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Phan Thành Nam

Publications and source records attributed to Phan Thành Nam.

At least 55 records · Page 3Linked to original sources
arXiv2021

Two-term expansion of the ground state one-body density matrix of a mean-field Bose gas

We consider the homogeneous Bose gas on a unit torus in the mean-field regime when the interaction strength is proportional to the inverse of the particle number. In the limit when the number of particles becomes large, we derive a two-term expansion of the one-body density matrix of the ground state. The proof is based on a cubic correction to Bogoliubov's approximation of the ground state energy and the ground state.

math-ph↗
arXiv2021

Correlation Energy of a Weakly Interacting Fermi Gas

We derive rigorously the leading order of the correlation energy of a Fermi gas in a scaling regime of high density and weak interaction. The result verifies the prediction of the random-phase approximation. Our proof refines the method of collective bosonization in three dimensions. We approximately diagonalize an effective Hamiltonian describing approximately bosonic collective excitations around the Hartree-Fock state, while showing that gapless and non-collective excitations have only a negligible effect on the ground state energy.

math-ph↗
arXiv2021

The Lieb-Thirring inequality for interacting systems in strong-coupling limit

We consider an analogue of the Lieb-Thirring inequality for quantum systems with homogeneous repulsive interaction potentials, but without the antisymmetry assumption on the wave functions. We show that in the strong-coupling limit, the Lieb-Thirring constant converges to the optimal constant of the one-body Gagliardo-Nirenberg interpolation inequality without interaction.

math-ph↗
arXiv2021

Optimal rate of condensation for trapped bosons in the Gross-Pitaevskii regime

We study the Bose-Einstein condensates of trapped Bose gases in the Gross-Pitaevskii regime. We show that the ground state energy and ground states of the many-body quantum system are correctly described by the Gross-Pitaevskii equation in the large particle number limit, and provide the optimal convergence rate. Our work extends the previous results of Lieb, Seiringer and Yngvason on the leading order convergence, and of Boccato, Brennecke, Cenatiempo and Schlein on the homogeneous gas. Our method relies on the idea of 'completing the square', inspired by recent works of Brietzke, Fournais and Solovej on the Lee-Huang-Yang formula, and a general estimate for Bogoliubov quadratic Hamiltonians on Fock space.

math-ph↗
arXiv2021

Existence and nonexistence in the liquid drop model

We revisit the liquid drop model with a general Riesz potential. Our new result is the existence of minimizers for the conjectured optimal range of parameters. We also prove a conditional uniqueness of minimizers and a nonexistence result for heavy nuclei.

math.AP↗
arXiv2020

Classical field theory limit of many-body quantum Gibbs states in 2D and 3D

We provide a rigorous derivation of nonlinear Gibbs measures in two and three space dimensions, starting from many-body quantum systems in thermal equilibrium. More precisely, we prove that the grand-canonical Gibbs state of a large bosonic quantum system converges to the Gibbs measure of a nonlinear Schr{ö}dinger-type classical field theory, in terms of partition functions and reduced density matrices. The Gibbs measure thus describes the behavior of the infinite Bose gas at criticality, that is, close to the phase transition to a Bose-Einstein condensate. The Gibbs measure is concentrated on singular distributions and has to be appropriately renormalized, while the quantum system is well defined without any renormalization. By tuning a single real parameter (the chemical potential), we obtain a counter-term for the diverging repulsive interactions which provides the desired Wick renormalization of the limit classical theory. The proof relies on a new estimate on the entropy relative to quasi-free states and a novel method to control quantum variances.

math.AP↗
arXiv2020

Lieb-Thirring inequalities for wave functions vanishing on the diagonal set

We propose a general strategy to derive Lieb-Thirring inequalities for scale-covariant quantum many-body systems. As an application, we obtain a generalization of the Lieb-Thirring inequality to wave functions vanishing on the diagonal set of the configuration space, without any statistical assumption on the particles.

