Benoît Grébert
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View article: On uniqueness of radial potentials for given Dirichlet spectra with distinct angular momenta
On uniqueness of radial potentials for given Dirichlet spectra with distinct angular momenta Open
We consider an inverse spectral problem for radial Schr\" odinger operators with singular potentials. First, we show that the knowledge of the Dirichlet spectra for infinitely many angular momenta~$\ell$ satisfying a Müntz-type condition u…
View article: Almost global existence for some nonlinear Schrödinger equations on $\mathbb{T}^d$ in low regularity
Almost global existence for some nonlinear Schrödinger equations on $\mathbb{T}^d$ in low regularity Open
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View article: On almost periodic solutions to NLS without external parameters
On almost periodic solutions to NLS without external parameters Open
In this note, we present a result established in [BGR24] where we prove that nonlinear Schrödinger equations on the circle, without external parameters, admit plenty of infinite dimensional non resonant invariant tori, or equivalently, ple…
View article: On almost periodic solutions to NLS without external parameters
On almost periodic solutions to NLS without external parameters Open
In this note, we present a result established in [BGR24] where we prove that nonlinear Schrodinger equations on the circle, without external parameters, admit plenty of infinite dimensional non resonant invariant tori, or equivalently, ple…
View article: Longtime dynamics for the Landau Hamiltonian with a time dependent magnetic field
Longtime dynamics for the Landau Hamiltonian with a time dependent magnetic field Open
We consider a modulated magnetic field, B(t) = B_{0} + \varepsilon f(\omega t) , perpendicular to a fixed plane, where B_{0} is constant, \varepsilon>0 and f a periodic function on the torus {\mathbb{T}}^{n} . Our aim is to study classical…
View article: Almost global existence for Hamiltonian PDEs on compact manifolds
Almost global existence for Hamiltonian PDEs on compact manifolds Open
We prove an abstract result of almost global existence of small solutions to semi-linear Hamiltonian partial differential equations satisfying very weak non resonance conditions and basic multilinear estimates. Thanks to works by Delort--S…
View article: Infinite dimensional invariant tori for nonlinear Schr\"odinger equations
Infinite dimensional invariant tori for nonlinear Schr\"odinger equations Open
We prove that nonlinear Schr\"odinger equations on the circle, without external parameters, admits plenty of almost periodic solutions. Indeed, we prove that arbitrarily close to most of the finite dimensional KAM tori constructed by Kuksi…
View article: Longtime dynamics for the Landau Hamiltonian with a time dependent magnetic field
Longtime dynamics for the Landau Hamiltonian with a time dependent magnetic field Open
We consider a modulated magnetic field, $B(t) = B_0 +\varepsilon f(ωt)$, perpendicular to a fixed plane, where $B_0$ is constant, $\varepsilon>0$ and $f$ a periodic function on the torus ${\mathbb T}^n$. Our aim is to study classical and q…
View article: Exponential stability of solutions to the Schrödinger–Poisson equation
Exponential stability of solutions to the Schrödinger–Poisson equation Open
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View article: Exponential stability of solutions to the Schr{ö}dinger-Poisson equation
Exponential stability of solutions to the Schr{ö}dinger-Poisson equation Open
We prove an exponential stability result for the small solutions of the Schr{ö}dinger-Poisson equation on the circle without exterior parameters in Gevrey class. More precisely we prove that for most of the initial data of Gevrey-norm smal…
View article: Long time solutions for quasilinear Hamiltonianperturbations of Schrödinger and Klein–Gordon equations on tori
Long time solutions for quasilinear Hamiltonianperturbations of Schrödinger and Klein–Gordon equations on tori Open
We consider quasi-linear, Hamiltonian perturbations of the cubic Schrödinger and of the cubic (derivative) Klein-Gordon equations on the $d$ dimensional torus. If $\varepsilon\ll1$ is the size of the initial datum, we prove that the lifesp…
View article: Dynamics of quintic nonlinear Schr{ö}dinger equations in $H^{2/5+}(\mathbb{T})$
Dynamics of quintic nonlinear Schr{ö}dinger equations in $H^{2/5+}(\mathbb{T})$ Open
In this paper, we succeed in integrating Strichartz estimates (encoding the dispersive effects of the equations) in Birkhoff normal form techniques. As a consequence, we deduce a result on the long time behavior of quintic NLS solutions on…
View article: Dynamics of nonlinear Klein–Gordon equations in low regularity on $\mathbb{S}^2$
Dynamics of nonlinear Klein–Gordon equations in low regularity on $\mathbb{S}^2$ Open
We describe the long-time behavior of small nonsmooth solutions to the nonlinear Klein–Gordon equations on the sphere \mathbb{S}^2 . More precisely, we prove that the low harmonic energies (also called super-actions) are almost preserved f…
