Towards the Theory of the Yukawa Potential

Using three different approaches, Perturbation Theory (PT), the Lagrange Mesh Method (Lag-Mesh) and the Variational Method (VM), we study the low-lying states of the Yukawa potential $V(r)=-(\lambda/r)e^{-\alpha r}\,$. First orders in PT in powers of $\alpha$ are calculated in the framework of the Non-Linerization Procedure. It is found that the Pad\'e approximants to PT series together with the Lag-Mesh provide highly accurate values of the energy and the positions of the radial nodes of the wave function. The most accurate results, at present, of the critical screening parameters ($\alpha_c$) for some low-lying states and the first coefficients in the expansion of the energy at $\alpha_c$ are presented. A locally-accurate and compact approximation for the eigenfunctions of the low-lying states for any $r\in [ 0,\infty)$ is discovered. This approximation used as a trial function in VM eventually leads to energies as precise as those of PT and Lag-Mesh. Finally, a compact analytical expression for the energy as a function of $\alpha$, that reproduce at least $6$ decimal digits in the entire physical range of $\alpha$, is found.