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Didier Robert

Publications and source records attributed to Didier Robert.

At least 19 recordsLinked to original sources
arXiv2024

Numerical approaches to compute spectra of non-self adjoint operators in dimensions two and three

In this article we are interested for the numerical computation of spectra of non-self adjoint quadratic operators, in two and three spatial dimensions. Indeed, in the multidimensional case very few results are known on the location of the eignevalues. This leads to solve nonlinear eigenvalue problems. In introduction we begin with a review of theoretical results and numerical results obtained for the one dimensional case. Then we present the numerical methods developed to compute the spectra (finite difference discretization) for the two and three dimensional cases. The numerical results obtained are presented and analyzed. One difficulty here is that we have to compute eigenvalues of strongly non-self-adjoint operators which are unstable. This work is in continuity of a previous work in one spatial dimension.

math.NA↗
arXiv2024

Spin-orbit interaction with large spin in the semi-classical regime

We consider the time dependent Schrödinger equation with a coupling spin-orbit in the semi-classical regime $\hbar\searrow 0$ and large spin number $\spin\rightarrow +\infty$ such that $\hbar^δ\spin=c$ where $c>0$ and $δ>0$ are constant. The initial state $Ψ(0)$ is a product of an orbital coherent state in $L^2(\R^d)$ and a spin coherent state in a spin irreducible representation space ${\mathcal H}_{2\spin +1}$. For $δ<1$, at the leading order in $\hbar$, the time evolution $Ψ(t)$ of $ Ψ(0)$ is well approximated by the product of an orbital and a spin coherent state. Nevertheless for $1/2<δ<1$ the quantum orbital leaves the classical orbital. For $δ=1$ we prove that this last claim is no more true when the interaction depends on the orbital variables. For the Dicke model, we prove that the orbital partial trace of the projector on $Ψ(t)$ is a mixed state in $L^2(\R)$ for small $t>0$.

math-ph↗
arXiv2024

Longtime dynamics for the Landau Hamiltonian with a time dependent magnetic field

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 and quantum dynamics for the corresponding Landau Hamiltonian. It turns out that the results depend strongly on the chosen gauge. For the Landau gauge the position observable is unbounded for "almost all" non resonant frequencies $\omega$. On the contrary, for the symmetric gauge we obtain that, for "almost all" non resonant frequencies $\omega$, the Landau Hamiltonian is reducible to a two dimensional harmonic oscillator and thus gives rise to bounded dynamics. The proofs use KAM algorithms for the classical dynamics. Quantum applications are given. In particular, the Floquet spectrum is absolutely continuous in the Landau gauge while it is discrete, of finite multiplicity, in symmetric gauge.

math.AP↗
arXiv2023

Asymptotic initial value representation of the solutions of semi-classical systems presenting smooth codimension one crossings

This paper is devoted to the construction of approximations of the propagator associated with a semi-classical matrix-valued Schr\"odinger operator with symbol presenting smooth eigenvalues crossings. Inspired by the approach of the theoretical chemists Herman and Kluk who propagated continuous superpositions of Gaussian wave-packets for scalar equations, we consider frozen and thawed Gaussian initial value representations that incorporate classical transport and branching processes along a hopping hypersurface. Based on the Gaussian wave-packet framework, our result relies on an accurate analysis of the solutions of the associated Schr\"odinger equation for data that are vector-valued wave-packets. We prove that these solutions are asymptotic to wavepackets at any order in terms of the semi-classical parameter.

math.AP↗
arXiv2023

When Poisson and Moyal Brackets are equal?

In the phase space $\R^{2d}$, let us denote $\{A,B\}$ the Poisson bracket of two smooth classical observables and $\{A, B\}_\circledast $ their Moyal bracket, defined as the Weyl symbol of $i[ A, B]$, where $ \hat A$ is the Weyl quantization of $A$ and $[ \hat A, \hat B]= \hat A \hat B- \hat B \hat A$ (commutator). In this note we prove that if a smooth Hamiltonian $H$ on the phase space $\R^{2d}$, with derivatives of moderate growth, satisfies $\{A,H\}= \{A, H\}_\circledast$ for any smooth and bounded observable $A$ then $H$ must be a polynomial of degree at most 2. This is related with the Groenewold-van Hove Theorem \cite{Gotay, Groen, vHove} concerning quantization of polynomial observables.

math-ph↗
arXiv2020

Adiabatic and non-adiabatic evolution of wave packets and applications to initial value representations