View article: Birkhoff normal forms for Hamiltonian PDEs in their energy space
Birkhoff normal forms for Hamiltonian PDEs in their energy space Open
We study the long time behavior of small solutions of semi-linear dispersive Hamiltonian partial differential equations on confined domains. Provided that the system enjoys a new non-resonance condition and a sufficiently strong energy est…
View article: Almost global existence for some nonlinear Schr{ö}dinger equations on $\mathbb{T}^d$ in low regularity
Almost global existence for some nonlinear Schr{ö}dinger equations on $\mathbb{T}^d$ in low regularity Open
We are interested in the long time behavior of solutions of the nonlinear Schr{ö}dinger equation on the $d$-dimensional torus in low regularity, i.e. for small initial data in the Sobolev space $H^{s_0}(\mathbb T^d)$ with $s_0>d/2$. We pro…
View article: Dynamics of nonlinear Klein-Gordon equations in low regularity on S^2
Dynamics of nonlinear Klein-Gordon equations in low regularity on S^2 Open
We describe the long time behavior of small non-smooth solutions to the nonlinear Klein-Gordon equations on the sphere S^2. More precisely, we prove that the low harmonic energies (also called super-actions) are almost preserved for times …
View article: Growth of Sobolev norms for abstract linear Schrödinger equations
Growth of Sobolev norms for abstract linear Schrödinger equations Open
We prove an abstract theorem giving a \langle t\rangle^\epsilon bound (for all \epsilon > 0 ) on the growth of the Sobolev norms in linear Schrödinger equations of the form \mathrm i \dot \psi = H_0 \psi + V(t)\psi as t \to \infty . The ab…
View article: Long time solutions for quasi-linear Hamiltonian perturbations of\n Schr\\"odinger and Klein-Gordon equations on tori
Long time solutions for quasi-linear Hamiltonian perturbations of\n Schr\\"odinger and Klein-Gordon equations on tori Open
We consider quasi-linear, Hamiltonian perturbations of the cubic\nSchr\\"odinger and of the cubic (derivative) Klein-Gordon equations on the $d$\ndimensional torus. If $\\varepsilon\\ll1$ is the size of the initial datum, we\nprove that th…
View article: Reducibility of Schrödinger equation on a Zoll manifold with unbounded potential
Reducibility of Schrödinger equation on a Zoll manifold with unbounded potential Open
In this article, we prove a reducibility result for the linear Schrödinger equation on a Zoll manifold with quasi-periodic in time pseudo-differential perturbation of order less than or equal to 1/2. As far as we know, this is the first re…
LONG TIME BEHAVIOR OF THE SOLUTIONS OF NLW ON THE -DIMENSIONAL TORUS Open
We consider the nonlinear wave equation (NLW) on the $d$ -dimensional torus $\mathbb{T}^{d}$ with a smooth nonlinearity of order at least 2 at the origin. We prove that, for almost any mass, small and smooth solutions of high Sobolev indic…
View article: Reducibility of Schrödinger Equation on the Sphere
Reducibility of Schrödinger Equation on the Sphere Open
In this article we prove a reducibility result for the linear Schrödinger equation on the sphere $\mathbb{S}^n$ with quasi-periodic in time perturbation. Our result includes the case of unbounded perturbation that we assume to be of order …
View article: Long time behavior of the solutions of NLW on the d-dimensional torus
Long time behavior of the solutions of NLW on the d-dimensional torus Open
We consider the non linear wave equation (NLW) on the d-dimensional torus with a smooth nonlinearity of order at least two at the origin. We prove that, for almost any mass, small and smooth solutions of high Sobolev indices are stable up …
View article: Stable and unstable time quasi periodic solutions for a system of coupled NLS equations
Stable and unstable time quasi periodic solutions for a system of coupled NLS equations Open
We prove that a system of coupled nonlinear Schr{ö}dinger equations on the torus exhibits both stable and unstable small KAM tori. In particular the unstable tori are related to a beating phenomena which has been proved recently in [6]. Th…
View article: A KAM theorem for space-multidimensional Hamiltonian PDEs
A KAM theorem for space-multidimensional Hamiltonian PDEs Open
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View article: On reducibility of Quantum Harmonic Oscillator on $\mathbb{R}^d$ with quasiperiodic in time potential
On reducibility of Quantum Harmonic Oscillator on $\mathbb{R}^d$ with quasiperiodic in time potential Open
We prove that a linear d-dimensional Schr{ö}dinger equation on $\mathbb{R}^d$ with harmonic potential $|x|^2$ and small t-quasiperiodic potential $i\partial\_t u -- Δu + |x|^2 u + εV (tω, x)u = 0, x \in \mathbb{R}^d$ reduces to an autonomo…
View article: KAM for the Klein Gordon equation on $\mathbb S^d$
KAM for the Klein Gordon equation on $\mathbb S^d$ Open
Recently the KAM theory has been extended to multidimensional PDEs. Nevertheless all these recent results concern PDEs on the torus, essentially because in that case the corresponding linear PDE is diagonalized in the Fourier basis and the…