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1. CJM 2014 (vol 66 pp. 1110)

Li, Dong; Xu, Guixiang; Zhang, Xiaoyi
 On the Dispersive Estimate for the Dirichlet SchrÃ¶dinger Propagator and Applications to Energy Critical NLS We consider the obstacle problem for the SchrÃ¶dinger 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 SchrÃ¶dinger 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 SchrÃ¶dinger propagator, dispersive estimate, Dirichlet boundary condition, scattering theory, energy criticalCategories:35P25, 35Q55, 47J35

2. CJM 2013 (vol 65 pp. 1095)

Sambou, Diomba
 RÃ©sonances prÃ¨s de seuils d'opÃ©rateurs magnÃ©tiques de Pauli et de Dirac Nous considÃ©rons les perturbations $H := H_{0} + V$ et $D := D_{0} + V$ des Hamiltoniens libres $H_{0}$ de Pauli et $D_{0}$ de Dirac en dimension 3 avec champ magnÃ©tique non constant, $V$ Ã©tant un potentiel Ã©lectrique qui dÃ©croÃ®t super-exponentiellement dans la direction du champ magnÃ©tique. Nous montrons que dans des espaces de Banach appropriÃ©s, les rÃ©solvantes de $H$ et $D$ dÃ©finies sur le demi-plan supÃ©rieur admettent des prolongements mÃ©romorphes. Nous dÃ©finissons les rÃ©sonances de $H$ et $D$ comme Ã©tant les pÃ´les de ces extensions mÃ©romorphes. D'une part, nous Ã©tudions la rÃ©partition des rÃ©sonances de $H$ prÃ¨s de l'origine $0$ et d'autre part, celle des rÃ©sonances de $D$ prÃ¨s de $\pm m$ oÃ¹ $m$ est la masse d'une particule. Dans les deux cas, nous obtenons d'abord des majorations du nombre de rÃ©sonances dans de petits domaines au voisinage de $0$ et $\pm m$. Sous des hypothÃ¨ses supplÃ©mentaires, nous obtenons des dÃ©veloppements asymptotiques du nombre de rÃ©sonances qui entraÃ®nent leur accumulation prÃ¨s des seuils $0$ et $\pm m$. En particulier, pour une perturbation $V$ de signe dÃ©fini, nous obtenons des informations sur la rÃ©partition des valeurs propres de $H$ et $D$ prÃ¨s de $0$ et $\pm m$ respectivement. Keywords:opÃ©rateurs magnÃ©tiques de Pauli et de Dirac, rÃ©sonancesCategories:35B34, 35P25

3. 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 transformCategory:35P25

4. CJM 2004 (vol 56 pp. 794)

Michel, Laurent
 Semi-Classical Behavior of the Scattering Amplitude for Trapping Perturbations at Fixed Energy We study the semi-classical behavior as $h\rightarrow 0$ of the scattering amplitude $f(\theta,\omega,\lambda,h)$ associated to a Schr\"odinger operator $P(h)=-\frac 1 2 h^2\Delta +V(x)$ with short-range trapping perturbations. First we realize a spatial localization in the general case and we deduce a bound of the scattering amplitude on the real line. Under an additional assumption on the resonances, we show that if we modify the potential $V(x)$ in a domain lying behind the barrier $\{x:V(x)>\lambda\}$, the scattering amplitude $f(\theta,\omega,\lambda,h)$ changes by a term of order $\O(h^{\infty})$. Under an escape assumption on the classical trajectories incoming with fixed direction $\omega$, we obtain an asymptotic development of $f(\theta,\omega,\lambda,h)$ similar to the one established in thenon-trapping case. Categories:35P25, 35B34, 35B40

5. CJM 2000 (vol 52 pp. 119)

Edward, Julian
 Corrigendum to Spectral Theory for the Neumann Laplacian on Planar Domains with Horn-Like Ends'' Errors to a previous paper (Canad. J. Math. (2) {\bf 49}(1997), 232--262) are corrected. A non-standard regularisation of the auxiliary operator $A$ appearing in Mourre theory is used. Categories:35P25, 58G25, 47F05

6. CJM 1997 (vol 49 pp. 232)

Edward, Julian
 Spectral theory for the Neumann Laplacian on planar domains with horn-like ends The spectral theory for the Neumann Laplacian on planar domains with symmetric, horn-like ends is studied. For a large class of such domains, it is proven that the Neumann Laplacian has no singular continuous spectrum, and that the pure point spectrum consists of eigenvalues of finite multiplicity which can accumulate only at $0$ or $\infty$. The proof uses Mourre theory. Categories:35P25, 58G25
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