The critical binding of quantum states in Screened Coulomb Potentials such as Yukawa/Debye, Hulthén, and ECSC (Exponential Cosine Screened Coulomb) potentials is of perennial interest and relevance in many fields of science, ranging from nuclear and particle physics; plasma physics, astrophysics, cosmology, and nuclear fusion; physical chemistry, condensed matter, and materials physics; to synthetic nanostructures and nanophotonics. The purpose of this paper is to heuristically explore two related mysteries, one new, the other more than 50 years old. The solutions to these mysteries have implications for a much broader class of potentials, those addressed by Klaus and Simon. In our recent paper we presented numerical calculations using the Phase Method (PM), which is accurate to 60 digits and to screening lengths D≤103 au and l=0–20 of the critical binding parameters for these potentials and, for Yukawa and ECSC, l=0–12 to D≤105 au, at 30 digits. In doing so, we discovered an anomalous period-40 sawtooth structure in the critical parameters of the ECSC potential that is not observed for the Yukawa potential. In this second paper, we quantitatively explain the origin and periodicity of this newly discovered structure. To do so, we use two complementary approaches: a “neoclassical” (NC) variant of conventional semiclassical phase-space quantization and the PM for very precise fully quantum calculations. The observed period-40 sawtooth structure is quantitatively explained in terms of a novel “tick-tock” mechanism. The periodicity is calculated in terms of the ratio of phase-space integrals for the primary and secondary potential wells. A quartic double-well potential is used as a simple model to further illustrate the tick-tock mechanism. Using the NC method, an approximate expression is derived to predict the locations of tick-tock glitches from higher-order wells; it is confirmed by a PM calculation up to D≤106 au. The second mystery is a strangely linear dependence of the total number of bound states vs. screening length for both the Yukawa and ECSC potentials. Using the PM, we confirm and extend these empirical relations. We show, using the PM, that an approximate trivariate linear relation between the square root of the critical screening length Dc, state number n, and angular momentum l applies to these potentials. This, plus a geometrical state accumulation argument, solve the second mystery. We show these properties derive from the scaling relation between screening length and coupling constant and, as such, are predicted to be applicable to the whole class of potentials. These results are expected to be of both theoretical interest and experimental relevance when interpreting spectra or calculating thermal properties. The significance of these results, and the applicability of these methods and conclusions to a vast array of related potentials, is briefly discussed.
Grant Bunker (Sun,) studied this question.