The semiclassical resolvent and the propagator for non-trapping scattering metrics

Abstract

Consider a compact manifold with boundary M with a scattering metric g or, equivalently, an asymptotically conic manifold (M • , g). (Euclidean R n , with a compactly supported metric perturbation, is an example of such a space.) Let ∆ be the positive Laplacian on (M, g), and V a smooth potential on M which decays to second order at infinity. In this paper we construct the kernel of the operator (h 2 ∆ + V − (λ 0 ± i0) 2 ) −1 , at a nontrapping energy λ 0 > 0, uniformly for h ∈ (0, h 0 ), h 0 > 0 small, within a class of Legendre distributions on manifolds with codimension three corners. Using this we construct the kernel of the propagator, e −it(∆/2+V ) , t ∈ (0, t 0 ) as a quadratic Legendre distribution. We also determine the global semiclassical structure of the spectral projector, Poisson operator and scattering matrix.

Approximation and Equidistribution of Phase Shifts: Spherical Symmetry

“…Indeed it is the graph of a symplectic transformation (ω, η) → (ω , η ), and the scattering matrix is a semiclassical Fourier integral operator associated to this symplectic graph [2], [10]. The sojourn time, however, carries extra information and is directly related to high-energy scattering asymptotics as observed in [16], [9], [10].…”

“…We use the fact, proven in [10], [2], that the integral kernel of S h is an oscillatory integral associated (in a manner we describe directly) to the Legendre submanifold L in (2.4). To be precise, the Schwartz kernel of S h can be decomposed following [10,Prop.…”

“…To be precise, the Schwartz kernel of S h can be decomposed following [10,Prop. 15] (with minor changes in notation) as…”

“…Melrose and Zworski [18] showed that for fixed h > 0 the absolute scattering matrix for a Schrödinger operator on a scattering, or asymptotically conic, manifold is an FIO associated to the geodesic flow on the manifold at infinity for time π. Alexandrova [2] studied the scattering matrix for a nontrapping semiclassical Schrödinger operator, and showed that localized to finite frequency, it is a semiclassical FIO associated to the limiting Hamilton flow relation at infinity, which includes the behavior of the Hamilton flow in compact sets. A more global description was given in Hassell-Wunsch [10] where the semiclassical asymptotics of the scattering matrix were unified with the singularities of the scattering matrix at fixed frequency (i.e. the Melrose-Zworski result [18]).…”

Strichartz estimates and nonlinear wave equation on nontrapping asymptotically conic manifolds

“…In the free Euclidean space, the half wave propagator has an explicit formula by using the Fourier transform, but in the asymptotically conical manifold it turns out to be quite complicated. From the results of [12,16], we have known that the Schwartz kernel of the spectral measure can be described as a Legendrian distribution on the compactification of the space M × M uniformly with respect to the spectral parameter λ. As pointed out in introduction, we really need to choose an operator partition of unity to microlocalize the spectral measure such that the spectral measure can be expressed in a formula capturing not only the size also the oscillatory behavior.…”