Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:math-ph/0204027

The Ground State of the Bose Gas

Élliott H. Lieb, Robert Seiringer, Jan Philip Solovej, Jakob Yngvason

Published Apr 11, 2002• 5 citations
arXivHTMLPDF

Abstract

Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density, $ρ$, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous -- and this fact is used in all modeling. In such cases, the leading term in the energy/particle is $2π\hbar^2 a ρ/m$ where $a$ is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange $N^{7/5}$ law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has recently been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two-dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, two other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the `traps' actually used in the experiments. For this system it is also possible to prove complete Bose condensation, as we have shown. Another topic is a proof that Foldy's 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy.

DOI
10.48550/arxiv.math-ph/0204027

Related papers

Ground State Energy of the Low Density Bose Gas

Élliott H. Lieb, Jakob Yngvason

Now that the properties of low temperature Bose gases at low density, $\ensuremath{\rho}$, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors four to five decades ago. One of these is that the leading term in the energy/particle is $2\ensuremath{\pi}{\ensuremath{\Elzxh}}^{2}\ensuremath{\rho}a/m$, where $a$ is the scattering length. Owing to the delicate and peculiar nature of bosonic correlations, four decades of research have failed to establish this plausible formula rigorously. The only known lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct bound is proved here.

arXiv:cond-mat/9712138•Physical Review Letters•Mar 23, 1998•181 citations

The Ground State Energy of a Dilute Bose Gas

Élliott H. Lieb, Jakob Yngvason

According to a formula that was put forward many decades ago the ground state energy per particle of an interacting, dilute Bose gas at density $ρ$ is $2π\hbar^2ρa/m$ to leading order in $ρa^3\ll 1$, where $a$ is the scattering length of the interaction potential and $m$ the particle mass. This result, which is important for the theoretical description of current experiments on Bose-Einstein condensation, has recently been established rigorously for the first time. We give here an account of the proof that applies to nonnegative, spherically symmetric potentials decreasing faster than $1/r^3$ at infinity.

arXiv:math-ph/9910033•Oct 20, 1999•9 citations

Bosons in a Trap: Asymptotic Exactness of the Gross-Pitaevskii Ground State Energy Formula

Robert Seiringer

Recent experimental breakthroughs in the treatment of dilute Bose gases have renewed interest in their quantum mechanical description, respectively in approximations to it. The ground state properties of dilute Bose gases confined in external potentials and interacting via repulsive short range forces are usually described by means of the Gross-Pitaevskii energy functional. In joint work with Elliott H. Lieb and Jakob Yngvason its status as an approximation for the quantum mechanical many-body ground state problem has recently been rigorously clarified. We present a summary of this work, for both the two-and three-dimensional case. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

arXiv:math-ph/0010006•Birkhäuser Basel eBooks•Jan 1, 2001•4 citations

Ground-state energy of the low-density Fermi gas

Élliott H. Lieb, Robert Seiringer, Jan Philip Solovej

Recent developments in the physics of low-density trapped gases make it worthwhile to verify old, well-known results that, while plausible, were based on perturbation theory and assumptions about pseudopotentials. We use and extend recently developed techniques to give a rigorous derivation of the asymptotic formula for the ground-state energy of a dilute gas of $N$ fermions interacting with a short-range, positive potential of scattering length $a$. For spin-$1∕2$ fermions, this is $E\ensuremath{\sim}{E}^{0}+({\ensuremath{\hbar}}^{2}∕2m)2\ensuremath{\pi}N\ensuremath{\varrho}a$, where ${E}^{0}$ is the energy of the noninteracting system and $\ensuremath{\varrho}$ is the density. A similar formula holds in two dimensions (2D), with $\ensuremath{\varrho}a$ replaced by $\ensuremath{\varrho}∕\ensuremath{\mid}\mathrm{ln}(\ensuremath{\varrho}{a}^{2})\ensuremath{\mid}$. Obviously this 2D energy is not the expectation value of a density-independent pseudopotential.

arXiv:math-ph/0412080•Physical Review A•May 13, 2005•54 citations

Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional

Élliott H. Lieb, Robert Seiringer, Jakob Yngvason

The ground-state properties of interacting Bose gases in external potentials, as considered in recent experiments, are usually described by means of the Gross-Pitaevskii energy functional. We present here a rigorous proof of the asymptotic exactness of this approximation for the ground-state energy and particle density of a dilute Bose gas with a positive interaction.