math-ph↗
arXiv2020

Improved stability for 2D attractive Bose gases

We study the ground-state energy of N attractive bosons in the plane. The interaction is scaled for the gas to be dilute, so that the corresponding mean-field problem is a local non-linear Schr{ö}dinger (NLS) equation. We improve the conditions under which one can prove that the many-body problem is stable (of the second kind). This implies, using previous results, that the many-body ground states and dynamics converge to the NLS ones for an extended range of diluteness parameters.

math.AP↗
arXiv2019

Derivation of 3D energy-critical nonlinear Schrödinger equation and Bogoliubov excitations for Bose gases

We derive the 3D quintic NLS as the mean field limit of a Bose gas with three-body interactions. The quintic NLS is energy-critical, leading to several new difficulties in comparison with the cubic NLS which emerges from Bose gases with pair-interactions. Our method is based on Bogoliubov's approximation, which also provides the information on the fluctuations around the condensate in terms of a norm approximation for the N-body wave function.

math-ph↗
arXiv2019

Optimal Upper Bound for the Correlation Energy of a Fermi Gas in the Mean-Field Regime

While Hartree-Fock theory is well established as a fundamental approximation for interacting fermions, it has been unclear how to describe corrections to it due to many-body correlations. In this paper we start from the Hartree-Fock state given by plane waves and introduce collective particle-hole pair excitations. These pairs can be approximately described by a bosonic quadratic Hamiltonian. We use Bogoliubov theory to construct a trial state yielding a rigorous Gell-Mann-Brueckner-type upper bound to the ground state energy. Our result justifies the random phase approximation in the mean-field scaling regime, for repulsive, regular interaction potentials.

math-ph↗
arXiv2018

Gibbs measures based on 1D (an)harmonic oscillators as mean-field limits

We prove that Gibbs measures based on 1D defocusing nonlinear Schr{ö}dinger functionals with sub-harmonic trapping can be obtained as the mean-field/large temperature limit of the corresponding grand-canonical ensemble for many bosons. The limit measure is supported on Sobolev spaces of negative regularity and the corresponding density matrices are not trace-class. The general proof strategy is that of a previous paper of ours, but we have to complement it with Hilbert-Schmidt estimates on reduced density matrices.

math-ph↗
arXiv2018

Fluctuations of $N$-particle quantum dynamics around the nonlinear Schrödinger equation

We consider a system of $N$ bosons interacting through a singular two-body potential scaling with $N$ and having the form $N^{3β-1} V (N^βx)$, for an arbitrary parameter $β\in (0,1)$. We provide a norm-approximation for the many-body evolution of initial data exhibiting Bose-Einstein condensation in terms of a cubic nonlinear Schrödinger equation for the condensate wave function and of a unitary Fock space evolution with a generator quadratic in creation and annihilation operators for the fluctuations.

math-ph↗
arXiv2018

Ground state energy of mixture of Bose gases

We consider the asymptotic behavior of a system of multi-component trapped bosons, when the total particle number $N$ becomes large. In the dilute regime, when the interaction potentials have the length scale of order $O(N^{-1})$, we show that the leading order of the ground state energy is captured correctly by the Gross-Pitaevskii energy functional and that the many-body ground state fully condensates on the Gross-Pitaevskii minimizers. In the mean-field regime, when the interaction length scale is $O(1)$, we are able to verify Bogoliubov's approximation and obtain the second order expansion of the ground state energy. While such asymptotic results have several precursors in the literature on one-component condensates, the adaption to the multi-component setting is non-trivial in various respects and the analysis will be presented in details.

math-ph↗
arXiv2018

Convergence of Levy-Lieb to Thomas-Fermi density functional

We prove that the Levy-Lieb density functional Gamma-converges to the Thomas-Fermi functional in the semiclassical mean-field limit. In particular, this aides an easy alternative proof of the validity of the atomic Thomas-Fermi theory which was first established by Lieb and Simon.

math-ph↗
arXiv2018

The Lieb-Thirring inequality revisited

We provide new estimates on the best constant of the Lieb-Thirring inequality for the sum of the negative eigenvalues of Schrödinger operators, which significantly improve the so far existing bounds.

math-ph↗