arXiv:1601.04677 · Journal of Physics A Mathematical and Theoretical

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

Craig A. Tracy, Harold Widom

Published Jun 6, 2016• 25 citations

Abstract

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.

DOI: 10.1088/1751-8113/49/29/294001

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arXiv:1601.04677 · Journal of Physics A Mathematical and Theoretical

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

Craig A. Tracy, Harold Widom

Published Jun 6, 2016• 25 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/1751-8113/49/29/294001

Related papers

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Abstract We prove the following formula for the ground state energy density of a dilute Bose gas with density

\rho

ρ in 2 dimensions in the thermodynamic limit

\begin{aligned} e^{\text {2D}}(\rho ) = 4\pi \rho ^2 Y\Big (1 - Y \vert \log Y \vert + \Big ( 2\Gamma + \frac{1}{2} + \log (\pi ) \Big ) Y \Big ) + o(\rho ^2 Y^{2}), \end{aligned}

e 2D ( ρ ) = 4 π ρ 2 Y ( 1 - Y | log Y | + ( 2 Γ + 1 2 + log ( π ) ) Y ) + o ( ρ 2 Y 2 ) , as

\rho a^2 \rightarrow 0

ρ a 2 → 0 . Here

Y= |\log (\rho a^2)|^{-1}

Y = | log ( ρ a 2 ) | - 1 and a is the scattering length of the two-body potential. This result in 2 dimensions corresponds to the famous Lee–Huang–Yang formula in 3 dimensions. The proof is valid for essentially all positive potentials with finite scattering length, in particular, it covers the crucial case of the hard core potential.