Energy trajectories for the N -boson problem by the method of potential envelopes

Abstract

This paper concerns the ground-state energy ${E}_{N}$ 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 $p={p}_{1}$ and $p={p}_{2}$. 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 $\ensuremath{\epsilon}={F}_{N}(v)$, $N=2, 3, 4, \dots{}$, 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 $p={p}_{1}, {p}_{2}$. 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 ${E}_{N}$ depends on $N$ and all the parameters of the pair potential.