How the inter-electronic potential Ans\"atze affect the bound state solutions of a planar two-electron quantum dot model

The model of a two-electron quantum dot, confined to move in a two dimensional flat space, in the presence of an external harmonic oscillator potential, is revisited for a specific purpose. Indeed, eigenvalues and eigenstates of the bound state solutions are obtained for any oscillation frequency considering both the $1/r$ and $\ln r$ Ans\"atze for inter-electronic Coulombic-like potentials in 2$D$. Then, it is pointed out that the significative difference between measurable quantities predicted from these two potentials can shed some light on the problem of space dimensionality as well as on the physical nature of the potential itself.