arXiv:math-ph/9908027•Physical Review A•Mar 6, 2000•375 citations

On the energy of dilute Bose gases

Søren Fournais, Jan Philip Solovej

A fundamental problem in quantum mechanics is to understand the structure and the energy of ground states of interacting systems of many particles. The quantum correlations in ground states or low lying energy states are supposed to explain phenomena such as superfluidity or superconductivity. A long-standing conjecture in mathematical physics has been to establish a universal two-term asymptotic formula for the ground state energy of a system of bosons in the dilute limit of low density predicted by the theory of superfluidity. We discuss a recent proof of this formula.

EMS Press eBooks•Dec 15, 2023•5 citations
Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density, ρρρ, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous -- and this fact is used in all modeling. In such cases, the leading term in the energy/particle is 2πℏ2aρ/m2π\hbar^2 a ρ/m2πℏ2aρ/m where aaa is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange N7/5N^{7/5}N7/5 law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has recently been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two-dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, two other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the `traps' actually used in the experiments. For this system it is also possible to prove complete Bose condensation, as we have shown. Another topic is a proof that Foldy's 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy.
Now that the properties of low temperature Bose gases at low density, $\ensuremath{\rho}$, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors four to five decades ago. One of these is that the leading term in the energy/particle is $2\ensuremath{\pi}{\ensuremath{\Elzxh}}^{2}\ensuremath{\rho}a/m$, where aaa is the scattering length. Owing to the delicate and peculiar nature of bosonic correlations, four decades of research have failed to establish this plausible formula rigorously. The only known lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct bound is proved here.
According to a formula that was put forward many decades ago the ground state energy per particle of an interacting, dilute Bose gas at density ρρρ is 2πℏ2ρa/m2π\hbar^2ρa/m2πℏ2ρa/m to leading order in ρa3≪1ρa^3\ll 1ρa3≪1, where aaa is the scattering length of the interaction potential and mmm the particle mass. This result, which is important for the theoretical description of current experiments on Bose-Einstein condensation, has recently been established rigorously for the first time. We give here an account of the proof that applies to nonnegative, spherically symmetric potentials decreasing faster than 1/r31/r^31/r3 at infinity.
Recent experimental breakthroughs in the treatment of dilute Bose gases have renewed interest in their quantum mechanical description, respectively in approximations to it. The ground state properties of dilute Bose gases confined in external potentials and interacting via repulsive short range forces are usually described by means of the Gross-Pitaevskii energy functional. In joint work with Elliott H. Lieb and Jakob Yngvason its status as an approximation for the quantum mechanical many-body ground state problem has recently been rigorously clarified. We present a summary of this work, for both the two-and three-dimensional case. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Recent developments in the physics of low-density trapped gases make it worthwhile to verify old, well-known results that, while plausible, were based on perturbation theory and assumptions about pseudopotentials. We use and extend recently developed techniques to give a rigorous derivation of the asymptotic formula for the ground-state energy of a dilute gas of NNN fermions interacting with a short-range, positive potential of scattering length aaa. For spin-1∕21∕21∕2 fermions, this is $E\ensuremath{\sim}{E}^{0}+({\ensuremath{\hbar}}^{2}∕2m)2\ensuremath{\pi}N\ensuremath{\varrho}a$, where E0{E}^{0}E0 is the energy of the noninteracting system and $\ensuremath{\varrho}$ is the density. A similar formula holds in two dimensions (2D), with $\ensuremath{\varrho}a$ replaced by $\ensuremath{\varrho}∕\ensuremath{\mid}\mathrm{ln}(\ensuremath{\varrho}{a}^{2})\ensuremath{\mid}$. Obviously this 2D energy is not the expectation value of a density-independent pseudopotential.
A fundamental problem in quantum mechanics is to understand the structure and the energy of ground states of interacting systems of many particles. The quantum correlations in ground states or low lying energy states are supposed to explain phenomena such as superfluidity or superconductivity. A long-standing conjecture in mathematical physics has been to establish a universal two-term asymptotic formula for the ground state energy of a system of bosons in the dilute limit of low density predicted by the theory of superfluidity. We discuss a recent proof of this formula.