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Search: All articles in the CJM digital archive with keyword scattering theory

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1. CJM Online first

Li, Dong; Xu, Guixiang; Zhang, Xiaoyi
On the Dispersive Estimate for the Dirichlet Schrödinger Propagator and Applications to Energy Critical NLS
We consider the obstacle problem for the Schrödinger evolution in the exterior of the unit ball with Dirichlet boundary condition. Under the radial symmetry we compute explicitly the fundamental solution for the linear Dirichlet Schrödinger propagator $e^{it\Delta_D}$ and give a robust algorithm to prove sharp $L^1 \rightarrow L^{\infty}$ dispersive estimates. We showcase the analysis in dimensions $n=5,7$. As an application, we obtain global well-posedness and scattering for defocusing energy-critical NLS on $\Omega=\mathbb{R}^n\backslash \overline{B(0,1)}$ with Dirichlet boundary condition and radial data in these dimensions.

Keywords:Dirichlet Schrödinger propagator, dispersive estimate, Dirichlet boundary condition, scattering theory, energy critical
Categories:35P25, 35Q55, 47J35

2. CJM 2011 (vol 63 pp. 961)

Bouclet, Jean-Marc
Low Frequency Estimates for Long Range Perturbations in Divergence Form
We prove a uniform control as $ z \rightarrow 0 $ for the resolvent $ (P-z)^{-1} $ of long range perturbations $ P $ of the Euclidean Laplacian in divergence form by combining positive commutator estimates and properties of Riesz transforms. These estimates hold in dimension $d \geq 3 $ when $ P $ is defined on $ \mathbb{R}^d $ and in dimension $ d \geq 2 $ when $ P $ is defined outside a compact obstacle with Dirichlet boundary conditions.

Keywords:resolvent estimates, thresholds, scattering theory, Riesz transform

3. CJM 2000 (vol 52 pp. 897)

Christiansen, T. J.; Joshi, M. S.
Higher Order Scattering on Asymptotically Euclidean Manifolds
We develop a scattering theory for perturbations of powers of the Laplacian on asymptotically Euclidean manifolds. The (absolute) scattering matrix is shown to be a Fourier integral operator associated to the geodesic flow at time $\pi$ on the boundary. Furthermore, it is shown that on $\Real^n$ the asymptotics of certain short-range perturbations of $\Delta^k$ can be recovered from the scattering matrix at a finite number of energies.

Keywords:scattering theory, conormal, Lagrangian

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