42 research outputs found
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 has only
two double eigenvalues and that degeneracies are due to a symmetry of
(theorem of Kramers). In this case, we can build a small 4-dimensional manifold
of stationary solutions tangent to the first eigenspace of . 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
perturbation of stationary solutions, with , 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.Comment: 37 page
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 by Cuccagna for the
nonlinear Schr\"odinger equation, because of the strong indefiniteness of the
energy.Comment: We have corrected the hypotheses adding an extra symmetry to our
class of solution
Virial identity and weak dispersion for the magnetic Dirac equation
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.Comment: 13 pages, typos in the statement of theorem 1.3 and clarification of
the proof in subsection 3.
Small time reachable set of bilinear quantum systems
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
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 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
Which notion of energy for bilinear quantum systems?
In this note we investigate what is the best L^p-norm in order to describe
the relation between the evolution of the state of a bilinear quantum system
with the L^p-norm of the external field. Although L^2 has a structure more easy
to handle, the L^1 norm is more suitable for this purpose. Indeed for every
p>1, it is possible to steer, with arbitrary precision, a generic bilinear
quantum system from any eigenstate of the free Hamiltonian to any other with a
control of arbitrary small L^p norm. Explicit optimal costs for the L^1 norm
are computed on an example