Toward the theory of the Yukawa potential

Abstract

Using three different approaches Perturbation Theory (PT), the Lagrange Mesh Method (LMM), and the variational method, we study the low-lying states of the Yukawa potential. First orders of PT, in powers of the screening parameter, are calculated in the framework of the non-linearization procedure. It is found that the Padé approximants to PT series together with the LMM provide highly accurate values of energy and the positions of the radial nodes of the wave function. The most accurate results, at present, of the critical screening parameters for some low-lying states and the first coefficients in the expansion of the energy at the critical parameter are presented. A locally accurate and compact trial function for the eigenfunctions of the low-lying states is discovered. This function used as a zeroth order entry in PT leads to energies as precise as those of Padé approximants and LMM. Finally, a compact analytical expression for the energy, that reproduces at least 6 decimal digits in the entire physical range of the screening parameter, is found.