We consider the dynamics of a 2D Bose gas with an interaction potential of the form N 2-1w (N center dot ) for E (0,. 3 ). The interaction may be chosen to be negative and large, leading to the instability regime 2 where the corresponding focusing cubic nonlinear Schr & ouml,.dinger equation (NLS) may blow up in finite time. We show that to leading order, the N-body quantum dynamics can be effectively described by the NLS prior to the blow-up time. Moreover, we prove the validity of the Bogoliubov approximation, where the excitations from the condensate are captured in a norm approximation of the many-body dynamics.

We give a new proof of the compactness of minimizing sequences of the Sobolev inequalities in the critical case. Our approach relies on a simplified version of the concentration-compactness principle, which does not require any refinement of the Sobolev embedding theorem.

We prove a Cwikel-Lieb-Rozenblum type inequality for the number of negative eigenvalues of the Hardy-Schr & ouml,.dinger operator -Delta-(d-2)(2)/(4|x|(2))-W(x) on L-2(R-d). The bound is given in terms of a weighted Ld/2-norm of W which is sharp in both large and small coupling regimes. We also obtain a similar bound for the fractional Laplacian. (c) 2024 The Author(s). Published by Elsevior Masson SAS. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

We extend the Moser-Trudinger inequality of one function to systems of orthogonal functions. Our results are asymptotically sharp when applied to the collective behavior of eigenfunctions of Schr delta dinger operators on bounded domains.

We consider an inhomogeneous system of N bosons in R3 confined by an external potential and interacting via a repulsive potential of the form N2V(N(x - y)). We prove that the low-energy excitation spectrum of the system is determined by the eigenvalues of an effective one-particle operator, which agrees with Bogoliubov's approximation. (c) 2023 Elsevier Masson SAS. All rights reserved.

"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."

We consider the low density Bose gas in the thermodynamic limit with a three-body interaction potential. We prove that the leading order of the ground state energy of the system is determined completely in terms of the scattering energy of the interaction potential. The corresponding result for two-body interactions was proved in seminal papers of Dyson [Phys. Rev. 106, 20-26 (1957)] and of Lieb and Yngvason Published under an exclusive license by AIP Publishing.

We review our recent study on the ground state energy of dilute Bose gases with three-body interactions. The main feature of our results is the emergence of the 3D energy-critical Schrodinger equation to describe the ground state energy of a Bose-Einstein condensate, where the nonlinearity strength is determined by a zero-scattering problem. Several open questions are also discussed. Published under an exclusive license by AIP Publishing.

We consider the quantum many-body evolution of a homogeneous Fermi gas in three dimensions in the coupled semiclassical and mean-field scaling regime. We study a class of initial data describing collective particle-hole pair excitations on the Fermi ball. Using a rigorous version of approximate bosonization, we prove that the many-body evolution can be approximated in Fock space norm by a quasi-free bosonic evolution of the collective particle-hole excitations.

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.

We consider the focusing nonlinear Schrodinger equation with the critical inverse square potential. We give the first proof of the uniqueness of the ground state solution. Consequently, we obtain a sharp Hardy-Gagliardo-Nirenberg interpolation inequality. Moreover, we provide a complete characterization for the minimal mass blow-up solutions to the time dependent problem. (C) 2021 Elsevier Inc. All rights reserved.

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.

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.

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 Schrodinger-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.

We study the collapse of a many-body system which is used to model two-component Bose-Einstein condensates with attractive intra-species interactions and either attractive or repulsive inter-species interactions. Such a system consists a mixture of two different species for N identical bosons in R2, interacting with potentials rescaled in the mean-field manner -N2 beta-1w(sigma)(N beta x), we first show that the leading order of the quantum energy is captured correctly by the Gross-Pitaevskii energy. Secondly, we investigate the blow-up behavior of the quantum energy as well as the ground states when N ->infinity, and either the total interaction strength of intra-species and inter-species or the strengths of intra-species interactions of each component approach sufficiently slowly a critical value, which is the critical strength for the focusing Gross-Pitaevskii functional. We prove that the many-body ground states fully condensate on the (unique) Gagliardo-Nirenberg solution.

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 Schrodinger (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. Published under license by AIP Publishing.