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Physical Review A

Gravitating boson systems

Richard L. Hall

Published Jun 1, 1992• 12 citations
Publisher

Abstract

We study a system composed of N identical bosons interacting in three dimensions via attractive Yukawa pair potentials V(r)=-${\mathit{e}}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}\mathit{r}}$/r. Four approaches to the problem are related: the ``equivalent two-body method,'' translation-invariant Gaussian trial functions, collective field theory, and Hartree trial functions. Upper and lower energy bounds are given and are compared to some recent results which were obtained independently by an optimized Hartree method [M. Membrado, F. Pacheco, and J. Sa\~nudo, Phys. Rev. A 39, 4207 (1989)]. In the pure gravitational limit \ensuremath{\lambda}\ensuremath{\rightarrow}0 the N-body energy is determined for all N and all values of the coupling with an error of less than 7.2%.

DOI
10.1103/physreva.45.7682

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We study a system composed of N identical bosons interacting in three dimensions via attractive Yukawa pair potentials V(r)=-e\ensuremath−\ensuremathλr{\mathit{e}}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}\mathit{r}}e\ensuremath−\ensuremathλr/r. Four approaches to the problem are related: the ``equivalent two-body method,'' translation-invariant Gaussian trial functions, collective field theory, and Hartree trial functions. Upper and lower energy bounds are given and are compared to some recent results which were obtained independently by an optimized Hartree method [M. Membrado, F. Pacheco, and J. Sa~nudo, Phys. Rev. A 39, 4207 (1989)]. In the pure gravitational limit \ensuremath{\lambda}\ensuremath{\rightarrow}0 the N-body energy is determined for all N and all values of the coupling with an error of less than 7.2%.
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> 3d>~3d> 3 for the two cases p=2p=2p=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 consider a system of N identical bosons which interact in one dimension via attractive pair potentials and obey nonrelativistic quantum mechanics. This system is studied from two complementary viewpoints. The equivalent two-body method approximates the system as a collection of (N-1) independent two-particle systems with coupling constants enhanced by the factor N/2, and this yields energy lower bounds. The collective-field method approaches the problem from the standpoint of the limit as N\ensuremath{\rightarrow}\ensuremath{\infty} and leads to energy upper bounds. We find that Gaussian trial functions for the N-particle Hamiltonian and Gaussian trial densities in the collective-field theory lead to precisely the same energy upper bounds. The upper bounds provided by field theory allow for systematic improvement via a variational principle. Details are worked out for two exactly soluble problems, namely, the harmonic oscillator and the attractive \ensuremath{\delta} potential.
Studies 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)= gamma f(x/a) and obey non-relativistic quantum mechanics. A recent energy lower bound based on the known exact solution for the delta potential is compared to an earlier bound provided by the equivalent two-body method. The delta bound is good whenever the potential is very narrow and deep. For any bounded potential, the earlier energy bound is always better for large N. Detailed results and graphs are given for the sech-squared potential and the square-well potential.
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{)}^{1/3})1/3(\ensuremath{\gamma}N/2)2/3{/2)}^{2/3}/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.