The Ground State Energy of Dilute Bose Gas in Potentials with Positive Scattering Length

The leading term of the ground state energy/particle of a dilute gas of bosons with mass $m$ in the thermodynamic limit is $2\pi \hbar^2 a \rho/m$ when the density of the gas is $\rho$, the interaction potential is non-negative and the scattering length $a$ is positive. In this paper, we generalize the upper bound part of this result to any interaction potential with positive scattering length, i.e, $a>0$ and the lower bound part to some interaction potentials with shallow and/or narrow negative parts.Comment: Latex 28 page

Free Energy of a Dilute Bose Gas: Lower Bound

A lower bound is derived on the free energy (per unit volume) of a homogeneous Bose gas at density $\rho$ and temperature $T$. In the dilute regime, i.e., when $a^3\rho \ll 1$, where $a$ denotes the scattering length of the pair-interaction potential, our bound differs to leading order from the expression for non-interacting particles by the term $4\pi a (2\rho^2 - [\rho-\rho_c]_+^2)$. Here, $\rho_c(T)$ denotes the critical density for Bose-Einstein condensation (for the non-interacting gas), and $[ ]_+$ denotes the positive part. Our bound is uniform in the temperature up to temperatures of the order of the critical temperature, i.e., $T \sim \rho^{2/3}$ or smaller. One of the key ingredients in the proof is the use of coherent states to extend the method introduced in [arXiv:math-ph/0601051] for estimating correlations to temperatures below the critical one.Comment: LaTeX2e, 53 page

Derivation of the Gross-Pitaevskii Hierarchy

We report on some recent results regarding the dynamical behavior of a trapped Bose-Einstein condensate, in the limit of a large number of particles. These results were obtained in \cite{ESY}, a joint work with L. Erd\H os and H.-T. Yau.Comment: 15 pages; for the proceedings of the QMath9 International Conference, Giens, France, Sept. 200

A Rigorous Derivation of the Gross-Pitaevskii Energy Functional for a Two-Dimensional Bose Gas

We consider the ground state properties of an inhomogeneous two-dimensional Bose gas with a repulsive, short range pair interaction and an external confining potential. In the limit when the particle number $N$ is large but $\bar\rho a^2$ is small, where $\bar\rho$ is the average particle density and $a$ the scattering length, the ground state energy and density are rigorously shown to be given to leading order by a Gross-Pitaevskii (GP) energy functional with a coupling constant $g\sim 1/|\ln(\bar\rho a^2)|$. In contrast to the 3D case the coupling constant depends on $N$ through the mean density. The GP energy per particle depends only on $Ng$. In 2D this parameter is typically so large that the gradient term in the GP energy functional is negligible and the simpler description by a Thomas-Fermi type functional is adequate.Comment: 14 pages, no figures, latex 2e. References, some clarifications and an appendix added. To appear in Commun. Math. Phy

The TF Limit for Rapidly Rotating Bose Gases in Anharmonic Traps

Starting from the full many body Hamiltonian we derive the leading order energy and density asymptotics for the ground state of a dilute, rotating Bose gas in an anharmonic trap in the ` Thomas Fermi' (TF) limit when the Gross-Pitaevskii coupling parameter and/or the rotation velocity tend to infinity. Although the many-body wave function is expected to have a complicated phase, the leading order contribution to the energy can be computed by minimizing a simple functional of the density alone

Ground state energy of the low density Hubbard model

We derive a lower bound on the ground state energy of the Hubbard model for given value of the total spin. In combination with the upper bound derived previously by Giuliani, our result proves that in the low density limit, the leading order correction compared to the ground state energy of a non-interacting lattice Fermi gas is given by $8\pi a \rho_u \rho_d$, where $\rho_{u(d)}$ denotes the density of the spin-up (down) particles, and $a$ is the scattering length of the contact interaction potential. This result extends previous work on the corresponding continuum model to the lattice case.Comment: LaTeX2e, 18 page

Free Energies of Dilute Bose gases: upper bound

We derive a upper bound on the free energy of a Bose gas system at density $\rho$ and temperature $T$. In combination with the lower bound derived previously by Seiringer \cite{RS1}, our result proves that in the low density limit, i.e., when $a^3\rho\ll 1$, where $a$ denotes the scattering length of the pair-interaction potential, the leading term of $\Delta f$ the free energy difference per volume between interacting and ideal Bose gases is equal to 4\pi a (2\rho^2-[\rho-\rhoc]^2_+). Here, \rhoc(T) denotes the critical density for Bose-Einstein condensation (for the ideal gas), and $[\cdot ]_+$ $=$ $\max\{\cdot, 0\}$ denotes the positive part.Comment: 56 pages, no figure

The Second Order Upper Bound for the Ground Energy of a Bose Gas

Consider $N$ bosons in a finite box $\Lambda= [0,L]^3\subset \mathbf R^3$ interacting via a two-body smooth repulsive short range potential. We construct a variational state which gives the following upper bound on the ground state energy per particle $$\bar\lim_{\rho\to0} \bar \lim_{L \to \infty, N/L^3 \to \rho} (\frac{e_0(\rho)- 4 \pi a \rho}{(4 \pi a)^{5/2}(\rho)^{3/2}})\leq \frac{16}{15\pi^2},$$ where $a$ is the scattering length of the potential. Previously, an upper bound of the form $C 16/15\pi^2$ for some constant $C > 1$ was obtained in \cite{ESY}. Our result proves the upper bound of the the prediction by Lee-Yang \cite{LYang} and Lee-Huang-Yang \cite{LHY}.Comment: 62 pages, no figure

Second-order corrections to mean field evolution for weakly interacting Bosons. I

Inspired by the works of Rodnianski and Schlein and Wu, we derive a new nonlinear Schr\"odinger equation that describes a second-order correction to the usual tensor product (mean-field) approximation for the Hamiltonian evolution of a many-particle system in Bose-Einstein condensation. We show that our new equation, if it has solutions with appropriate smoothness and decay properties, implies a new Fock space estimate. We also show that for an interaction potential $v(x)= \epsilon \chi(x) |x|^{-1}$, where $\epsilon$ is sufficiently small and $\chi \in C_0^{\infty}$, our program can be easily implemented locally in time. We leave global in time issues, more singular potentials and sophisticated estimates for a subsequent part (part II) of this paper

The Free Energy of the Quantum Heisenberg Ferromagnet at Large Spin

We consider the spin-S ferromagnetic Heisenberg model in three dimensions, in the absence of an external field. Spin wave theory suggests that in a suitable temperature regime the system behaves effectively as a system of non-interacting bosons (magnons). We prove this fact at the level of the specific free energy: if $S \to \infty$ and the inverse temperature $\beta \to 0$ in such a way that $\beta S$ stays constant, we rigorously show that the free energy per unit volume converges to the one suggested by spin wave theory. The proof is based on the localization of the system in small boxes and on upper and lower bounds on the local free energy, and it also provides explicit error bounds on the remainder.Comment: 11 pages, pdfLate