In these lecture notes we describe the propagation of singularities of tempered distributional solutions $u\in {𝒮}^{\text{'}}$ of $(H-\lambda )u=0$, where $H$ is a many-body hamiltonian $H=\Delta +V$, $\Delta \ge 0$, $V={\sum }_a{V}_a$, and $\lambda$ is not a threshold of $H$, under the assumption that the inter-particle (e.g. two-body) interactions $V_a$ are real-valued polyhomogeneous symbols of order $-1$ (e.g. Coulomb-type with the singularity at the origin removed). Here the term “singularity” provides a microlocal description of the lack of decay at infinity. Our result is then that the...

For fixed magnetic quantum number m results on spectral properties and scattering theory are given for the three-dimensional Schrödinger operator with a constant magnetic field and an axisymmetrical electric potential V. In various, mostly singular settings, asymptotic expansions for the resolvent of the Hamiltonian H m+Hom+V are deduced as the spectral parameter tends to the lowest Landau threshold. Furthermore, scattering theory for the pair (H m, H om) is established and asymptotic expansions...

In these notes, we will describe recent work on globally solving quasilinear wave equations in the presence of trapped rays, on Kerr-de Sitter space, and obtaining the asymptotic behavior of solutions. For the associated linear problem without trapping, one would consider a global, non-elliptic, Fredholm framework; in the presence of trapping the same framework is available for spaces of growing functions only. In order to solve the quasilinear problem we thus combine these frameworks with the normally...

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...