented.) for some (Formula presented.)), then the functions (Formula presented.) converge in another weighted L² space (Formula presented.) to the unique solution u of the Helmholtz equation (Formula presented.) which obeys the Sommerfeld outgoing radiation condition. In this paper, we investigate more quantitative (or effective) versions of this principle, for the Schrödinger operator on asymptotically conic manifolds with short-range potentials, and in particular consider estimates of the form(Formula presented.)We are particularly interested in the exact nature of the dependence of the constants (Formula presented.) and H. It turns out that the answer to this question is quite subtle, with distinctions being made between low energies (Formula presented.), medium energies (Formula presented.), 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 Schrödinger 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.