We review some recent results obtained for the time evolution of wave packets for systems of equations of pseudo-differential type, including Schr{ö}dinger ones, and discuss their application to the approximation of the associated unitary propagator. We start with scalar equations, propagation of coherent states, and applications to the Herman-Kluk approximation. Then we discuss the extension of these results to systems with eigenvalues of constant multiplicity or with smooth crossings.

math.AP↗
arXiv2017

Growth of Sobolev norms for abstract linear Schrödinger Equations

We prove an abstract theorem giving a $\langle t\rangle^ε$ bound ($\forall ε>0$) on the growth of the Sobolev norms in linear Schrödinger equations of the form $i \dot ψ= H_0 ψ+ V(t) ψ$ when the time $t \to \infty$. The abstract theorem is applied to several cases, including the cases where (i) $H_0$ is the Laplace operator on a Zoll manifold and $V(t)$ a pseudodifferential operator of order smaller then 2; (ii) $H_0$ is the (resonant or nonresonant) Harmonic oscillator in $R^d$ and $V(t)$ a pseudodifferential operator of order smaller then $H_0$ depending in a quasiperiodic way on time. The proof is obtained by first conjugating the system to some normal form in which the perturbation is a smoothing operator and then applying the results of \cite{MaRo}.

math.AP↗
arXiv2016

On time dependent Schr{ö}dinger equations: global well-posedness and growth of Sobolev norms

In this paper we consider time dependent Schr{ö}dinger linear PDEs of the form i$\partial$t$ψ$ = L(t)$ψ$, where L(t) is a continuous family of self-adjoint operators. We give conditions for well-posedness and polynomial growth for the evolution in abstract Sobolev spaces. If L(t) = H + V (t) where V (t) is a perturbation smooth in time and H is a self-adjoint positive operator whose spectrum can be enclosed in spectral clusters whose distance is increasing, we prove that the Sobolev norms of the solution grow at most as t $ε$ when t $\rightarrow$ $\infty$, for any $ε$ \textgreater{} 0. If V (t) is analytic in time we improve the bound to (log t) $γ$ , for some $γ$ \textgreater{} 0. The proof follows the strategy, due to Howland, Joye and Nenciu, of the adiabatic approximation of the flow. We recover most of known results and obtain new estimates for several models including 1-degree of freedom Schr{ö}dinger operators on R and Schr{ö}dinger operators on Zoll manifolds.

math.AP↗
arXiv2016

Numerical approaches for some Nonlinear Eigenvalue Problems

In this article we are interested for the numerical study of nonlinear eigenvalue problems. We begin with a review of theoretical results obtained by functional analysis methods, especially for the Schrodinger pencils. Some recall are given for the pseudospectra. Then we present the numerical methods and results obtained for eigenvalues computation with spectral methods and finite difference discretization, in infinite or bounded domains. Comparison with theoretical results is done. The main difficulty here is that we have to compute eigenvalues of strongly non-self-adjoint operators which are very unstable.

math.NA↗
arXiv2016

Irregular time dependent perturbations of quantum Hamiltonians

Our main goal in this paper is to prove existence (and uniqueness) of the quantum propagator for time dependent quantum Hamiltonians $\hat H(t)$ when this Hamiltonian is perturbed with a quadratic white noise $\dotβ\hat K$. $β$ is a continuous function in time $t$, $\dot β$ its time derivative and $K$ is a quadratic Hamiltonian. $\hat K$ is the Weyl quantization of $K$. For time dependent quadratic Hamiltonians $H(t)$ we recover, under less restrictive assumptions, the results obtained in \cite{bofu, du}.In our approach we use an exact Hermann Kluk formula \cite{ro2} to deduce a Strichartz estimate for the propagator of $\hat H(t) +\dot βK$. This is applied to obtain local and global well posedness for solutions for non linear Schrödinger equations with an irregular time dependent linear part.

math-ph↗
arXiv2015

Random weighted Sobolev inequalities and application to quantum ergodicity

This paper is a continuation of Poiret-Robert-Thomann (2013) where we studied a randomisation method based on the Laplacian with harmonic potential. Here we extend our previous results to the case of any polynomial and confining potential $V$ on $\mathbb{R}^d$. We construct measures, under concentration type assumptions, on the support of which we prove optimal weighted Sobolev estimates on $\mathbb{R}^d$. This construction relies on accurate estimates on the spectral function in a non-compact configuration space. Then we prove random quantum ergodicity results without specific assumption on the classical dynamics. Finally, we prove that almost all basis of Hermite functions is quantum uniquely ergodic.

