Effective Limiting Absorption Principles, and Applications

Abstract

The limiting absorption principle asserts that if H is a suitable Schrodinger operator, and f lives in a suitable weighted L (2) space (namely for some ), then the functions converge in another weighted L (2) space to the unique solution u of the Helmholtz equation which obeys the Sommerfeld outgoing radiation condition. In this paper, we investigate more quantitative (or effective) versions of this principle, for the Schrodinger operator on asymptotically conic manifolds with short-range potentials, and in particular consider estimates of the form parallel to R (lambda + i epsilon) f parallel to (H)0,0-1/2-sigma <= C (lambda, H) parallel to f parallel to (H0,1/2+0) We are particularly interested in the exact nature of the dependence of the constants on both and H. It turns out that the answer to this question is quite subtle, with distinctions being made between low energies , medium energies , and large energies , 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 Schrodinger 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.