The weakly bound states in Gaussian wells: From the binding energy of deuteron to the electronic structure of quantum dots

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. To conclude our study, we test the ability of our Ansatz to describe relativistic corrections into a quantum dot and its performance as an orbital in the exploration of the electronic structure of a two-electron quantum dot.