We consider perturbations of a stratified medium ${ℝ}_x^{n-1}\times {ℝ}_y$, where the operator studied is $c^2(x,y)\Delta$. The function $c$ is a perturbation of $c_0\left(y\right)$, which is constant for sufficiently large $\left|y\right|$ and satisfies some other conditions. Under certain restrictions on the perturbation $c$, we give results on the Fourier integral operator structure of the scattering matrix. Moreover, we show that we can recover the asymptotic expansion at infinity of $c$ from knowledge of $c_0$ and the singularities of the scattering matrix at fixed energy....

The problem of recovering the singularities of a potential from backscattering data is studied. Let $\Omega$ be a smooth precompact domain in ${ℝ}^n$ which is convex (or normally accessible). Suppose $V_i=v+w_i$ with $v\in C_c^{\infty }({ℝ}^n)$ and $w_i$ conormal to the boundary of $\Omega$ and supported inside $\bar{\Omega }$ then if the backscattering data of $V_1$ and $V_2$ are equal up to smoothing, we show that $w_1-w_2$ is smooth.

We consider Schrödinger operators $H$ on ${ℝ}^n$ with variable coefficients. Let $H_0=-\frac{1}{2}▵$ be the free Schrödinger operator and we suppose $H$ is a “short-range” perturbation of $H_0$. Then, under the nontrapping condition, we show that the time evolution operator: $e^{-itH}$ can be written as a product of the free evolution operator $e^{-it{H}_0}$ and a Fourier integral operator $W\left(t\right)$ which is associated to the canonical relation given by the classical mechanical scattering. We also prove a similar result for the wave operators. These results...

2000 Mathematics Subject Classification: 35P25, 81U20, 35S30, 47A10, 35B38.We study the microlocal structure of the resolvent of the semiclassical Schrödinger operator with short range potential at an energy which is a unique non-degenerate global maximum of the potential. We prove that it is a semiclassical Fourier integral operator quantizing the incoming and outgoing Lagrangian submanifolds associated to the fixed hyperbolic point. We then discuss two applications of this result to describing...

We consider the 3D Schrödinger operator $H=H_0+V$ where $H_0={(-i\nabla -A)}^2-b$, $A$ is a magnetic potential generating a constant magneticfield of strength $b\>0$, and $V$ is a short-range electric potential which decays superexponentially with respect to the variable along the magnetic field. We show that the resolvent of $H$ admits a meromorphic extension from the upper half plane to an appropriate Riemann surface $ℳ$, and define the resonances of $H$ as the poles of this meromorphic extension. We study their distribution near any fixed...

Nous étudions les résonances de Rayleigh créées par un obstacle strictement convexe à bord analytique en dimension 2. Nous montrons qu’il existe exactement deux suites de résonances $\left(z_{k,+}\right)$ et $\left(z_{k,-}\right)$ convergeant exponentiellement vite vers l’axe réel dans un voisinage polynomial de l’axe réel, et exponentiellement proches d’une suite de quasimodes réels. De plus, $k^{-1}\Re z_{k,\pm }$ est un symbole analytique d’ordre 0 en la variable $k^{-1}$ dont on donne le premier terme du développement. Nous construisons pour cela des quasimodes...