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%.
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.
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 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.
This paper concerns the ground-state energy
EN of a system of
N identical bosons interacting via the attractive central pair potential $V({r}_{\mathrm{ij}})=\ensuremath{-}{V}_{0}f(\frac{{r}_{\mathrm{ij}}}{a})$ and obeying nonrelativistic quantum mechanics. It is assumed that the potential shape
f is decreasing and can be represented as the envelope of each of two complementary families of power-law potentials $\ensuremath{\alpha}+\ensuremath{\beta}{r}^{p}$ (one family is above
f and the other below) for suitable fixed
p=p1 and
p=p2. If $\ensuremath{\epsilon}=\ensuremath{-}\frac{m{a}^{2}{E}_{N}}{(N\ensuremath{-}1){\ensuremath{\hbar}}^{2}}$ and $v=\frac{\mathrm{Nm}{V}_{0}{a}^{2}}{2{\ensuremath{\hbar}}^{2}}$, then it is proved that the entire collection of nonintersecting energy trajectories $\ensuremath{\epsilon}={F}_{N}(v)$,
N=2,3,4,…, is bounded between the fixed curves $(v, \ensuremath{\epsilon})=(\ensuremath{-}\ensuremath{\gamma}(p){[{s}^{3}{f}^{\ensuremath{'}}(s)]}^{\ensuremath{-}1}, (\frac{v}{2})[2f(s)+s{f}^{\ensuremath{'}}(s)])$, where the curve parameter
s>0, and
p=p1,p2. Potentials, for example, with shapes $f(r)=\frac{{\ensuremath{\alpha}}_{1}}{r}+\frac{{\ensuremath{\alpha}}_{2}}{(r+{\ensuremath{\alpha}}_{3})}\ensuremath{-}{\ensuremath{\alpha}}_{4}\mathrm{ln}r\ensuremath{-}{\ensuremath{\alpha}}_{5}\mathrm{sgn}(q){r}^{q}$, where ${\ensuremath{\alpha}}_{i}\ensuremath{\ge}0$ and $|q|\ensuremath{\le}1$, have the $\ensuremath{\gamma}$ numbers $\ensuremath{\gamma}(\ensuremath{-}1)=2$ and $\ensuremath{\gamma}(1)=\frac{12}{\ensuremath{\pi}}$. The appropriate $\ensuremath{\gamma}$ numbers are provided for other classes of potential shape including perturbed harmonic oscillators, and also for problems in one spatial dimension. The method yields in effect a recipe for the way
EN depends on
N and all the parameters of the pair potential.
Close upper and lowei bounds are found on the binding energy of a system of N identical bosons interacting in one dimension by an attractive square-well pair potential. In the limit in which the potential has the shape of a δ function the bounds are found to be consistent with McGuire's exact solution. In three dimensions the limit of vanishingly small range, in which the two-particle binding energy is kept constant, gives infinite binding to the N -particle system, in agreement with Thomas's theorem Consideration of other limiting Proc.esses eliminates an apparent contradiction between Thomas and Svartholm.