This paper is concerned with the internal distributed control problem for the 1D Schrödinger equation, i ut(x,t) = −uxx+α(x) u+m(u) u, that arises in quantum semiconductor models. Here m(u) is a non local Hartree–type nonlinearity stemming from the coupling with the 1D Poisson equation, and α(x) is a regular function with linear growth at infinity, including constant electric fields. By means of both the Hilbert Uniqueness Method and the contraction mapping theorem it is shown that for initial and...

We consider a quantum particle in a 1D infinite square potential well with variable length. It is a nonlinear control system in which the state is the wave function ϕ of the particle and the control is the length l(t) of the potential well. We prove the following controllability result : given ${\phi }_0$ close enough to an eigenstate corresponding to the length l = 1 and ${\phi }_f$ close enough to another eigenstate corresponding to the length l=1, there exists a continuous function $l:[0,T]\to {ℝ}_{+}^{*}$ with T > 0, such that l(0)...

In the recent literature, the phenomenon of phase separation for binary mixtures of Bose–Einstein condensates can be understood, from a mathematical point of view, as governed by the asymptotic limit of the stationary Gross–Pitaevskii system $-\Delta u+u^3+\beta u{v}^2=\lambda u,-\Delta v+v^3+\beta u^{2}v=\mu v,u,v\in H_0^1\left(\Omega \right),u,v>0$, as the interspecies scattering length $\beta$ goes to $+\infty$. For this system we consider the associated energy functionals $J_{\beta },\beta \in (0,+\infty )$, with $L^2$-mass constraints, which limit $J_{\infty }$ (as $\beta \to +\infty$) is strongly irregular. For such functionals, we construct multiple critical points via a common...

Nonlinear Schrödinger equations (NLS)${}_a$ with strongly singular potential ${a\left|x\right|}^{-2}$ on a bounded domain $\Omega$ are considered. If $\Omega ={ℝ}^N$ and $a>-{(N-2)}^2/4$, then the global existence of weak solutions is confirmed by applying the energy methods established by N. Okazawa, T. Suzuki, T. Yokota (2012). Here $a=-{(N-2)}^2/4$ is excluded because $D\left(P_{a\left(N\right)}^{1/2}\right)$ is not equal to $H^1\left({ℝ}^N\right)$, where $P_{a\left(N\right)}:=-\Delta -{(N-2)}^2/{\left(4\right|x|}^2)$ is nonnegative and selfadjoint in $L^2\left({ℝ}^N\right)$. On the other hand, if $\Omega$ is a smooth and bounded domain with $0\in \Omega$, the Hardy-Poincaré inequality is proved in J. L. Vazquez, E. Zuazua (2000)....

2000 Mathematics Subject Classification: 35Q55,42B10.In this paper, we study the Schrödinger equation associated with the Dunkl operators, we study the dispersive phenomena and we prove the Strichartz estimates for this equation. Some applications are discussed.

Using the Maxwell-Higgs model, we prove that linearly unstable symmetric vortices in the Ginzburg-Landau theory are dynamically unstable in the H1 norm (which is the natural norm for the problem).In this work we study the dynamic instability of the radial solutions of the Ginzburg-Landau equations in R2 (...)

Given the plane triangle with vertices (0,0), (0,4) and (4,0) and the transformation F: (x,y) ↦ (x(4-x-y),xy) introduced by A. N. Sharkovskiĭ, we prove the existence of the following objects: a unique invariant curve of spiral type, a periodic trajectory of period 4 (given explicitly) and a periodic trajectory of period 5 (described approximately). Also, we give a decomposition of the triangle which helps to understand the global dynamics of this discrete system which is linked with the behavior...

This is a report on recent progress concerning the global well-posedness problem for energy-critical nonlinear Schrödinger equations posed on specific Riemannian manifolds $M$ with small initial data in $H^1\left(M\right)$. The results include small data GWP for the quintic NLS in the case of the $3d$ flat rational torus $M={𝕋}^3$ and small data GWP for the corresponding cubic NLS in the cases $M={ℝ}^2\times {𝕋}^2$ and $M={ℝ}^3\times 𝕋$. The main ingredients are bi-linear and tri-linear refinements of Strichartz estimates which obey the critical scaling, as well...

L’objet de cet exposé est de montrer comment l’évolution de Schrödinger pour le problème à $N$ corps quantique est approchée, lorsque $N$ tend vers l’infini, dans un régime convenable, par une évolution non-linéaire en dimension trois d’espace. On traitera le cas des bosons, qui conduit à l’équation de Schrödinger-Poisson, et celui des fermions, qui débouche sur le système de Hartree-Fock.

The Coupled Cluster (CC) method is a widely used and highly successful high precision method for the solution of the stationary electronic Schrödinger equation, with its practical convergence properties being similar to that of a corresponding Galerkin (CI) scheme. This behaviour has for the discrete CC method been analyzed with respect to the discrete Galerkin solution (the “full-CI-limit”) in [Schneider, 2009]. Recently, we globalized the CC formulation to the full continuous space, giving a root...

In this proceedings article we shall survey a series of results on the stability of self-similar solutions of the vortex filament equation. This equation is a geometric flow for curves in ${ℝ}^3$ and it is used as a model for the evolution of a vortex filament in fluid mechanics. The main theorem give, under suitable assumptions, the existence and description of solutions generated by curves with a corner, for positive and negative times. Its companion theorem describes the evolution of perturbations...