arXiv subjects

Éric Séré

Publications and source records attributed to Éric Séré.

7 recordsLinked to original sources

arXiv2025

Peierls Instability in Hexagonal Lattices

We investigate a conventional tight-binding model for graphene, where distortion of the honeycomb lattice is allowed, but penalized by a quadratic energy. We prove that the optimal 3-periodic lattice configuration has Kekulé O-type symmetry, and that for a sufficiently small elasticity parameter, the minimizer is not translation-invariant. Conversely, we prove that for a large elasticity parameter the translation-invariant configuration is the unique minimizer.

math-ph↗

arXiv2024

Exponential decay of the critical points in a discrete model of polyacetylene

In this paper we consider stationary states of the SSH model for infinite polyacetylene chains that are homoclinic or heteroclinic connections between two-periodic dimerized states. We prove that such connections converge exponentially fast to the corresponding asymptotic periodic states.

math-ph↗

arXiv2024

A local surjection theorem with continuous inverse in Banach spaces

In this paper we prove a local surjection theorem with continuous right-inverse for maps between Banach spaces, and we apply it to a class of inversion problems with loss of derivatives.

math.FA↗

arXiv2023

Asymptotic estimates for the wave functions of the Dirac-Coulomb operator and applications

In this paper we prove some uniform asymptotic estimates for confluent hypergeometric functions making use of the steepest-descent method. As an application, we obtain Strichartz estimates with loss of angular derivatives for the massless Dirac-Coulomb equation in $3D$.

math.AP↗

arXiv2021

Dirac-Coulomb operators with general charge distribution. I. Distinguished extension and min-max formulas

This paper is the first of a series where we study the spectral properties of Dirac operators with the Coulomb potential generated by any finite signed charge distribution $μ$. We show here that the operator has a unique distinguished self-adjoint extension under the sole condition that $μ$ has no atom of weight larger than or equal to one. Then we discuss the case of a positive measure and characterize the domain using a quadratic form associated with the upper spinor, following earlier works by Esteban and Loss. This allows us to provide min-max formulas for the eigenvalues in the gap. In the event that some eigenvalues have dived into the negative continuum, the min-max formulas remain valid for the remaining ones. At the end of the paper we also discuss the case of multi-center Dirac-Coulomb operators corresponding to $μ$ being a finite sum of deltas.

math.SP↗

arXiv2020

Dirac-Coulomb operators with general charge distribution. II. The lowest eigenvalue

Consider the Coulomb potential $-μ\ast|x|^{-1}$ generated by a non-negative finite measure $μ$. It is well known that the lowest eigenvalue of the corresponding Schrödinger operator $-Δ/2-μ\ast|x|^{-1}$ is minimized, at fixed mass $μ(\mathbb{R}^3)=ν$, when $μ$ is proportional to a delta. In this paper we investigate the conjecture that the same holds for the Dirac operator $-iα\cdot\nabla+β-μ\ast|x|^{-1}$. In a previous work on the subject we proved that this operator is self-adjoint when $μ$ has no atom of mass larger than or equal to 1, and that its eigenvalues are given by min-max formulas. Here we consider the critical mass $ν_1$, below which the lowest eigenvalue does not dive into the lower continuum spectrum for all $μ\geq0$ with $μ(\mathbb{R}^3)<ν_1$. We first show that $ν_1$ is related to the best constant in a new scaling-invariant Hardy-type inequality. Our main result is that for all $0\leqν<ν_1$, there exists an optimal measure $μ\geq0$ giving the lowest possible eigenvalue at fixed mass $μ(\mathbb{R}^3)=ν$, which concentrates on a compact set of Lebesgue measure zero. The last property is shown using a new unique continuation principle for Dirac operators. The existence proof is based on the concentration-compactness principle.

math.SP↗

arXiv2017

Derivation of the magnetic Euler-Heisenberg energy

In quantum field theory, the vacuum is a fluctuating medium which behaves as a nonlinear polarizable material. In this article, we perform the first rigorous derivation of the magnetic Euler-Heisenberg effective energy, a nonlinear functional that describes the effective fluctuations of the quantum vacuum in a classical magnetic field. We start from a classical magnetic field in interaction with a quan-tized Dirac field in its ground state, and we study a limit in which the classical magnetic field is slowly varying. After a change of scales, this is equivalent to a semi-classical limit $\hbar\to0$, with a strong magnetic field of order $1/\hbar$. In this regime, we prove that the energy of Dirac's polarized vacuum converges to the Euler-Heisenberg functional. The model has ultraviolet divergences, which we regularize using the Pauli-Villars method. We also discuss how to remove the regularization of the Euler-Heisenberg effective Lagrangian, using charge renormaliza-tion, perturbatively to any order of the coupling constant.

math-ph↗