math.AP↗
arXiv2014

Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator

Thanks to an approach inspired from Burq-Lebeau \cite{bule}, we prove stochastic versions of Strichartz estimates for Schrödinger with harmonic potential. As a consequence, we show that the nonlinear Schrödinger equation with quadratic potential and any polynomial non-linearity is almost surely locally well-posed in $L^{2}(\R^{d})$ for any $d\geq 2$. Then, we show that we can combine this result with the high-low frequency decomposition method of Bourgain to prove a.s. global well-posedness results for the cubic equation: when $d=2$, we prove global well-posedness in $\H^{s}(\R^{2})$ for any $s>0$, and when $d=3$ we prove global well-posedness in $\H^{s}(\R^{3})$ for any $s>1/6$, which is a supercritical regime. Furthermore, we also obtain almost sure global well-posedness results with scattering for NLS on $\R^{d}$ without potential. We prove scattering results for $L^2-$supercritical equations and $L^2-$subcritical equations with initial conditions in $L^2$ without additional decay or regularity assumption.

math.AP↗
arXiv2014

On random Hermite series

We study integrability and continuity properties of random series of Hermite functions. We get optimal results which are analogues to classical results concerning Fourier series, like the Paley-Zygmund or the Salem-Zygmund theorems. We also consider the case of series of radial Hermite functions, which are not so well-behaved. In this context, we prove some L^p bounds of radial Hermite functions, which are optimal when p is large.

math.AP↗
arXiv2013

Random weighted Sobolev inequalities on $\mathbb{R}^d$ and application to Hermite functions

We extend a randomisation method, introduced by Shiffman-Zelditch and developed by Burq-Lebeau on compact manifolds for the Laplace operator, to the case of $\mathbb{R}^d$ with the harmonic oscillator. We construct measures, thanks to probability laws which satisfy the concentration of measure property, on the support of which we prove optimal weighted Sobolev estimates on $\mathbb{R}^d$. This construction relies on accurate estimates on the spectral function in a non-compact configuration space. As an application, we show that there exists a basis of Hermite functions with good decay properties in $L^{\infty}(\mathbb{R}^d$)$, when $d\geq 2$.

math.AP↗
arXiv2012

Time Evolution of States for Open Quantum Systems. The quadratic case

Our main goal in this paper is to extend to any system of coupled quadratic Hamiltonians some properties known for systems of quantum harmonic oscillators related with the Brownian Quantum Motion model. In a first part we get a rather general formula for the purity (or the linear entropy) in a short time approximation. In a second part we establish a master equation (or a Fokker-Planck type equation) for the time evolution of the reduced matrix density for bilinearly coupled quadratic Hamiltonians. The Hamiltonians and the bilinear coupling can be time dependent. Moreover we give an explicit formula for the solution of this master equation so that the time evolution of the reduced density at time $t$ is connected with the reduced density at initial time $t_0$ for $t_0 \leq t <t_0 +t_c$ where $t_c\in ]0, \infty]$ is a critical time but reversibility is lost for $t \geq t_0 +t_c$.

math-ph↗
arXiv2009

On the Herman-Kluk Semiclassical Approximation

For a subquadratic symbol $H$ on $\R^d\times\R^d = T^*(\R^d)$, the quantum propagator of the time dependent Schrödinger equation $i\hbar\frac{\partialψ}{\partial t} = \hat Hψ$ is a Semiclassical Fourier-Integral Operator when $\hat H=H(x,\hbar D_x)$ ($\hbar$-Weyl quantization of $H$). Its Schwartz kernel is describe by a quadratic phase and an amplitude. At every time $t$, when $\hbar$ is small, it is "essentially supported" in a neighborhood of the graph of the classical flow generated by $H$, with a full uniform asymptotic expansion in $\hbar$ for the amplitude. In this paper our goal is to revisit this well known and fondamental result with emphasis on the flexibility for the choice of a quadratic complex phase function and on global $L^2$ estimates when $\hbar$ is small and time $t$ is large. One of the simplest choice of the phase is known in chemical physics as Herman-Kluk formula. Moreover we prove that the semiclassical expansion for the propagator is valid for $| t| << \frac{1}{4δ}|\log\hbar|$ where $δ>0$ is a stability parameter for the classical system.

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