arXiv subjects
Publications and source records attributed to Phan Thành Nam.
We announce a new theorem bearing on high-temperature 2D Bose gases. In a certain mean-field-like regime, the grand-canonical quantum Gibbs state reduces to a nonlinear Gibbs measure constructed from a renormalized mean-field energy functional. This establishes a rigorous connection between quantum fields and (singular) classical fields in the regime of interest.
We consider the Gross-Pitaevskii equation describing an attractive Bose gas trapped to a quasi 2D layer by means of a purely harmonic potential, and which rotates at a fixed speed of rotation $Ω$. First we study the behavior of the ground state when the coupling constant approaches $a\_*$ , the critical strength of the cubic nonlinearity for the focusing nonlinear Schr{ö}dinger equation. We prove that blow-up always happens at the center of the trap, with the blow-up profile given by the Gagliardo-Nirenberg solution. In particular, the blow-up scenario is independent of $Ω$, to leading order. This generalizes results obtained by Guo and Seiringer (Lett. Math. Phys., 2014, vol. 104, p. 141--156) in the non-rotating case. In a second part we consider the many-particle Hamiltonian for $N$ bosons, interacting with a potential rescaled in the mean-field manner $--a\_N N^{2β--1} w(N^β x), with $w$ a positive function such that $\int\_{\mathbb{R}^2} w(x) dx = 1$. Assuming that $β< 1/2$ and that $a\_N \to a\_*$ sufficiently slowly, we prove that the many-body system is fully condensed on the Gross-Pitaevskii ground state in the limit $N \to \infty$.
We study the norm approximation to the Schrödinger dynamics of $N$ bosons in $\mathbb{R}^d$ ($d=1,2$) with an interaction potential of the form $N^{dβ-1}w(N^β(x-y))$. Here we are interested in the focusing case $w\le 0$. Assuming that there is complete Bose-Einstein condensation in the initial state, we show that in the large $N$ limit, the evolution of the condensate is effectively described by a nonlinear Schrödinger equation and the evolution of the fluctuations around the condensate is governed by a quadratic Hamiltonian, resulting from Bogoliubov approximation. Our result holds true for all $β>0$ when $d=1$ and for all $0<β<1$ when $d=2$.
We consider the dynamics of a large quantum system of $N$ identical bosons in 3D interacting via a two-body potential of the form $N^{3β-1} w(N^β(x-y))$. For fixed $0\leq β<1/3$ and large $N$, we obtain a norm approximation to the many-body evolution in the $N$-particle Hilbert space. The leading order behaviour of the dynamics is determined by Hartree theory while the second order is given by Bogoliubov theory.
In 1975, Lieb and Thirring derived a semiclassical lower bound on the kinetic energy for fermions, which agrees with the Thomas-Fermi approximation up to a constant factor. Whenever the optimal constant in their bound coincides with the semiclassical one is a long-standing open question. We prove an improved bound with the semiclassical constant and a gradient error term which is of lower order.
We compute the energy needed to add or remove one particle from a homogeneous system of $N$ bosons on a torus. We focus on the mean-field limit when $N$ becomes large and the strength of particle interactions is proportional to $N^{-1}$, which allows us to justify Bogoliubov's approximation.
We prove that in Thomas-Fermi-Dirac-von Weizsäcker theory, a nucleus of charge $Z>0$ can bind at most $Z+C$ electrons, where $C$ is a universal constant. This result is obtained through a comparison with Thomas-Fermi theory which, as a by-product, gives bounds on the screened nuclear potential and the radius of the minimizer. A key ingredient of the proof is a novel technique to control the particles in the exterior region, which also applies to the liquid drop model with a nuclear background potential.
We study the norm approximation to the Schrödinger dynamics of $N$ bosons in $\mathbb{R}^3$ with an interaction potential of the form $N^{3β-1}w(N^β(x-y))$. Assuming that in the initial state the particles outside of the condensate form a quasi-free state with finite kinetic energy, we show that in the large $N$ limit, the fluctuations around the condensate can be effectively described using Bogoliubov approximation for all $0\le β<1/2$. The range of $β$ is expected to be optimal for this large class of initial states.
