Semiclassical Estimates in Asymptotically Euclidean Scattering

The purpose of this note is to obtain semiclassical resolvent estimates for long range perturbations of the Laplacian on asymptotically Euclidean manifolds. For an estimate which is uniform in the Planck constant h we need to assume that the energy level is non-trapping. In the high energy limit (that is, when we consider ∆ − λ, as λ → ∞, which is equivalent to h∆ − 1, h → 0), this corresponds to the global assumption that the geodesic flow is non-trapping. We note here that a sufficiently small neighbourhood of infinity is always non-trapping. The resolvent estimates in the classical (h = 1) and semi-classical cases have a long tradition going back to the limiting absorption principle – see [1] and references given there. Various variants of the theorem we present were proved in Euclidean potential scattering by JensenMourre-Perry [9], Robert-Tamura [14], Gérard-Martinez [5], Gérard [4] and Wang [15], and for more general elliptic operators by Robert [12],[13]. The proofs were based on Mourre theory whose underlying feature is the positive commutator method accompanied by functional analytic techniques for obtaining a resolvent estimate. While the work of Gérard-Martinez [5] explains the role of geometry in the positive commutator estimate itself, it refers to Mourre’s work for the functional analytic argument. We adopt a completely geometric approach based on direct microlocal ideas. The classical version of the estimate on asymptotically Euclidean manifolds (h = 1 in which case there is no need for the non-trapping assumption) is essentially in Melrose’s original paper on the subject [10] in which he introduced a fully microlocal point of view to scattering. However, the proof presented here is somewhat different in spirit: a global positive commutator argument is used to derive an estimate on the resolvent directly. Referring to (2.3) below for the definition of a scattering metric, to (2.4),(2.5) for the definition of a long range semi-classical perturbation, and (2.7) for the definition of a non-trapping energy, we state our main