724 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
Generalised Weyl theorems and spectral pollution in the Galerkin method
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.Comment: The new version deals with Galerkin sequences which are dense in the
form domain of A, when A is bounded from belo
Approximate controllability of the Schr\"{o}dinger Equation with a polarizability term in higher Sobolev norms
This analysis is concerned with the controllability of quantum systems in the
case where the standard dipolar approximation, involving the permanent dipole
moment of the system, is corrected with a polarizability term, involving the
field induced dipole moment. Sufficient conditions for approximate
controllability are given. For transfers between eigenstates of the free
Hamiltonian, the control laws are explicitly given. The results apply also for
unbounded or non-regular potentials
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.
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