We study a bound system of N identical bosons interacting by model pair potentials of the form $V(r)=\ensuremath{\lambda}\mathrm{sgn}{(p)r}^{p}+\ensuremath{\mu}{/r}^{2},$ $\ensuremath{\lambda}>0,$ $\ensuremath{\mu}>~0.$ By using a variational trial function and the ``equivalent two-body method,'' we find explicit upper and lower bound formulas for the N-particle ground-state energy in arbitrary spatial dimensions $d>~3$ for the two cases $p=2$ and $p=\ensuremath{-}1.$ It is demonstrated that the upper bound can be systematically improved with the aid of a special large-N limit in collective field theory.We study a bound system of N identical bosons interacting by model pair potentials of the form $V(r)=\ensuremath{\lambda}\mathrm{sgn}{(p)r}^{p}+\ensuremath{\mu}{/r}^{2},$ $\ensuremath{\lambda}>0,$ $\ensuremath{\mu}>~0.$ By using a variational trial function and the ``equivalent two-body method,'' we find explicit upper and lower bound formulas for the N-particle ground-state energy in arbitrary spatial dimensions $d>~3$ for the two cases $p=2$ and $p=\ensuremath{-}1.$ It is demonstrated that the upper bound can be systematically improved with the aid of a special large-N limit in collective field theory.

arXiv:math-ph/0008021Physical Review AOct 13, 20002 citations

Ground state energy of a many-particle boson system

M. E. Grypeos

International Journal of Theoretical PhysicsDec 1, 19721 citation

Complementary energy bounds forN-boson systems with linear pair potentials

Richard L. Hall

We study the ground-state energy of a system of N identical bosons, each having mass m, which interact in one dimension via the pair potential V(x)=\ensuremath{\gamma}\ensuremath{\Vert}x\ensuremath{\Vert} and obey nonrelativistic quantum mechanics. It is shown that the energy is given by the formula \ensuremath{\varepsilon}=C(N)(N-1)(${\ensuremath{\Elzxh}}^{2}$/ m ${)}^{1/3}$ (\ensuremath{\gamma}N ${/2)}^{2/3}$ , where 1.01879(N)1.02333 for all N\ensuremath{\ge}2. The lower bound is provided by the ``equivalent two-body method'' whereas the upper bound is derived by the use of collective field theory. The general relation between these complementary theories is investigated.We study the ground-state energy of a system of N identical bosons, each having mass m, which interact in one dimension via the pair potential V(x)=\ensuremath{\gamma}\ensuremath{\Vert}x\ensuremath{\Vert} and obey nonrelativistic quantum mechanics. It is shown that the energy is given by the formula \ensuremath{\varepsilon}=C(N)(N-1)(${\ensuremath{\Elzxh}}^{2}$/ m${)}^{1/3}$(\ensuremath{\gamma}N${/2)}^{2/3}$, where 1.01879(N)1.02333 for all N\ensuremath{\ge}2. The lower bound is provided by the ``equivalent two-body method'' whereas the upper bound is derived by the use of collective field theory. The general relation between these complementary theories is investigated.

Physical Review AOct 1, 19875 citations

Ground State Energy for a Many Boson System: a New Microscopic Approach.

Christina Keller

Jan 1, 19880 citations

An Expression of the Ground State Energy of the Spin-Boson Model

Masao Hirokawa

Journal of Functional AnalysisFeb 1, 199922 citations

A second order upper bound on the ground state energy of a Bose gas beyond the Gross–Pitaevskii regime

Giulia Basti

We consider a system of N bosons in a unitary box in the grand-canonical setting interacting through a potential with the scattering length scaling as N−1+κ, κ ∈ (0, 2/3). This regimes interpolate between the Gross–Pitaevskii regime (κ = 0) and the thermodynamic limit (κ = 2/3). In the work of Basti et al. [Forum Math., Sigma 9, E74 (2021)], as an intermediate step to prove an upper bound in agreement with the Lee–Huang–Yang formula in the thermodynamic limit, a second order upper bound on the ground state energy for κ < 5/9 was obtained. In this paper, thanks to a more careful analysis of the error terms, we extend the mentioned result to κ < 7/12.We consider a system of N bosons in a unitary box in the grand-canonical setting interacting through a potential with the scattering length scaling as N−1+κ, κ ∈ (0, 2/3). This regimes interpolate between the Gross–Pitaevskii regime (κ = 0) and the thermodynamic limit (κ = 2/3). In the work of Basti et al. [Forum Math., Sigma 9, E74 (2021)], as an intermediate step to prove an upper bound in agreement with the Lee–Huang–Yang formula in the thermodynamic limit, a second order upper bound on the ground state energy for κ < 5/9 was obtained. In this paper, thanks to a more careful analysis of the error terms, we extend the mentioned result to κ < 7/12.

arXiv:2203.01841Journal of Mathematical PhysicsJul 1, 20224 citations

d>~3

for the two cases

p=2

and $p=\ensuremath{-}1.$ It is demonstrated that the upper bound can be systematically improved with the aid of a special large-N limit in collective field theory.

{)}^{1/3}

(\ensuremath{\gamma}N

{/2)}^{2/3}

, where 1.01879(N)1.02333 for all N\ensuremath{\ge}2. The lower bound is provided by the ``equivalent two-body method'' whereas the upper bound is derived by the use of collective field theory. The general relation between these complementary theories is investigated.