Semiclassical estimates in asymptotically Euclidean scattering

Abstract

In this note we obtain semiclassical resolvent estimates for non-trapping long range perturbations of the Laplacian on asymptotically Euclidean manifolds. Our proof is based on a positive commutator argument which differs from Mourre-type estimates by making the commutant also positive. The resolvent estimates, including the weighting of the Sobolev spaces in the estimates, are an immediate consequence.

11 pages, 4 figures, AMS Latex

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• Gluing semiclassical resolvent estimates via propagation of singularities
• A sharp version of Price's law for wave decay on asymptotically flat spacetimes
• Local smoothing for scattering manifolds with hyperbolic trapped sets
• Semiclassical resolvent estimates for Schroedinger operators with Coulomb singularities
• Smoothing effect for Schrödinger boundary value problems
• Analyticity of quasinormal modes in the Kerr and Kerr-de Sitter spacetimes
• Sharp Strichartz estimates on non-trapping asymptotically conic manifolds
• Semi-classical resolvent estimates for l $\infty$ potentials on Riemannian manifolds
• Eigenvalue bounds for non-self-adjoint Schrödinger operators with non-trapping metrics
• Fractal upper bounds on the density of semiclassical resonances
• From resolvent estimates to damped waves
• Equidistribution of phase shifts in semiclassical potential scattering
• Resolvent estimates on asymptotically cylindrical manifolds and on the half line
• Fractal Weyl laws for asymptotically hyperbolic manifolds
• Strichartz estimates without loss outside two strictly convex obstacles
• Resonance free regions for nontrapping manifolds with cusps
• The heat kernel on an asymptotically conic manifold
• Semiclassical propagation through cone points
• Helmholtz and dispersive equations with variable coefficients on exterior domains
• Sharp norm estimates of layer potentials and operators at high frequency
• Positive commutators at the bottom of the spectrum
• The semiclassical resolvent and the propagator for nontrapping scattering metrics
• Exponential decay of eigenfunctions in many-body type scattering with second order perturbations
• Pseudodifferential operators on manifolds with a Lie structure at infinity
• Structure of the Semi-Classical Amplitude for General Scattering Relations
• Global-in-time Strichartz estimates for Schrodinger on scattering manifolds