Short-distance regularization of a resonant point scatterer

An outgoing three-dimensional Green function behaves as $e^{ikr}/r=1/r+ik+O(r)$. Evaluating a point-coupled discrete-level equation at cutoff distance $\delta$ produces a real shift proportional to $1/\delta$ and an imaginary outgoing-wave term proportional to $ik$. The bare level must depend on the cutoff to keep the measured detuning finite.