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Ground State Energy of 1D Attractive δ-Function Interacting Fermi Gas

Yahui Wang, Zhong-Qi Ma

The ground state energy of one-dimensional two-component Fermi gas with the attractive δ-function interaction is fitted with a simple approximate function.

Chinese Physics LettersAug 1, 20122 citations

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

尤亦庄

Chinese Physics LettersJan 1, 20100 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.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 ReviewApr 20, 196418 citations

Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State

Élliott H. Lieb, Werner Liniger

A gas of one-dimensional Bose particles interacting via a repulsive delta-function potential has been solved exactly. All the eigenfunctions can be found explicitly and the energies are given by the solutions of a transcendental equation. The problem has one nontrivial coupling constant, $\ensuremath{\gamma}$. When $\ensuremath{\gamma}$ is small, Bogoliubov's perturbation theory is seen to be valid. In this paper, we explicitly calculate the ground-state energy as a function of $\ensuremath{\gamma}$ and show that it is analytic for all $\ensuremath{\gamma}$, except $\ensuremath{\gamma}=0$. In Part II, we discuss the excitation spectrum and show that it is most convenient to regard it as a double spectrum---not one as is ordinarily supposed.A gas of one-dimensional Bose particles interacting via a repulsive delta-function potential has been solved exactly. All the eigenfunctions can be found explicitly and the energies are given by the solutions of a transcendental equation. The problem has one nontrivial coupling constant, $\ensuremath{\gamma}$. When $\ensuremath{\gamma}$ is small, Bogoliubov's perturbation theory is seen to be valid. In this paper, we explicitly calculate the ground-state energy as a function of $\ensuremath{\gamma}$ and show that it is analytic for all $\ensuremath{\gamma}$, except $\ensuremath{\gamma}=0$. In Part II, we discuss the excitation spectrum and show that it is most convenient to regard it as a double spectrum---not one as is ordinarily supposed.

Physical ReviewMay 15, 19632,707 citations

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

Craig A. Tracy, Harold Widom

Trends in mathematicsJan 1, 201811 citations

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

Craig A. Tracy, Harold Widom

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.

arXiv:1604.02440Journal of Mathematical PhysicsOct 1, 201614 citations

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

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

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