Critical screening parameters and critical behaviors of one-electron systems with screened-type Coulomb potentials

The critical screening parameters for one-electron systems screened by Hulthen, Debye-Huckel, and exponential cosine screened Coulomb potentials are calculated with an accuracy close to the precision of numerical arithmetic. The results for a H atom with an infinitely heavy nucleus are reported from the ground to high-lying excited states, and those for arbitrary two-body charged systems are derived from the Zm-scaling law. A thorough comparison of the critical screening parameters for the ground and the first p-wave excited states with previous predictions is made to demonstrate the accuracy of our calculations. The critical behaviors of system bound and pseudo-continuum eigenenergies for s- and non-s-wave states are shown to follow the quadratic and linear laws, respectively. The variation of corresponding wave functions are analyzed in detail. For systems with non-zero orbital angular momenta, the bound states would convert into shape-type resonances when the screening parameter exceeds the critical value. The resonance energy shares the same linear law as the pseudo-continuum state, while the resonance width varies by an l-dependent power law. It is further shown that the different asymptotic behaviors of resonance energy and width are consistent with the complex analog of the Hellmann–Feynman theorem.