arXiv subjects
Publications and source records attributed to Phan Thành Nam.
We consider N bosons on the unit torus $Λ= [0,1]^3$ in the Gross-Pitaevski regime where the interaction potential scales as $N^2 V (N(x -y))$. We prove that the thermal equilibrium at low temperatures exhibits the Bose-Einstein condensation in a strong sense, namely the probability of having $n$ particles outside of the condensation decays exponentially in $n$.
We prove an analogue of the Lieb--Thirring inequality for many-body quantum systems with the kinetic operator $\sum_i (-Δ_i)^s$ and the interaction potential of the form $\sum_i δ_i^{-2s}$ where $δ_i$ is the nearest-neighbor distance to the point $x_i$. Our result extends the standard Lieb--Thirring inequality for fermions and applies to quantum systems without the anti-symmetry assumption on the wave functions. Additionally, we derive similar results for the Hardy--Lieb--Thirring inequality and obtain the asymptotic behavior of the optimal constants in the strong coupling limit.
We derive a family of interpolation estimates which improve Hardy's inequality and cover the Sobolev critical exponent. We also determine all optimizers among radial functions in the endpoint case and discuss open questions on nonrestricted optimizers.
We consider a translation-invariant system of $N$ bosons in $\mathbb{T}^{3}$ that interact through a repulsive two-body potential with scattering length of order $N^{-1}$ in the limit $N\to \infty$. We derive second order expressions for the one- and two-particle reduced density matrix matrices of the Gibbs state at fixed positive temperatures, thus obtaining a justification of Bogoliubov's prediction on the fluctuations around the condensate.
We prove a Cwikel-Lieb-Rozenblum type inequality for the number of negative eigenvalues of the Hardy-Schrödinger operator $-Δ- (d-2)^2/(4|x|^2) -W(x)$ on $L^2(\mathbb{R}^d)$. The bound is given in terms of a weighted $L^{d/2}-$norm of $W$ which is sharp in both large and small coupling regimes. We also obtain a similar bound for the fractional Laplacian.
We consider a low density Bose gas interacting through a repulsive potential in the thermodynamic limit. We justify, as a rigorous lower bound, a Lee--Huang--Yang type formula for the free energy at suitably low temperatures, where the modified excitation spectrum leads to a second order correction of the same order as the Lee--Huang--Yang correction to the ground state energy.
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 Schrödinger operators on bounded domains.
We present a general approach to justify the random phase approximation for the homogeneous Fermi gas in three dimensions in the mean-field scaling regime. We consider a system of $N$ fermions on a torus, interacting via a two-body repulsive potential proportional to $N^{-\frac{1}{3}}$. In the limit $N\to\infty$, we derive the exact leading order of the correlation energy and the bosonic elementary excitations of the system, which are consistent with the prediction of the random phase approximation in the physics literature.
We consider the dynamics of a 2D Bose gas with an interaction potential of the form $N^{2β-1}w(N^β\cdot)$ for $β\in (0,3/2)$. The interaction may be chosen to be negative and large, leading to the instability regime where the corresponding focusing cubic nonlinear Schr{ö}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 consider an inhomogeneous system of $N$ bosons in $\mathbb{R}^3$ confined by an external potential and interacting via a repulsive potential of the form $N^2 V(N(x-y))$. 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.
We prove a rigorous upper bound on the correlation energy of interacting fermions in the mean-field regime for a wide class of interaction potentials. Our result covers the Coulomb potential, and in this case we obtain the analogue of the Gell-Mann$-$Brueckner formula $c_{1}ρ\log\left(ρ\right)+c_{2}ρ$ in the high density limit. We do this by refining the analysis of our bosonization method to deal with singular potentials, and to capture the exchange contribution which is absent in the purely bosonic picture.
We give a proof of the Lieb-Thirring inequality on the kinetic energy of orthonormal functions by using a microlocal technique, in which the uncertainty and exclusion principles are combined through the use of the Besicovitch covering lemma.
We consider an effective quasi-bosonic Hamiltonian of the electron gas which emerges naturally from the random phase approximation and describes the collective excitations of the gas. By a rigorous argument, we explain how the plasmon modes can be interpreted as a special class of approximate eigenstates of this model.
We consider a trapped dilute gas of $N$ bosons in $\mathbb{R}^3$ interacting via a three-body interaction potential of the form $N\, V(N^{1/2}(x-y,x-z))$. In the limit $N\to \infty$, we prove that every approximate ground state of the system is a convex superposition of minimizers of a 3D energy-critical nonlinear Schrödinger functional where the nonlinear coupling constant is proportional to the scattering energy of the interaction potential. In particular, the $N$-body ground state exhibits complete Bose--Einstein condensation if the nonlinear Schrödinger minimizer is unique up to a complex phase.
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 (1957) and Lieb--Yngvason (1998).
While it is well-known experimentally that a neutral atom can bind at most one or two extra electrons, deriving this fact rigorously from first principles of quantum mechanics remains a very challenging problem, often referred to as the ionization conjecture. We will review some of Elliott H. Lieb's fundamental contributions to this topic and discuss their impacts on several recent developments.
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 Schrödinger 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 will also be discussed.
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 quasifree bosonic evolution of the collective particle-hole excitations.