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 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.
A system of N identical particles with masses m1 interact with each other via a pair potential V and with an additional particle of mass m0 by the pair potential U, and the system obeys nonrelativistic quantum mechanics. A general lower bound formula is derived for the ground-state energy, which reduces to Coleman’s result in the special case m0=∞. The lower bound is compared with the exact energy for the harmonic oscillator problem V (x) =k21x2 and U (x) =k02x2, which has recently been solved exactly (Hall).
Suppose
EN is the lowest energy of a translation-invariant system of
N identical bosons which interact by attractive central pair potentials and obey nonrelativistic quantum mechanics. Sufficient conditions are prescribed on the potential to prove that
EN is bounded above by $({N}{2}){({K}{2})}^{\ensuremath{-}1}{E}_{K}$, where $N\ensuremath{\ge}\ensuremath{\mu}K$. The $\ensuremath{\mu}$ numbers are specified for various classes of potential shape. In cases where $\ensuremath{\mu}\ensuremath{\le}\frac{3}{2}$, it follows that ${E}_{N}\ensuremath{\le}\frac{1}{2}N(N\ensuremath{-}1){E}_{2}$ for $N\ensuremath{\ge}3$.