We review some recent results on the norm approximation to the Schrödinger dynamics. We consider $N$ bosons in $\mathbb{R}^3$ with an interaction potential of the form $N^{3β-1}w(N^β(x-y))$ with $0\le β<1/2$, and show that in the large $N$ limit, the fluctuations around the condensate can be effectively described using Bogoliubov approximation.
We study the ionization problem in the Thomas-Fermi-Dirac-von Weizsäcker theory for atoms and molecules. We prove the nonexistence of minimizers for the energy functional when the number of electrons is large and the total nuclear charge is small. This nonexistence result also applies to external potentials decaying faster than the Coulomb potential. In the case of arbitrary nuclear charges, we obtain the nonexistence of stable minimizers and radial minimizers.
We consider an atom described by Müller theory, which is similar to Hartree-Fock theory, but with a modified exchange term. We prove that a nucleus of charge Z can bind at most Z+C electrons, where C is a universal constant. Our proof proceeds by comparison with Thomas-Fermi theory and a key ingredient is a novel bound on the number of electrons far from the nucleus.
We give a simplified proof of the nonexistence of large nuclei in the liquid drop model and provide an explicit bound. Our bound is within a factor of 2.3 of the conjectured value and seems to be the first quantitative result.
We summarize recent results on positive temperature equilibrium states of large bosonic systems. The emphasis will be on the connection between bosonic grand-canonical thermal states and the (semi-) classical Gibbs measures on one-body quantum states built using the corresponding mean-field energy functionals. An illustrative comparison with the case of "distinguishable" particles (boltzons) is provided.
We provide general conditions for which bosonic quadratic Hamiltonians on Fock spaces can be diagonalized by Bogoliubov transformations. Our results cover the case when quantum systems have infinite degrees of freedom and the associated one-body kinetic and paring operators are unbounded. Our sufficient conditions are optimal in the sense that they become necessary when the relevant one-body operators commute.
We consider a 2D quantum system of $N$ bosons in a trapping potential $|x|^s$, interacting via a pair potential of the form $N^{2β-1} w(N^βx)$. We show that for all $0 \textless{} β\textless{} (s+1)/(s+2)$, the leading order behavior of ground states of the many-body system is described in the large $N$ limit by the corresponding cubic nonlinear Schr{ö}dinger energy functional. Our result covers the focusing case ($w \leq 0$) where even the stability of the many-body system is not obvious. This answers an open question mentioned by X.~Chen and J.~Holmer for harmonic traps ($s=2$). Together with the BBGKY hierarchy approach used by these authors, our result implies the convergence of the many-body quantum dynamics to the focusing NLS equation with harmonic trap for all $0 \textless{} β\textless{} 3/4$.
We study the ground state of a trapped Bose gas, starting from the full many-body Schr{ö}dinger Hamiltonian, and derive the nonlinear Schr{ö}dinger energy functional in the limit of large particle number, when the interaction potential converges slowly to a Dirac delta function. Our method is based on quantitative estimates on the discrepancy between the full many-body energy and its mean-field approximation using Hartree states. These are proved using finite dimensional localization and a quantitative version of the quantum de Finetti theorem. Our approach covers the case of attractive interactions in the regime of stability. In particular, our main new result is a derivation of the 2D attractive nonlinear Schr{ö}dinger ground state.
We study the ground state of a dilute Bose gas in a scaling limit where the Gross-Pitaevskii functional emerges. This is a repulsive non-linear Schrödinger functional whose quartic term is proportional to the scattering length of the interparticle interaction potential. We propose a new derivation of this limit problem, with a method that bypasses some of the technical difficulties that previous derivations had to face. The new method is based on a combination of Dyson's lemma, the quantum de Finetti theorem and a second moment estimate for ground states of the effective Dyson Hamiltonian. It applies equally well to the case where magnetic fields or rotation are present.
We prove analogues of the Lieb-Thirring and Hardy-Lieb-Thirring inequalities for many-body quantum systems with fractional kinetic operators and homogeneous interaction potentials, where no anti-symmetry on the wave functions is assumed. These many-body inequalities imply interesting one-body interpolation inequalities, and we show that the corresponding one- and many-body inequalities are actually equivalent in certain cases.