Mathieu Lewin
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View article: Liquid Drop Model for Nuclear Matter in the Low Density Limit
Liquid Drop Model for Nuclear Matter in the Low Density Limit Open
We consider the liquid drop model with a positive background density in the thermodynamic limit. We prove a two-term asymptotics for the ground state energy per unit volume in the dilute limit. Our proof justifies the expectation that opti…
View article: Some open mathematical problems concerning charged quantum particles
Some open mathematical problems concerning charged quantum particles Open
I present some open mathematical problems concerning electrons in quantum mechanics and charged particles in general. After discussing the Schrödinger Hamiltonian describing atoms and molecules with classical nuclei, I turn to infinite sys…
View article: Positive-density ground states of theGross–Pitaevskii equation
Positive-density ground states of theGross–Pitaevskii equation Open
International audience
View article: Ground State Energy Is Not Always Convex in the Number of Electrons
Ground State Energy Is Not Always Convex in the Number of Electrons Open
We provide the first counterexample showing that the ground state energy of electrons in an external Coulomb potential is not always a convex function of the number of electrons. This convexity has been conjectured for decades and plays an…
View article: Stability estimate for the Lane-Emden inequality
Stability estimate for the Lane-Emden inequality Open
The Lane-Emden inequality controls $\iint_{\mathbb{R}^{2d}}ρ(x)ρ(y)|x-y|^{-λ}\,dx\,dy$ in terms of the $L^1$ and $L^p$ norms of $ρ$. We provide a remainder estimate for this inequality in terms of a suitable distance of $ρ$ to the manifold…
View article: Mean-field limits for quantum systems and nonlinear Gibbs measures
Mean-field limits for quantum systems and nonlinear Gibbs measures Open
We consider the linear Schrödinger equation describing N quantum (bosonic) particles at equilibrium and study its behavior as N tends to infinity. We place the system in the mean-field regime, in which the particles are very tightly packed…
View article: Classical Density Functional Theory: The Local Density Approximation
Classical Density Functional Theory: The Local Density Approximation Open
We prove that the lowest free energy of a classical interacting system at temperature $T$ with a prescribed density profile $ρ(x)$ can be approximated by the local free energy $\int f_T(ρ(x))dx$, provided that $ρ$ varies slowly over suffic…
View article: Positive-density ground states of the Gross-Pitaevskii equation
Positive-density ground states of the Gross-Pitaevskii equation Open
We consider the nonlinear Gross-Pitaevskii equation at positive density, that is, for a bounded solution not tending to 0 at infinity. We focus on infinite ground states, which are by definition minimizers of the energy under local perturb…
View article: Classical Density Functional Theory: Representability and Universal Bounds
Classical Density Functional Theory: Representability and Universal Bounds Open
We provide upper and lower bounds on the lowest free energy of a classical system at given one-particle density $$\rho (x)$$ . We study both the canonical and grand-canonical cases, assuming the particles interact with a pair potential wh…
View article: The Nonlinear Schrödinger Equation for Orthonormal Functions : Existence of Ground States
The Nonlinear Schrödinger Equation for Orthonormal Functions : Existence of Ground States Open
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 Nj tending to inf…
View article: Classical Density Functional Theory: Representability and Universal Bounds
Classical Density Functional Theory: Representability and Universal Bounds Open
We provide upper and lower bounds on the lowest free energy of a classical system at given one-particle density $ρ(x)$. We study both the canonical and grand-canonical cases, assuming the particles interact with a pair potential which deca…
View article: DFT Exchange: Sharing Perspectives on the Workhorse of Quantum Chemistry and Materials Science
DFT Exchange: Sharing Perspectives on the Workhorse of Quantum Chemistry and Materials Science Open
In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practiti…
View article: DFT Exchange: Sharing Perspectives on the Workhorse of Quantum Chemistry and Materials Science
DFT Exchange: Sharing Perspectives on the Workhorse of Quantum Chemistry and Materials Science Open
In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practiti…
View article: Coulomb and Riesz gases: The known and the unknown
Coulomb and Riesz gases: The known and the unknown Open
We review what is known, unknown, and expected about the mathematical properties of Coulomb and Riesz gases. Those describe infinite configurations of points in Rd interacting with the Riesz potential ±|x|−s (respectively, −log |x| for s =…
View article: Improved Lieb-Oxford bound on the indirect and exchange energies
Improved Lieb-Oxford bound on the indirect and exchange energies Open
The Lieb-Oxford inequality provides a lower bound on the Coulomb energy of a classical system of $N$ identical charges only in terms of their one-particle density. We prove here a new estimate on the best constant in this inequality. Numer…
View article: Grand-Canonical Optimal Transport
Grand-Canonical Optimal Transport Open
We study a generalization of the multi-marginal optimal transport problem, which has no fixed number of marginals $N$ and is inspired of statistical mechanics. It consists in optimizing a linear combination of the costs for all the possibl…
View article: DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science
DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science Open
In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practiti…
View article: Dirac–Coulomb operators with general charge distribution I. Distinguished extension and min-max formulas
Dirac–Coulomb operators with general charge distribution I. Distinguished extension and min-max formulas Open
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 . We show here that the operator has a unique distinguished self-ad…
View article: Optimizers for the finite-rank Lieb-Thirring inequality
Optimizers for the finite-rank Lieb-Thirring inequality Open
The finite-rank Lieb-Thirring inequality provides an estimate on a Riesz sum of the $N$ lowest eigenvalues of a Schrödinger operator $-Δ-V(x)$ in terms of an $L^p(\mathbb{R}^d)$ norm of the potential $V$. We prove here the existence of an …
View article: Partial Differential Equations, Spectral Theory, and Mathematical Physics
Partial Differential Equations, Spectral Theory, and Mathematical Physics Open
We investigate spectral and asymptotic properties of the particular Schr\"odinger operator (also known as the Bloch-Torrey operator), $-\Delta + i g x$, in infinite periodically perforated domains of $\mathbb R^d$. We consider Dirichlet re…
View article: The periodic Lieb–Thirring inequality
The periodic Lieb–Thirring inequality Open
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 demonstr…
View article: The Hartree and Vlasov equations at positive density
The Hartree and Vlasov equations at positive density Open
We consider the nonlinear Hartree and Vlasov equations around a translation-invariant (homogeneous) stationary state in infinite volume, for a short range interaction potential. For both models, we consider time-dependent solutions which h…
View article: Universal Functionals in Density Functional Theory
Universal Functionals in Density Functional Theory Open
In this chapter we first review the Levy-Lieb functional, which gives the lowest kinetic and interaction energy that can be reached with all possible quantum states having a given density. We discuss two possible convex generalizations of …
View article: Floating Wigner crystal with no boundary charge fluctuations
Floating Wigner crystal with no boundary charge fluctuations Open
International audience
View article: Derivation of renormalized Gibbs measures from equilibrium many-body quantum Bose gases
Derivation of renormalized Gibbs measures from equilibrium many-body quantum Bose gases Open
We review our recent result on the rigorous derivation of the renormalized Gibbs measure from the many-body Gibbs state in 1D and 2D. The many-body renormalization is accomplished by simply tuning the chemical potential in the grand-canoni…