Related papers: Wigner Measure Propagation and Conical Singularity…
We study propagation of the Gabor wave front set for a Schr\"odinger equation with a Hamiltonian that is the Weyl quantization of a quadratic form with non-negative real part. We point out that the singular space associated to the quadratic…
We introduce new models for Schr\"odinger-type equations, which generalize standard NLS and for which different dispersion occurs depending on the directions. Our purpose is to understand dispersive properties depending on the directions of…
We study the propagation in time of $1/2$-Gelfand-Shilov singularities, i.e. global analytic singularities, of tempered distributional solutions of the initial value problem \begin{align*} \begin{cases} u_t + q^w(x,D) u = 0 \\ u|_{t=0} =…
We study the propagation of singularities for semilinear Schrodinger equations with quadratic Hamiltonians, in particular for the semilinear harmonic oscillator. We show that the propagation still occurs along the flow the Hamiltonian flow,…
Let $X$ be a manifold with boundary, endowed with a metric with conic singularities at the boundary components of $X$. Let $u$ be a solution to the wave equation on $\mathbb{R} \times X$. When a singularity of $u$ strikes a cone point of…
This paper is devoted to the investigation of propagation of singularities in hyperbolic equations with non-smooth oefficients, using the Colombeau theory of generalized functions. As a model problem, we study the Cauchy problem for the…
We prove unique continuation properties for solutions of the evolution Schr\"odinger equation with time dependent potentials. As an application of our method we also obtain results concerning the possible concentration profiles of blow up…
In this paper we describe the propagation of smooth (C^\infty) and Sobolev singularities for the wave equation on smooth manifolds with corners M equipped with a Riemannian metric g. That is, for X=MxR, P=D_t^2-\Delta_M, and u locally in…
We consider the logarithmic Schr{\"o}dinger equation in a semiclassical scaling, in the presence of a smooth, at most quadratic, external potential. For initial data under the form of a single coherent state, we identify the notion of…
In this article we prove global propagation of analyticity in finite time for solutions of semilinear Schr\"odinger equations with analytic nonlinearity from a region $\omega$ where the Geometric Control Condition holds. Our approach…
The dynamics of Schr\"odinger equation with time dependent potentials of general time dependence is considered. It is shown that for localized in space potentials, there is propagation of regularity which is uniformly bounded in higher…
We study the phase-space concentration of the so-called generalized metaplectic operators whose main examples are Schr\"odinger equations with bounded perturbations. To reach this goal, we perform a so-called $\mathcal{A}$-Wigner analysis…
We show global asymptotic stability of solitary waves of the nonlinear Schr\"odinger equation in space dimension 1. Furthermore, the radiation is shown to exhibit long range scattering if the nonlinearity is cubic at the origin, or standard…
In this paper, we investigate properties of unique continuation for hyperbolic Schr\"odinger equations with time-dependent complex-valued electric fields and time-independent real magnetic fields. We show that positive masses inside of a…
We derive dispersion estimates for solutions of the one-dimensional discrete perturbed Schr\"odinger and wave equations. In particular, we improve upon previous works and weaken the conditions on the potentials. To this end we also provide…
We consider the Schr\"odinger equation \begin{equation*} i \displaystyle\frac{\partial u}{\partial t} +Hu=0,\quad H=a(x,D), \end{equation*} where the Hamiltonian $a(z)$, $z=(x,\xi)$, is assumed real-valued and smooth, with bounded…
We numerically investigate the propagation of plane gravitational waves in the form of an initial boundary value problem with de Sitter initial data. The full non-linear Einstein equations with positive cosmological constant $\lambda$ are…
The universality for the eigenvalue spacing statistics of generalized Wigner matrices was established in our previous work \cite{EYY} under certain conditions on the probability distributions of the matrix elements. A major class of…
In this article we study limits of Wigner distributions (the so-called semiclassical measures) corresponding to sequences of solutions to the semiclassical Schroedinger equation at times scales $\alpha_{h}$ tending to infinity as the…
Sparsity properties for phase-space representations of several types of operators have been extensively studied in recent papers, including pseudodifferential, Fourier integral and metaplectic operators, with applications to time-frequency…