Nabile Boussaid

Stable Directions for Small Nonlinear Dirac Standing Waves

We prove that for a Dirac operator, with no resonance at thresholds nor eigenvalue at thresholds, the propagator satisfies propagation and dispersive estimates. When this linear operator has only two simple eigenvalues sufficiently close to each other, we study an associated class of nonlinear Dirac equations which have stationary solutions. As an application of our decay estimates, we show that these solutions have stable directions which are tangent to the subspaces associated with the continuous spectrum of the Dirac operator. This result is the analogue, in the Dirac case, of a theorem by Tsai and Yau about the Schrödinger equation. To our knowledge, the present work is the first mathematical study of the stability problem for a nonlinear Dirac equation

On stability of standing waves of nonlinear Dirac equations

We consider the stability problem for standing waves of nonlinear Dirac models. Under a suitable definition of linear stability, and under some restriction on the spectrum, we prove at the same time orbital and asymptotic stability. We are not able to get the full result proved in [24] for the nonlinear Schrödinger equation, because of the strong indefiniteness of the energy.

Weakly Coupled Systems in Quantum Control

Weakly coupled systems are a class of infinite dimensional conservative bilinear control systems with discrete spectrum. An important feature of these systems is that they can be precisely approached by finite dimensional Galerkin approximations. This property is of particular interest for the approximation of quantum system dynamics and the control of the bilinear Schrödinger equation. The present study provides rigorous definitions and analysis of the dynamics of weakly coupled systems and gives sufficient conditions for an infinite dimensional quantum control system to be weakly coupled. As an illustration we provide examples chosen among common physical systems.

Non-variational computation of the eigenstates of Dirac operators with radially symmetric potentials

We discuss a novel strategy for computing the eigenvalues and eigenfunctions of the relativistic Dirac operator with a radially symmetric potential. The virtues of this strategy lie on the fact that it avoids completely the phenomenon of spectral pollution and it always provides two-side estimates for the eigenvalues with explicit error bounds on both eigenvalues and eigenfunctions. We also discuss convergence rates of the method as well as illustrate our results with various numerical experiments.

Limiting Absorption Principle for Some Long Range Perturbations of Dirac Systems at Threshold Energies

We establish a limiting absorption principle for some long range perturbations of the Dirac systems at threshold energies. We cover multi-center interactions with small coupling constants. The analysis is reduced to studying a family of non-self-adjoint operators. The technique is based on a positive commutator theory for non-self-adjoint operators, which we develop in the Appendix. We also discuss some applications to the dispersive Helmholtz model in the quantum regime.

On the Asymptotic Stability of Small Nonlinear Dirac Standing Waves in a Resonant Case

We study the behavior of perturbations of small nonlinear Dirac standing waves. We assume that the linear Dirac operator of reference

Generalized Weyl theorems and spectral pollution in the Galerkin method

H=D_m+V

Periodic control laws for bilinear quantum systems with discrete spectrum

has only two double eigenvalues and that degeneracies are due to a symmetry of H (theorem of Kramers). In this case, we can build a small four-dimensional manifold of stationary solutions tangent to the first eigenspace of H. Then we assume that a resonance condition holds, and we build a center manifold of real codimension 8 around each stationary solution. Inside this center manifold any

Existence and Stability of Standing Waves for Supercritical NLS with a Partial Confinement

H^s

Finite element eigenvalue enclosures for the Maxwell operator

perturbation of stationary solutions, with