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

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_{ij}) = - V_{0} f (\frac{r_{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

α + β r^{p}

(one family is above

f

and the other below) for suitable fixed

p = p_{1}

and

p = p_{2}

. If

ε = - \frac{m a^{2} E_{N}}{(N - 1) ℏ^{2}}

and

v = \frac{Nm V_{0} a^{2}}{2 ℏ^{2}}

, then it is proved that the entire collection of nonintersecting energy trajectories

ε = F_{N} (v)

N = 2, 3, 4, \dots

, is bounded between the fixed curves

(v, ε) = (- γ (p) {[s^{3} f^{'} (s)]}^{- 1}, (\frac{v}{2}) [2 f (s) + s f^{'} (s)])

, where the curve parameter

s > 0

, and

p = p_{1}, p_{2}

. Potentials, for example, with shapes

f (r) = \frac{α_{1}}{r} + \frac{α_{2}}{(r + α_{3})} - α_{4} \ln r - α_{5} sgn (q) r^{q}

, where

α_{i} \geq 0

and

| q | \leq 1

, have the

γ

numbers

γ (- 1) = 2

and

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