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Accueil > Les conférences du LMB > Colloques, journées > Equation de Dirac et interactions 2017 > Rencontres autour de l’équation de Dirac avec des interactions (...)

Rencontres autour de l’équation de Dirac avec des interactions singulières

We announce a four days meeting on questions linked to the Dirac equation involving singular type interactions.

The focus of the meeting is mainly prospective and aims at presenting open and challenging questions on the definition and properties of the models.

The meeting is organized around four talks with scheduled discussion sessions.

It will be held at the Laboratoire de mathématiques de Besançon (Room 316) from July, 10th to 13th, 2017. The invited speakers are :

- Claudio Cacciapuoti (Università dell’Insubria)
- Raffaele Carlone (Università degli Studi di Napoli Federico II)
- Andrew Comech (Texas A&M University & St. Petersburg State University).
- Diego Noja (Universita` di Milano Bicocca)
- Andrea Posilicano (Università dell’Insubria).

Timetable

Monday 10th, 14:00 Room 316B

Andrea Posilicano (Disat - Università dell’Insubria)

Nonlinear maximal monotone extensions of symmetric operators.

Given a linear semi-bounded symmetric operator S≥−ω, we explicitly define, and provide their nonlinear resolvents, nonlinear maximal monotone operators AΘ of type λ>ω (i.e., generators of one-parameter continuous nonlinear semigroups of contractions of type λ) which coincide with the Friedrichs extension of S on a convex set containing the domain of S. The extension parameter Θ⊂𝔥×𝔥 ranges over the set of nonlinear maximal monotone relations in an auxiliary Hilbert space 𝔥 isomorphic to the deficiency subspace of S. Moreover, AΘ+λ is a sub-potential operator (i.e., the sub-differential of a lower semicontinuous convex function) whenever Θ is sub-potential. Applications to Laplacians with nonlinear singular perturbations supported on null sets and to Laplacians with nonlinear boundary conditions on a bounded set are given.

Tuesday 11th, 11:00 Room 316B

Andrew Comech (Texas A&M University & St. Petersburg State University)

Global attraction to solitary waves : compactness of the spectrum for omega-limit trajectories

In a very few systems, one can obtain a very accurate description of a (weak) global attractor of all finite energy solutions, for example, proving that such an attractor is formed by solitary waves.

We will go through an elementary proof that the (time) spectrum of omega-limit trajectory, defined as the solution pattern that one can observe for large times, is compact. From here, in some cases, one can use the Titchmarsh convolution theorem to deduce that the solution has discrete frequencies.

Tuesday 11th, 14:00 Room 316B

Diego Noja (Università di Milano Bicocca)

A Dirac electron coupled with a classical nucleus

We consider the coupled dynamics of a Dirac electron in the Coulomb field generated by a nucleus which in turn is moving under the action of the Dirac spinor itself. Beside the coupling, a nonlocal term of Hartree type is a further source of nonlinearity. The model is a first attempt to study a relativistic coupled model of a heavy nucleus. The Schroedinger counterpart is well known and studied previously by Cances and Le Bris. Here we show local well posedness for the model in high regularity setting. The preliminary linear analysis, namely the properties of the propagator of the equation i∂tu = Du + V (· − q(t)) where D is the Dirac operator, V is the Coulomb potential and q(t) is a fixed trajectory, is a new result of independent interest.
Joint work with Federico Cacciafesta and Anne-Sophie de Suzzoni

Wednesday 12th, 14:00 Room 316B

Raffaele Carlone (Università Federico II Napoli)

Well-posedness of the two-dimensional nonlinear Schrodinger equation with concentrated nonlinearity

We consider a two-dimensional non linear Schrodinger equation with concentrated nonlinearity. In both the focusing and defocusing case we prove local well-posedness, i.e., existence and uniqueness of the solution for short times, as well as energy and mass conservation. In addition, we prove that this implies global existence in the defocusing case, irrespective of the power of the nonlinearity, while in the focusing case blowing-up solutions may arise.

joint work with L. Tentarelli, M.Correggi, A. Fiorenza, R. Adami.

Thursday 13th, 11:00 Room 316B

Claudio Cacciapuoti (Disat - Università dell’Insubria)

The one-dimensional Dirac Equation with concentrated nonlinearity

We define and study the Cauchy problem for a one-dimensional (1-D) nonlinear Dirac equation with nonlinearities concentrated at one point. Global well-posedness is provided and conservation laws for mass and energy are shown. Several examples, including nonlinear Gesztesy– Šeba models and the concentrated versions of the Bragg resonance and 1-D Soler (also known as massive Gross–Neveu) type models, all within the scope of the present paper, are given. The key point of the proof consists in the reduction of the original equation to a nonlinear integral equation for an auxiliary, space-independent variable.

joint work with Raffaele Carlone, Diego Noja and Andrea Posilicano