Abstracts of the talks

William Borrelli - Spectral properties of relativistic quantum waveguides

We make a spectral analysis of the Dirac operator in a tubular neighborhood of an unbounded planar curve, subject to infinite mass boundary conditions. Under general assumptions, we locate the essential spectrum and derive an effective Hamiltonian in the thin-strip limit. We also study the existence of eigenvalues in that regime, and in the non-relativistic limit.

Joint works with P. Briet, N. Kerraoui, D. Krejcirik and T. Ourmières-Bonafos.

Raffaele Carlone - 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 discussed. In the case of the nonlinear Dirac equation with Soler-type nonlinearity I will  present a detailed study of the spectrum of linearization at solitary waves.  Joint work with A. Comech, N. Boussaid, C.Cacciapuoti, A. Posilicano, D. Noja

Biagio Cassano - Sharp exponential decay for solutions to the Dirac equation

We determine the largest rate of exponential decay at infinity for non-trivial solutions to the stationary Dirac equation in presence of a (possibly non-Hermitian) matrix-valued perturbation V such that |V(x)| ~ |x|^{−ε} at infinity, for −∞ < ε < 1. Also, we show that our results are sharp for n = 2, 3, providing explicit examples of solutions that have the prescribed decay, in presence of a potential with the related behaviour at infinity. In this sense, our work is a result of unique continuation from infinity for the Dirac operator. Finally, we discuss the connections of this problem with the analogous one for the Laplace equation, the so called Landis’ conjecture.

Lucrezia Cossetti - A limiting absorption principle for time-harmonic isotropic Maxwell and Dirac equation

 In this talk we investigate the L^p − L^q mapping properties of the resolvent associated with the time-harmonic isotropic Maxwell and perturbed Dirac operator. As spectral parameters close to the spectrum are also covered by our analysis, we establish a L^p − L^q type limiting absorption principle for these operators. Our analysis relies on new results for Helmholtz systems with zero order non-Hermitian perturbations. The talk is based on a joint work with R. Mandel and on an ongoing project with R. Mandel and R. Schippa.

Elena Danesi - Strichartz estimates for the 2D and 3D massless Dirac-Coulomb equations.

The massless Dirac equation with a Coulomb potential is interesting both from a physical and a mathematical point of view; it appears in some physical models, for instance the 2D equation is used to describe the dynamics of carbon atoms in a sheet of non-perfect graphene, and on the mathematical side the homogeneity of degree -1 of the potential seems to have a critical behavior, as |x| goes to infinity, since Strichartz estimates are known to hold for potentials that decay faster and there are examples of potentials decaying slower such that the corresponding flows does not disperse.

In this talk I will present a recent result concerning Strichartz estimates for the solutions of the massless Dirac-Coulomb equation in 2 and 3 dimension with additional angular regularity. It extends the result on R^3 of Cacciafesta-Séré-Zhang and provides completely new estimates on R^2. As an application we will discuss a local well-posedness result for a nonlinear system.

Matteo Gallone - Self-adjointnt extensions of the Dirac-Coulomb operators

In this talk I will review the problem of constructing self-adjoint extensions of the Dirac-Coulomb operator. In the case of the operator on R^3, the parameter tuning the number of different self-adjoint extensions of the operator is the coupling constant (i.e. the charge of the nucleus). For very small values of the coupling constant, the problem was already solved by Kato in the '50s, while the complete range of essential self-adjointness was identified only in the '70s. In the regime where the operator is not essentialy self-adjoint anymore, a definition of distinguished extension was given by Schmincke followed by Nenciu and Wust. The classification of self-adjoint extensions is a more recent achievement, as it was done independently by Gallone-Michelangeli and Cassano-Pizzichillo. Last, I will discuss the Dirac-Coulomb operator in an infinite sector to show how the coupling constant of the potential interacts with the aperture of the angle in the self-adjointness problem.

This talk is based on a series of works with A. Michelangeli, B. Cassano and F. Pizzichillo

Jonas Lampart - The strong-coupling limit of the Dirac-Klein-Gordon system

I will discuss how the Dirac-Klein-Gordon system gives rise to the (cubic) nonlinear Dirac equation

 in the limit of strong coupling and large (field) mass. Joint work with Loic Le Treust, Simona Rota Nodari and Julien Sabin.

Loïc Le Treust - The Dirac bag model in strong magnetic field

We study Dirac operators on two-dimensional domains coupled to a magnetic field perpendicular to the plane. We focus on the infinite-mass boundary condition (also called MIT bag condition). In the case of bounded domains, we establish the asymptotic behavior of the low-lying (positive and negative) energies in the limit of strong magnetic field.

Albert Mas - Spectral analysis of a confinement model in relativistic quantum mechanics

In this talk we will focus on the Dirac operator on domains of R^3 with confining boundary conditions of scalar and electrostatic type. This operator is a generalization of the MIT-bag operator, which is used as a simplified model for the confinement of quarks in hadrons that has interested many scientists in the last decades. It is conjectured that, under a volume constraint, the ball is the domain which has the smallest first positive eigenvalue of the MIT-bag operator. I will describe our results -in collaboration with N. Arrizabalaga (U. País Vasco), T. Sanz-Perela (U. Autónoma de Madrid), and L. Vega (U. País Vasco and BCAM)- on the spectral analysis of the generalized operator. I will discuss on the parameterization of the eigenvalues, their symmetry and monotonicity properties, the optimality of the ball for large values of the parameter, and the connection to boundary Hardy spaces.

Fabio Pizzichillo - Keller estimates of the eigenvalues in the gap of Dirac operators

This talk aims to present estimates on the lowest eigenvalue in the gap of a Dirac operator in terms of a Lebesgue norm of the potential. Domain, self-adjointness, optimality and critical values of the norms are addressed, while the optimal potential is given by a Dirac equation with a Kerr nonlinearity. A new critical bound appears, which is the smallest value of the norm of the potential for which eigenvalues may reach

the bottom of the gap in the essential spectrum. Most of our result are established in the Birman-Schwinger reformulation of the problem. This is a collaboration work with Jean Dolbeault and David Gontier (University

Paris-Dauphine), and Hanne Van Den Bosch (University of Chile).