  Gaussian potentials serve as a valuable tool for the comprehensive modeling
of short-range interactions, spanning applications from nuclear physics to the
artificial confinement of electrons within quantum dots. This study focuses on
examining the lowest states within Gaussian wells, with particular emphasis on
the weakly bound regime. The analysis delves into the asymptotic behavior of
the exact wave function at both small and large distances, motivating the
development of a few-parametric Ansatz which is locally accurate and yields to
a fast convergent basis set. To validate its efficacy, we assess its
convergence rate using a toy model of Nuclear Physics, specifically for
Deuteron. Furthermore, we employ the expansion of the energy close to the
threshold to derive an analytical formula for the binding energy of the
deuteron whose accuracy improves as the effective parameter approaches the
critical. In concluding our investigation, we evaluate the performance of our
Ansatz as an orbital in the exploration of the electronic structure of a
two-electron quantum dot.
