Related papers: The scattering map for the Schrodinger operator on…
We consider the scattering theory for the Schr\"odinger operator $-\Dc_x^2+V(x)$ on graphs made of one-dimensional wires connected to external leads. We derive two expressions for the scattering matrix on arbitrary graphs. One involves…
This monograph is an expanded version of the preprint arXiv:1402.1716 or hal-00943396v1.It is devoted to the dynamics on Sobolev spaces of the cubic Szeg{\"o} equation on the circle ${\mathbb S} ^1$,$$ i\partial \_t u=\Pi (\vert u\vert…
The inverse scattering approach for the defocusing Davey-Stewartson II equation is given by a system of D-bar equations. We present a numerical approach to semi-classical D-bar problems for real analytic rapidly decreasing potentials. We…
Using the recent analysis of the output of the low-energy resolvent of Schr\"odinger operators on asymptotically conic manifolds (including Euclidean space) when the potential is short-range, we produce detailed asymptotic expansions for…
The first target of this article is the local well-posedness question for 1D quasilinear Schr\"odinger equations with cubic nonlinearities. The study of this class of problems, in all dimensions, was initiated in pioneering work of…
We consider the Chern-Simons-Schr\"odinger model in 1+2 dimensions, and prove scattering for small solutions of the Cauchy problem in the Coulomb gauge. This model is a gauge covariant Schr\"odinger equation, with a potential decaying like…
In this paper, partly based on Zachos' PhD thesis, we show that the geodesic X-ray transform is stably invertible near infinity on a class of asymptotically conic manifolds which includes perturbations of Euclidean space. In particular…
Evaluation of a product integral with values in the Lie group SU(1,1) yields the explicit solution to the impedance form of the Schr\"odinger equation. Explicit formulas for the transmission coefficient and $S$-matrix of the classical…
We consider the cubic nonlinear Schr\"odinger equation with long-range linear potentials in one space dimension, and prove the modified scattering in the energy space for the associated final state problem with a prescribed small asymptotic…
Let -Delta+V be the Schrodinger operator acting on L^2(R^d,C) with d odd larger than 2. Here V is a bounded real- or complex-valued function vanishing outside the closed ball of center 0 and radius a. If V belongs to the class of potentials…
In this paper we study microlocal singularities of solutions to Schrodinger equations on scattering manifolds, i.e., noncompact Riemannian manifolds with asymptotically conic ends. We characterize the wave front set of the solutions in…
We present an inverse scattering construction of generalised point interactions (GPI) -- point-like objects with non-trivial scattering behaviour. The construction is developed for single centre $S$-wave GPI models with rational…
We discuss resonances for Schr\"odinger operators on metric graphs which consists of a finite compact part and a finite number of halflines attached to it; the vertex coupling is assumed to be of the $\delta$-type or certain modifications…
For the solution of the cubic nonlinear Schr\"odinger equation in one space dimension, we propose and analyse a fully discrete low-regularity integrator. The scheme is explicit and can easily be implemented using the fast Fourier transform…
In this paper, we study the long time behavior of solutions to the defocusing Calogero--Moser derivative nonlinear Schr\"odinger equation (CM-DNLS). Using the G\'erard-type explicit formula, we prove the scattering result of solutions to…
We consider the Schr\"odinger operator with a periodic potential on quasi-1D models of armchair single-wall nanotubes. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite…
For any positive real number $s$, we study the scattering theory in a unified way for the fractional Schr\"{o}dinger operator $H=H_0+V$, where $H_0=(-\Delta)^\frac s2$ and the real-valued potential $V$ satisfies short range condition. We…
We analyze properties of semigroups generated by Schr\"odinger operators $-\Delta+V$ or polyharmonic operators $-(-\Delta)^m$, on metric graphs both on $L^p$-spaces and spaces of continuous functions. In the case of spatially constant…
This study is devoted to the asymptotic spectral analysis of multiscale Schr\"odinger operators with oscillating and decaying electric potentials. Different regimes, related to scaling considerations, are distinguished. By means of a normal…
In this paper we consider scattering theory on manifolds with special cusp-like metric singularities of warped product type g=dx^2 + x^(-2a)h, where a>0. These metrics form a natural subset in the class of metrics with warped product…