34 research outputs found
A variational study of some hadron bag models
Quantum chromodynamics (QCD) is the theory of strong interaction and accounts
for the internal structure of hadrons. Physicists introduced phe- nomenological
models such as the M.I.T. bag model, the bag approximation and the soliton bag
model to study the hadronic properties. We prove, in this paper, the existence
of excited state solutions in the symmetric case and of a ground state solution
in the non-symmetric case for the soliton bag and the bag approximation models
thanks to the concentration compactness method. We show that the energy
functionals of the bag approximation model are Gamma -limits of sequences of
soliton bag model energy functionals for the ground and excited state problems.
The pre- compactness, up to translation, of the sequence of ground state
solutions associated with the soliton bag energy functionals in the
non-symmetric case is obtained combining the Gamma -convergence theory and the
concentration-compactness method. Finally, we give a rigorous proof of the
original derivation of the M.I.T. bag equations done by Chodos, Jaffe, Johnson,
Thorn and Weisskopf via a limit of bag approximation ground state solutions in
the spherical case. The supersymmetry property of the Dirac operator is the key
point in many of our arguments
Self-Adjointness of Dirac Operators with Infinite Mass Boundary Conditions in Sectors
This paper deals with the study of the two-dimensional Dirac operatorwith
infinite mass boundary condition in a sector. We investigate the question
ofself-adjointness depending on the aperture of the sector: when the sector is
convexit is self-adjoint on a usual Sobolev space whereas when the sector is
non-convexit has a family of self-adjoint extensions parametrized by a complex
number of theunit circle. As a byproduct of this analysis we are able to give
self-adjointnessresults on polygones. We also discuss the question of
distinguished self-adjointextensions and study basic spectral properties of the
operator in the sector
Uniformly accurate time-splitting methods for the semiclassical linear Schr{\"o}dinger equation
This article is devoted to the construction of numerical methods which remain
insensitive to the smallness of the semiclassical parameter for the linear
Schr{\"o}dinger equation in the semiclassical limit. We specifically analyse
the convergence behavior of the first-order splitting. Our main result is a
proof of uniform accuracy. We illustrate the properties of our methods with
simulations
Uniformly accurate time-splitting methods for the semiclassical Schrödinger equationPart 1 : Construction of the schemes and simulations
This article is devoted to the construction of new numerical methods for the semiclassical Schrödinger equation. A phase-amplitude reformulation of the equation is described where the Planck constant epsilon is not a singular parameter. This allows to build splitting schemes whose accuracy is spectral in space, of up to fourth order in time, and independent of epsilon before the caustics. The second-order method additionally preserves the L^2-norm of the solution just as the exact flow does. In this first part of the paper, we introduce the basic splitting scheme in the nonlinear case, reveal our strategy for constructing higher-order methods, and illustrate their properties with simulations. In the second part, we shall prove a uniform convergence result for the first-order splitting scheme applied to the linear Schrödinger equation with a potential
Existence of nodal solutions for Dirac equations with singular nonlinearities
We prove, by a shooting method, the existence of infinitely many solutions of
the form of the nonlinear Dirac
equation {equation*} i\underset{\mu=0}{\overset{3}{\sum}} \gamma^\mu
\partial_\mu \psi- m\psi - F(\bar{\psi}\psi)\psi = 0 {equation*} where
is compactly supported and \[F(x) = \{{array}{ll}
p|x|^{p-1} & \text{if} |x|>0 0 & \text{if} x=0 {array}.] with
under some restrictions on the parameters and We study also the
behavior of the solutions as tends to zero to establish the link between
these equations and the M.I.T. bag model ones
Boundary states of the Robin magnetic Laplacian
This article tackles the spectral analysis of the Robin Laplacian on a smooth
bounded two-dimensional domain in the presence of a constant magnetic field. In
the semiclassical limit, a uniform description of the spectrum located between
the Landau levels is obtained. The corresponding eigenfunctions, called edge
states, are exponentially localized near the boundary. By means of a microlocal
dimensional reduction, our unifying approach allows on the one hand to derive a
very precise Weyl law and a proof of quantum magnetic oscillations for excited
states, and on the other hand to refine simultaneously old results about the
low-lying eigenvalues in the Robin case and recent ones about edge states in
the Dirichlet case
On the Dirac bag model in strong magnetic fields
In this work we study two-dimensional Dirac operators on bounded domains
coupled to a magnetic field perpendicular to the plane. We focus on the MIT bag
boundary condition and provide accurate asymptotic estimates for the low-lying
(positive and negative) energies in the limit of a strong magnetic field
On the semiclassical spectrum of the Dirichlet-Pauli operator
International audienceThis paper is devoted to semiclassical estimates of the eigenvalues of the Pauli operator on a bounded open set whose boundary carries Dirichlet conditions. Assuming that the magnetic field is positive and a few generic conditions, we establish the simplicity of the eigenvalues and provide accurate asymptotic estimates involving Segal-Bargmann and Hardy spaces associated with the magnetic field
Variational and topological methods for the study of nonlinear models from relativistic quantum mechanics.
Cette thèse porte sur l'étude de modèles non linéaires issus de la mécanique quantique relativiste.Dans la première partie, nous démontrons à l'aide d'une méthode de tir l'existence d'une infinité de solutions d'équations de Dirac non linéaires provenant d'un modèle de hadrons et d'un modèle de la physique des noyaux.Dans la seconde partie, nous prouvons par des méthodes variationnelles l'existence d'un état fondamental et d'états excités pour deux modèles de la physique des hadrons. Par la suite, nous étudions la transition de phase reliant les deux modèles grâce à la Gamma-convergence.La dernière partie est consacrée à l'étude d'un autre modèle de hadrons dans lequel les fonctions d'onde des quarks sont parfaitement localisées. Nous énonçons quelques résultats préliminaires que nous avons obtenus.This thesis is devoted to the study of nonlinear models from relativistic quantum mechanics.In the first part, we show thanks to a shooting method, the existence of infinitely many solutions of nonlinear Dirac equations of two models from the physics of hadrons and the physics of the nucleus.In the second part, we prove thanks to variational methods the existence of a ground state and excited states for two models of the physics of hadrons. Next, we study the phase transition which links the models thanks to the -convergence.The last part is devoted to the study of another model from the physics of hadrons in which the wave functions are perfectly confined. We give some preliminary results