##### Uniqueness of ground state and minimal-mass blow-up solutions for focusing NLS with Hardy potential

D. Mukherjee, P.T. Nam, P.-T. Nguyen

Journal of Functional Analysis 281, 109092 (2021).

We consider the focusing nonlinear Schrödinger 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.

##### Correlation energy of a weakly interacting Fermi gas

N. Benedikter, P.T. Nam, M. Porta, B. Schlein, R. Seiringer

Inventiones mathematicae 225, 885–979 (2021).

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.

##### The Lieb–Thirring Inequality for Interacting Systems in Strong-Coupling Limit

K. Kögler, P.T. Nam

Archive for Rational Mechanics and Analysis 240, 1169–1202 (2021).

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.

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

M. Lewin, P.T. Nam, N. Rougerie

Inventiones Mathematicae (2021).

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.

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

M. Lewin, P.T. Nam, N. Rougerie

Inventiones mathematicae 224, 315–444 (2021).

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 Schrödinger-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.

##### Improved stability for 2D attractive Bose gases

P.T. Nam, N. Rougerie

Journal of Mathematical Physics 61, 21901 (2020).

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 Schrödinger (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.