Naoufel Ben Abdallah

The Nonlinear Schrödinger Equation with a Strongly Anisotropic Harmonic Potential

The nonlinear Schrodinger equation with general nonlinearity of polynomial growth and harmonic confining potential is considered. More precisely, the confining potential is strongly anisotropic; i.e., the trap frequencies in different directions are of different orders of magnitude. The limit as the ratio of trap frequencies tends to zero is carried out. A concentration of mass on the ground state of the dominating harmonic oscillator is shown to be propagated, and the lower-dimensional modulation wave function again satisfies a nonlinear Schrodinger equation. The main tools of the analysis are energy and Strichartz estimates, as well as two anisotropic Sobolev inequalities. As an application, the dimension reduction of the three-dimensional Gross-Pitaevskii equation is discussed, which models the dynamics of Bose-Einstein condensates.

On a multidimensional Schrödinger–Poisson scattering model for semiconductors

We consider a stationary Schrodinger–Poisson problem modeling a self-consistent transport in a quantum coupler. The Schrodinger equation is set on a bounded domain with transparent boundary conditions describing incoming scattering states of the Schrodinger operator. The coupling with the Poisson equation is done thanks to a nonlinear limiting absorption procedure. The charge density of the limit potential is shown to be equal to the sum of the scattering state and bound state densities.

Multiscale simulation of transport in an open quantum system: Resonances and WKB interpolation

A numerical scheme for the one-dimensional stationary Schrodinger-Poisson model is described. The scheme is used to simulate a resonant tunneling diode and provides an important reduction of the simulation time. The improvement is twofold. First the grid spacing in the position variable is made coarser by using oscillating interpolation functions derived from the WKB asymptotics. Then the discretization of the energy variable, which is a parameter for the Schrodinger equation, is improved by approaching the wavefunctions in the double barrier region by its projection on the resonant states (following the work of Presilla-Sjostrand and Jona-Lasinio [On Schrodinger equations with concentrated non-linearities, Ann. Phys. 240 (1995) 1-21]).

On a Vlasov–Schrödinger–Poisson Model

Abstract A Vlasov–Schrödinger–Poisson system is studied, modeling the transport and interactions of electrons in a bidimensional electron gas. The particles are assumed to have a wave behaviour in the confinement direction (z) and to behave like point particles in the directions parallel to the electron gas (x). For each fixed x and at each time t, the eigenfunctions and the eigenenergies of the Schrödinger operator in the z are computed. The occupation number of each eigenfunction is computed through the resolution of a Vlasov equation in the x direction, the force field being the gradient of the eigenenergy. The whole system is coupled to the Poisson equation for the electrostatic interaction. Existence of weak solutions is shown for boundary value problems in the stationary and time-dependent regimes.

The Child-Langmuir law for the Boltzmann equation of semiconductors

We investigate the so-called Child-Langmuir asymptotics of the one-dimensional stationary Boltzmann-Poisson system. The asymptotics apply when the lattice temperature is small and leads to a singular perturbation problem. We derive the limit problem associated with these asymptotics, and prove the existence of the Child-Langmuir current.

Gross–Pitaevskii–Poisson Equations for Dipolar Bose–Einstein Condensate with Anisotropic Confinement

Ground states and dynamical properties of a dipolar Bose–Einstein condensate are analyzed based on the Gross–Pitaevskii–Poisson system (GPPS) and its dimension reduction models under an anisotropic confining potential. We begin with the three-dimensional (3D) GPPS and review its quasi-two-dimensional (2D) approximate equations when the trap is strongly confined in the z-direction and quasi-one-dimensional (1D) approximate equations when the trap is strongly confined in the x- and y-directions. In fact, in the quasi-2D equations, a fractional Poisson equation with the operator

The Child-Langmuir law as a model for electron transport in semiconductors

(-\Delta)^{1/2}

ANOMALOUS DIFFUSION LIMIT FOR KINETIC EQUATIONS WITH DEGENERATE COLLISION FREQUENCY

is involved which brings significant difficulties into the analysis. Existence and uniqueness as well as nonexistence of the ground state under different parameter regimes are established for the quasi-2D and quasi-1D equations. Well-posedness of the Cauchy problem for both types of equations and finite time blow-up in two dimensions are analyzed. Finally, we rigorously prove the convergence with linear convergence ...

Diffusion Approximation and Homogenization of the Semiconductor Boltzmann Equation

A model for space-charge limited electron transport in an N+-N-N+ device and a Schottky diode is proposed. It is based on the smallness of the lattice temperature with respect to the applied voltage. This model, derived from the stationary Boltzmann equation of semiconductors, has a much lower computational cost and leads to a good prediction for the I-V curve as well as for the built-in potential.

Relative entropies for the Vlasov–Poisson system in bounded domains

This paper is devoted to hydrodynamic limits for collisional linear kinetic equations. It is a classical result that under certain conditions on the collision operator, the long time/small mean free path asymptotic behavior of the density of particles can be described by diffusion-type equations. We are interested in situations in which the degeneracy of the collision frequency for small velocities causes this limit to break down. We show that the appropriate asymptotic analysis leads to an anomalous diffusion regime.