Scaling behavior of the Yukawa potential in two and three dimensions: a comparative study

By using the Laguerre basis in two and three dimensions, we report the scaling behavior of the bound state energies associated with the Yukawa potential. The energy spectrum crossover phenomenon is shown to exist through the scaling law for bound state energies $|n\ell ⟩,$ at different principal quantum numbers $n$ and the angular momentum $\ell ,$ as we approach the continuum. It is found that the bound/resonance phase transitions in $|n1⟩$ and $|n2⟩$ states are of the first order, while it is higher in the case of $|n0⟩$ states. Few physical phenomena, such as the dipole matrix element, the oscillator strength and the transition probabilities for the Yukawa potential are calculated in both two and three dimensions, and then compared with the analytic Coulomb potential. A comparison is made between the two- and three-dimensional cases. This method has a potential application in predicting stable and metastable nuclear, molecular, atomic and graphene states. (paper)