arXiv subjects

David Gontier

Publications and source records attributed to David Gontier.

At least 37 records · Page 2Linked to original sources

arXiv2021

The nonlinear Schrödinger equation for orthonormal functions: I. Existence of ground states

We study the nonlinear Schrödinger equation for systems of $N$ orthonormal functions. We prove the existence of ground states for all $N$ when the exponent $p$ of the non linearity is not too large, and for an infinite sequence $N_j$ tending to infinity in the whole range of possible $p$'s, in dimensions $d\geq1$. This allows us to prove that translational symmetry is broken for a quantum crystal in the Kohn-Sham model with a large Dirac exchange constant.

math.AP↗

arXiv2020

The periodic Lieb-Thirring inequality

We discuss the Lieb-Thirring inequality for periodic systems, which has the same optimal constant as the original inequality for finite systems. This allows us to formulate a new conjecture about the value of its best constant. To demonstrate the importance of periodic states, we prove that the 1D Lieb-Thirring inequality at the special exponent $γ=3/2$ admits a one-parameter family of periodic optimizers, interpolating between the one-bound state and the uniform potential. Finally, we provide numerical simulations in 2D which support our conjecture that optimizers could be periodic.

math-ph↗

arXiv2020

The nonlinear Schrödinger equation for orthonormal functions: II. Application to Lieb-Thirring inequalities

In this paper we disprove part of a conjecture of Lieb and Thirring concerning the best constant in their eponymous inequality. We prove that the best Lieb-Thirring constant when the eigenvalues of a Schrödinger operator $-Δ+V(x)$ are raised to the power $κ$ is never given by the one-bound state case when $κ>\max(0,2-d/2)$ in space dimension $d\geq1$. When in addition $κ\geq1$ we prove that this best constant is never attained for a potential having finitely many eigenvalues. The method to obtain the first result is to carefully compute the exponentially small interaction between two Gagliardo-Nirenberg optimisers placed far away. For the second result, we study the dual version of the Lieb-Thirring inequality, in the same spirit as in Part I of this work (D. Gontier, M. Lewin & F.Q. Nazar, arXiv:2002.04963). In a different but related direction, we also show that the cubic nonlinear Schrödinger equation admits no orthonormal ground state in 1D, for more than one function.

math.AP↗

arXiv2019

Edge states in ordinary differential equations for dislocations

In this article, we study Schrödinger operators on the real line, when the external potential represents a dislocation in a periodic medium. We study how the spectrum varies with the dislocation parameter. We introduce several integer-valued indices, including Chern number for bulk indices, and various spectral flows for edge indices. We prove that all these indices coincide, providing a proof a bulk-edge correspondence in this case. The study is also made for dislocations in Dirac models on the real line. We prove that 0 is always an eigenvalue of such operators.

math-ph↗

arXiv2019

Spin symmetry breaking in the translation-invariant Hartree-Fock electron gas

We study the breaking of spin symmetry for the nonlinear Hartree-Fock model describing an infinite translation-invariant interacting quantum gas (fluid phase). At zero temperature and for the Coulomb interaction in three space dimensions, we can prove the existence of a unique first order transition between a pure ferromagnetic phase at low density and a paramagnetic phase at high density. Multiple first or second order transitions can happen for other interaction potentials, as we illustrate on some examples. At positive temperature $T>0$ we compute numerically the phase diagram in the Coulomb case. We find the paramagnetic phase at high temperature or high density and a region where the system is ferromagnetic. We prove that the equilibrium state is unique and paramagnetic at high temperature or high density.

math-ph↗

arXiv2019

The reduced Hartree-Fock model with self-generated magnetic fields

We study the well-posedness of the reduced Hartree-Fock model for molecules and perfect crystals when taking into account a self-generated magnetic field. We exhibit a critical value $α_c > 0$ such that, if the fine structure constant $α$ is smaller than $α_c$, then the corresponding system is stable, whereas if $α$ is greater than $α_c$, it is unstable. We give an explicit characterisation of $α_c$ as a minimisation problem over the set of zero-modes, and we prove that the critical values for the molecular case and the periodic case coincide. Finally, we prove the existence of minimisers when the system is neutral or positively charged.

math-ph↗

arXiv2019

Lower Bound on the Hartree-Fock Energy of the Electron Gas

The Hartree-Fock ground state of the Homogeneous Electron Gas is never translation invariant, even at high densities. As proved by Overhauser, the (paramagnetic) free Fermi Gas is always unstable under the formation of spin or charge density waves. We give here the first explicit bound on the energy gain due to the breaking of translational symmetry. Our bound is exponentially small at high density, which justifies posteriori the use of the non-interacting Fermi Gas as a reference state in the large-density expansion of the correlation energy of the Homogeneous Electron Gas. We are also able to discuss the positive temperature phase diagram and prove that the Overhauser instability only occurs at temperatures which are exponentially small at high density. Our work sheds a new light on the Hartree-Fock phase diagram of the Homogeneous Electron Gas.

cond-mat.str-el↗

arXiv2018

Numerical construction of Wannier functions through homotopy

We provide a mathematically proven, simple and efficient algorithm to build localised Wannier functions, with the only requirement that the system has vanishing Chern numbers. Our algorithm is able to build localised Wannier for topological insulators such as the Kane-Mele model. It is based on an explicit and constructive proof of homotopies for the unitary group. We provide numerical tests validating the methods for several systems, including the Kane-Mele model.

