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:1604.02440 · Journal of Mathematical Physics

On the ground state energy of the delta-function Fermi gas

Craig A. Tracy, Harold Widom

Published Oct 1, 2016• 14 citations
arXivHTMLPDF

Abstract

The weak coupling asymptotics to order γ of the ground state energy of the delta-function Fermi gas, derived heuristically in the literature, is here made rigorous. Further asymptotics are in principle computable. The analysis applies to the Gaudin integral equation, a method previously used by one of the authors for the asymptotics of large Toeplitz matrices.

DOI
10.1063/1.4964252

Related papers

On the ground state energy of the δ -function Bose gas*

Craig A. Tracy, Harold Widom

Abstract The weak coupling asymptotics, to order ( c / ρ ) 2 , of the ground state energy of the delta-function Bose gas is derived. Here 2 c ≥ 0 is the delta-function potential amplitude and ρ the density of the gas in the thermodynamic limit. The analysis uses the electrostatic interpretation of the Lieb–Liniger integral equation.

arXiv:1601.04677•Journal of Physics A Mathematical and Theoretical•Jun 6, 2016•25 citations

Exact Analysis of δ-Function Attractive Fermions and Repulsive Bosons in One-Dimension

Toshiaki Iida, Miki Wadati

The Gaudin integral equation for the ground state of a one-dimensional delta-function attractive spin-1/2 fermions is solved in the form of power series. The first few terms of the asymptotic expansions for both strong and weak coupling cases are calculated analytically. The physical quantities such as the ground state energy are expressed in terms of a single dimensionless parameter gamma =c/D, where c is the coupling constant and D is the number density. The results agree with those obtained from the perturbation calculations, which include the one in the classical electrostatics originally by Kirchhoff. In the strong coupling limit, the connection to the solutions of the Lieb-Liniger integral equation for the ground state of a one-dimensional delta-function repulsive bose gas is shown explicitly.

arXiv:cond-mat/0503033•Journal of the Physical Society of Japan•Jun 1, 2005•45 citations

Ground State Energy of One-Dimensional δ-Function Interacting Bose and Fermi Gas

You Yi-Zhuang

The ground state energy curves of one-dimensional δ-function interacting bose and fermi gas are fitted with simple algebraic approximations according to their asymptotic behaviors.

Chinese Physics Letters•Aug 1, 2010•7 citations

Simplified Approach to the Ground-State Energy of an Imperfect Bose Gas. III. Application to the One-Dimensional Model

Élliott H. Lieb, Werner Liniger

We continue the study of the integrodifferential equation proposed previously for the evaluation of the ground-state energy of an imperfect Bose gas. We apply it here to the one-dimensional delta-function gas where the exact result is known for all values of the coupling constant $\ensuremath{\gamma}$. The results are: (i) For small $\ensuremath{\gamma}$, the equation gives the correct first two terms in an asymptotic series; (ii) a numerical solution of the equation shows that the maximum relative error occurs for $\ensuremath{\gamma}=\ensuremath{\infty}$ in which case it is 19%; (iii) for $\ensuremath{\gamma}=\ensuremath{\infty}$ we are able to compare the exact two-particle distribution function with that given by the equation. The agreement is quite good.

Physical Review•Apr 20, 1964•18 citations

One-dimensional multicomponent fermions with δ -function interaction in strong- and weak-coupling limits: Two-component Fermi gas

Xi‐Wen Guan, Zhong-Qi Ma

The Fredholm equations for one-dimensional two-component fermions with repulsive and with attractive $\ensuremath{\delta}$-function interactions are solved by an asymptotic expansion for strong repulsion, weak repulsion, weak attraction, and strong attraction. Consequently, we obtain the first few terms of the expansion of the ground-state energy for the Fermi gas with polarization for these regimes. We also prove that the two sets of Fredhom equations for weakly repulsive and attractive interactions are identical as long as the integration boundaries match each other between the two types. Thus the asymptotic expansions of the energies of repulsive and attractive fermions are identical to all orders in this region. The identity of the asymptotic expansions may not mean that the energy connects analytically.

arXiv:1110.2821•Physical Review A•Mar 26, 2012•34 citations

On the Ground State Energy of the Delta-function Fermi Gas II: Further Asymptotics

Craig A. Tracy, Harold Widom

Trends in mathematics•Jan 1, 2018•11 citations
The weak coupling asymptotics to order γ of the ground state energy of the delta-function Fermi gas, derived heuristically in the literature, is here made rigorous. Further asymptotics are in principle computable. The analysis applies to the Gaudin integral equation, a method previously used by one of the authors for the asymptotics of large Toeplitz matrices.
The Gaudin integral equation for the ground state of a one-dimensional delta-function attractive spin-1/2 fermions is solved in the form of power series. The first few terms of the asymptotic expansions for both strong and weak coupling cases are calculated analytically. The physical quantities such as the ground state energy are expressed in terms of a single dimensionless parameter gamma =c/D, where c is the coupling constant and D is the number density. The results agree with those obtained from the perturbation calculations, which include the one in the classical electrostatics originally by Kirchhoff. In the strong coupling limit, the connection to the solutions of the Lieb-Liniger integral equation for the ground state of a one-dimensional delta-function repulsive bose gas is shown explicitly.
We continue the study of the integrodifferential equation proposed previously for the evaluation of the ground-state energy of an imperfect Bose gas. We apply it here to the one-dimensional delta-function gas where the exact result is known for all values of the coupling constant $\ensuremath{\gamma}$. The results are: (i) For small $\ensuremath{\gamma}$, the equation gives the correct first two terms in an asymptotic series; (ii) a numerical solution of the equation shows that the maximum relative error occurs for $\ensuremath{\gamma}=\ensuremath{\infty}$ in which case it is 19%; (iii) for $\ensuremath{\gamma}=\ensuremath{\infty}$ we are able to compare the exact two-particle distribution function with that given by the equation. The agreement is quite good.
The Fredholm equations for one-dimensional two-component fermions with repulsive and with attractive $\ensuremath{\delta}$-function interactions are solved by an asymptotic expansion for strong repulsion, weak repulsion, weak attraction, and strong attraction. Consequently, we obtain the first few terms of the expansion of the ground-state energy for the Fermi gas with polarization for these regimes. We also prove that the two sets of Fredhom equations for weakly repulsive and attractive interactions are identical as long as the integration boundaries match each other between the two types. Thus the asymptotic expansions of the energies of repulsive and attractive fermions are identical to all orders in this region. The identity of the asymptotic expansions may not mean that the energy connects analytically.