Generalized saddle-point method for Feshbach resonances

The mini-max principle is extended to work for the approximations to resonances in the square-integrable function space. The hole-projection (or saddle-point) technique for Feshbach resonances, introduced previously by Chung [Phys. Rev. A 20, 1743 (1979)], is derived from the mini-max principle. Limits of applicability of the method are discussed and its generalization based on the Feshbach-type projector technique is given. The generalized method is applied to the case of the 1s2s2$p^2$P° resonance of ${\mathrm{He}}^{\mathrm{-}}$.