Bound states near the limit of the lower continuum (boson case)

The well-known effective range approximation in nonrelativistic scattering theory is generalized to the case of scalar particles satisfying the Klein–Gordon equation. Exact expressions are obtained for the parameters of the expansion of the $S$-matrix and the energy of levels near the limit of the lower continuum in terms of the wave hmction at the point at which a bound state arises for antiparticles (i.e., at $\varepsilon=-mc^2$). These expressions are used to investigate the motion of the levels near the limit $\varepsilon=-mc^2$ for different values of the angular momentum $l$. It is shown that the $p$-level curve can “bend” for potentials with sharp edge; this was known previously only for $s$-levels. A number of exactly solvable examples is considered. In particular, the exact solution of the Klein–Gordon equation for the $s$-levels in the Hulthen potential is investigated in detail, together with the passage to the limit of unscreened Coulomb potential.