arXiv subjects
Publications and source records attributed to Nabile Boussaid.
This note presents an example of bilinear conservative system in an infinite dimensional Hilbert space for which approximate controllability in the Hilbert unit sphere holds for arbitrary small times. This situation is in contrast with the finite dimensional case and is due to the unboundedness of the drift operator.
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 by Cuccagna for the nonlinear Schrödinger equation, because of the strong indefiniteness of the energy.
We consider a general framework for investigating spectral pollution in the Galerkin method. We show how this phenomenon is characterised via the existence of particular Weyl sequences which are singular in a suitable sense. For a semi-bounded selfadjoint operator A we identify relative compactness conditions on a selfadjoint perturbation B ensuring that the limiting set of spectral pollution of A and B coincide. Our results show that, under perturbation, this limiting set behaves in a similar fashion as the essential spectrum.
In this paper we study the error in the approximate simultaneous controllability of the bilinear Schrodinger equation. We provide estimates based on a tracking algorithm for general bilinear quantum systems and on the study of the finite dimensional Galerkin approximations for a particular class of quantum systems, weakly-coupled systems. We then present two physical examples: the perturbed quantum harmonic oscillator and the infinite potential well.
We analyze the dispersive properties of a Dirac system perturbed with a magnetic field. We prove a general virial identity; as applications, we obtain smoothing and endpoint Strichartz estimates which are optimal from the decay point of view. We also prove a Hardy-type inequality for the perturbed Dirac operator.
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 study 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 appendix. We also discuss some applications to the dispersive Helmholzt model in the quantum regime.
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.
We study the behavior of perturbations of small nonlinear Dirac standing waves. We assume that the linear Dirac operator of reference $H=D_m+V$ 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 4-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 $H^{s}$ perturbation of stationary solutions, with $s>2$, stabilizes towards a standing wave. We also build center-stable and center-unstable manifolds each one of real codimension 4. Inside each of these manifolds, we obtain stabilization towards the center manifold in one direction of time, while in the other, we have instability. Eventually, outside all these manifolds, we have instability in the two directions of time. For localized perturbations inside the center manifold, we obtain a nonlinear scattering result.
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 close enough, 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.