Coulomb problem with short-range interaction: Exactly solvable model

The nonrelativistic Coulomb problem with short-range interaction is considered. The strong potential $V_s(r)$ is modeled by a delta-function interaction on a sphere $r=r_0$. Exact solutions to the Schrödinger equation are obtained for states with arbitrary angular momentum $l$ together with explicit analytic expressions for the scattering lengths, effective ranges, etc. Comparison of the exact solutions with the approximate formulas established earlier [1–4] for arbitrary short-range potential $V_s(r)$ makes it possible to determine the limits of applicability of these approximations.