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Jean-Marie Barbaroux

Publications and source records attributed to Jean-Marie Barbaroux.

At least 19 recordsLinked to original sources
arXiv2025

Green's functions for magnetic Dirac operators and bulk-edge correspondence

We study the resolvent kernel (Green's function) of magnetic Dirac operators on a half-plane with boundary conditions interpolating between infinite mass and zigzag cases, excluding the latter. We show that these kernels have all the required properties so that the proof of bulk-edge correspondence from the infinite mass case shown recently can be repeated ad litteram. The zigzag case exhibits qualitatively different behavior and will be addressed in future work.

math-ph↗
arXiv2024

Magnetic Dirac systems: Violation of bulk-edge correspondence in the zigzag limit

We consider a Dirac operator with constant magnetic field defined on a half-plane with boundary conditions that interpolate between infinite mass and zigzag. By a detailed study of the energy dispersion curves we show that the infinite mass case generically captures the profile of these curves, which undergoes a continuous pointwise deformation into the topologically different zigzag profile. Moreover, these results are applied to the bulk-edge correspondence. In particular, by means of a counterexample, we show that this correspondence does not always hold true in the zigzag case.

math-ph↗
arXiv2021

Van der Waals-London interaction of atoms with pseudo-relativistic kinetic energy

We consider a multiatomic system where the nuclei are assumed to be point charges at fixed positions. Particles interact via Coulomb potential and electrons have pseudo-relativistic kinetic energy. We prove the van der Waals-London law, which states that the interaction energy between neutral atoms decays as the sixth power of the distance $|D|$ between the atoms. We rigorously compute all the terms in the binding energy up to the order $|D|^{-9}$ with error term of order $\mathcal{O}(|D|^{-10})$ . As intermediate steps we prove exponential decay of eigenfunctions of multiparticle Schrödinger operators with permutation symmetry imposed by the Pauli principle and new estimates of the localization error.

math-ph↗
arXiv2021

On the Dirac bag model in strong magnetic fields

In this work 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. Moreover, for a constant magnetic field $B$, we study the problem on the half-plane and find that the Dirac operator has continuous spectrum except for a gap of size $a\_0\sqrt{B}$, where $a\_0\in (0,\sqrt{2})$ is a universal constant. Remarkably, this constant characterizes certain energies of the system in a bounded domain as well. We discuss how these findings, together with our previous work, give a fairly complete description of the eigenvalue asymptotics of magnetic two-dimensional Dirac operators under general boundary conditions.

math.AP↗
arXiv2020

On the semiclassical spectrum of the Dirichlet-Pauli operator

This 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.

math.SP↗
arXiv2018

Localization for gapped Dirac Hamiltionians with random perturbations: Application to graphene antidot lattices

In this paper we study random perturbations of first order elliptic operators with periodic potentials. We are mostly interested in Hamiltonians modeling graphene antidot lattices with impurities. The unperturbed operator $H_0 := D_S + V_0$ is the sum of a Dirac-like operator $D_S$ plus a periodic matrix valued potential $V_0$, and is assumed to have an open gap. The random potential $V_ω$ is of Anderson-type with independent, identically distributed coupling constants and moving centers, with absolutely continuous probability distributions. We prove band edge localization, namely that there exists an interval of energies in the unperturbed gap where the almost sure spectrum of the family $H_ω:= H_0 + V_ω$ is dense pure point, with exponentially decaying eigenfunctions, that give rise to dynamical localization.

math-ph↗
arXiv2018

Resolvent convergence to Dirac operators on planar domains

Consider a Dirac operator defined on the whole plane with a mass term of size m supported outside a domain Omega. We give a simple proof for the norm resolvent convergence, as m goes to infinity, of this operator to a Dirac operator defined on Omega with infinite mass boundary conditions. The result is valid for bounded and unbounded domains and gives estimates on the speed of convergence. Moreover, the method easily extends when adding external matrix-valued potentials.

math-ph↗
arXiv2017

Spectral gaps in graphene antidot lattices

We consider the gap creation problem in an antidot graphene lattice, i.e. a sheet of graphene with periodically distributed obstacles. We prove several spectral results concerning the size of the gap and its dependence on different natural parameters related to the antidot lattice.

math-ph↗
arXiv2016

Local Decay for Weak Interactions with Massless Particles

We consider a mathematical model for the weak decay of the intermediate boson $Z^0$ into neutrinos and antineutrinos. We prove that the total Hamiltonian has a unique ground state in Fock space and we establish a limiting absorption principle, local decay and a property of relaxation to the ground state for initial states and observables suitably localized in energy and position. Our proofs rest, in particular, on Mourre's theory and a low-energy decomposition.

math-ph↗
arXiv2015

Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction

We study weak solutions of the homogeneous Boltzmann equation for Maxwellian molecules with a logarithmic singularity of the collision kernel for grazing collisions. Even though in this situation the Boltzmann operator enjoys only a very weak coercivity estimate, it still leads to strong smoothing of weak solutions in accordance to the smoothing expected by an analogy with a logarithmic heat equation.

math.AP↗
arXiv2015

Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules

It has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplacian. This has led to the hope that the homogenous Boltzmann equation enjoys similar regularity properties as the heat equation with a fractional Laplacian. In particular, the weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation with initial datum in $L^1_2(\mathbb{R}^d)\cap L\log L(\mathbb{R}^d)$, i.e., finite mass, energy and entropy, should immediately become Gevrey regular for strictly positive times. We prove this conjecture for Maxwellian molecules.

math.AP↗
arXiv2014

Spectral Theory near Thresholds for Weak Interactions with Massive Particles

We consider a Hamiltonian describing the weak decay of the massive vector boson Z0 into electrons and positrons. We show that the spectrum of the Hamiltonian is composed of a unique isolated ground state and a semi-axis of essential spectrum. Using a suitable extension of Mourre's theory, we prove that the essential spectrum below the boson mass is purely absolutely continuous.

math-ph↗
arXiv2010

Quantitative estimates on the Hydrogen ground state energy in non-relativistic QED

In this paper, we determine the exact expression for the hydrogen binding energy in the Pauli-Fierz model up to the order $O(α^5\logα^{-1})$, where $α$ denotes the finestructure constant, and prove rigorous bounds on the remainder term of the order $o(α^5\logα^{-1})$. As a consequence, we prove that the binding energy is not a real analytic function of $α$, and verify the existence of logarithmic corrections to the expansion of the ground state energy in powers of $α$, as conjectured in the recent literature.

math-ph↗
arXiv2007

On the ground state energy of the translation invariant Pauli-Fierz model

In this note, we determine the ground state energy of the translation invariant Pauli-Fierz model to subleading order $O(α^3)$ with respect to powers of the finestructure constant $α$, and prove rigorous error bounds of order $O(α^{4})$. A main objective of our argument is its brevity.

math-ph↗