  In this article, we illustrate the scaling properties of a family of
solutions for N attractive bosonic atoms in the limit of large $N$. These
solutions represent the quantized dynamics of solitonic degrees of freedom in
atomic droplets. In dimensions lower than two, or $d=2-\epsilon$, we
demonstrate that the number of isotropic droplet states scales as
$N^{3/2}/\epsilon^{1/2}$, and for $\epsilon=0$, or $d=2$, scales as ${N^2}$.
The ground state energies scale as $N^{2 / \epsilon + 1}$ in $d=2-\epsilon$,
and when $d=2$, scale as an exponential function of N. We obtain the universal
energy spectra and the generalized Tjon relation; their scaling properties are
uniquely determined by the asymptotic freedom of quantum bosonic fields at
short distances, a distinct feature in low dimensions. We also investigate the
effect of quantum loop corrections that arise from various virtual processes
and show that the resultant lifetime for a wide range of excited states scales
as $N^{\epsilon/2}E^{1-\epsilon/2}$.