math-ph↗

arXiv2017

Localised Wannier functions in metallic systems

The existence and construction of exponentially localised Wannier functions for insulators is a well-studied problem. In comparison, the case of metallic systems has been much less explored, even though localised Wannier functions constitute an important and widely used tool for the numerical band interpolation of metallic condensed matter systems. In this paper we prove that, under generic conditions, $N$ energy bands of a metal can be exactly represented by $N+1$ Wannier functions decaying faster than any polynomial. We also show that, in general, the lack of a spectral gap does not allow for exponential decay.

math-ph↗

arXiv2017

Numerical reconstruction of the first band(s) in an inverse Hill's problem

This paper concerns an inverse band structure problem for one dimensional periodic Schrödinger operators (Hill's operators). Our goal is to find a potential for the Hill's operator in order to reproduce as best as possible some given target bands, which may not be realisable. We recast the problem as an optimisation problem, and prove that this problem is well-posed when considering singular potentials (Borel measures). We then propose different algorithms to tackle the problem numerically.

math.OC↗

arXiv2017

Sub-wavelength focusing of acoustic waves in bubbly media

The purpose of this paper is to investigate acoustic wave scattering by a large number of bubbles in a liquid at frequencies near the Minnaert resonance frequency. This bubbly media has been exploited in practice to obtain super-focusing of acoustic waves. Using layer potential techniques we derive the scattering function for a single spherical bubble excited by an incident wave in the low frequency regime. We then derive the point scatter approximation for multiple scattering by N bubbles. We describe several numerical experiments based on the point scatterer approximation that demonstrate the possibility of achieving super-focusing using bubbly media.

math.AP↗

arXiv2017

Supercell calculations in the reduced Hartree-Fock model for crystals with local defects

In this article, we study the speed of convergence of the supercell reduced Hartree-Fock~(rHF) model towards the whole space rHF model in the case where the crystal contains a local defect. We prove that, when the defect is charged, the defect energy in a supercell model converges to the full rHF defect energy with speed $L^{-1}$, where $L^3$ is the volume of the supercell. The convergence constant is identified as the Makov-Payne correction term when the crystal is isotropic cubic. The result is extended to the non-isotropic case.

math-ph↗

arXiv2016

A mathematical and numerical framework for bubble meta-screens

The aim of this paper is to provide a mathematical and numerical framework for the analysis and design of bubble meta-screens. An acoustic meta-screen is a thin sheet with patterned subwavelength structures, which nevertheless has a macroscopic effect on the acoustic wave propagation. In this paper, periodic subwavelength bubbles mounted on a reflective surface (with Dirichlet boundary condition) is considered. It is shown that the structure behaves as an equivalent surface with Neumann boundary condition at the Minnaert resonant frequency which corresponds to a wavelength much greater than the size of the bubbles. Analytical formula for this resonance is derived. Numerical simulations confirm its accuracy and show how it depends on the ratio between the periodicity of the lattice, the size of the bubble, and the distance from the reflective surface. The results of this paper formally explain the super-absorption behavior observed in [V. Leroy et al., Phys. Rev. B, 2015].

math.AP↗

arXiv2016

Minnaert resonances for acoustic waves in bubbly media

Through the application of layer potential techniques and Gohberg-Sigal theory we derive an original formula for the Minnaert resonance frequencies of arbitrarily shaped bubbles. We also provide a mathematical justification for the monopole approximation of scattering of acoustic waves by bubbles at their Minnaert resonant frequency. Our results are complemented by several numerical examples which serve to validate our formula in two dimensions.

math.AP↗

arXiv2015

A mathematical analysis of the GW0 method for computing electronic excited energies of molecules

This paper analyses the GW method for finite electronic systems. In a first step, we provide a mathematical framework for the usual one-body operators that appear naturally in many-body perturbation theory. We then discuss the GW equations which construct an approximation of the one-body Green's function, and give a rigorous mathematical formulation of these equations. Finally, we study the well-posedness of the GW0 equations, proving the existence of a unique solution to these equations in a perturbative regime.

math-ph↗

arXiv2015

Convergence rates of supercell calculations in the reduced Hartree-Fock model

This article is concerned with the numerical simulations of perfect crystals. We study the rate of convergence of the reduced Hartree-Fock (rHF) model in a supercell towards the periodic rHF model in the whole space. We prove that, whenever the crystal is an insulator or a semi-conductor, the supercell energy per unit cell converges exponentially fast towards the periodic rHF energy per unit cell, with respect to the size of the supercell.

math-ph↗

arXiv2015

Pure-state $N$-representability in current-spin-density-functional theory

This paper is concerned with the pure-state $N$-representability problem for systems under a magnetic field. Necessary and sufficient conditions are given for a spin-density $2 \times 2$ matrix $R$ to be representable by a Slater determinant. We also provide sufficient conditions on the paramagnetic current $\mathbb{j}$ for the pair $(R, \mathbb{j})$ to be Slater-representable in the case where the number of electrons $N$ is greater than 12. The case $N < 12$ is left open.

quant-ph↗

arXiv2014

Existence of minimizers for Kohn-Sham within the Local Spin Density Approximation

The purpose of this article is to extend the work by Anantharaman and Cancès [1], and prove the existence of minimizers for the spin-polarized Kohn-Sham model in the presence of a magnetic field within the local spin density approximation. We show that for any magnetic field that vanishes at infinity, the existence of minimizers is ensured for neutral or positively charged systems. The proof relies on classical concentration-compactness techniques.

math-ph↗