In the neighborhood of scattering resonances, the plane-wave solutions are strongly altered and low-order perturbation theory breaks down. Instead of calculating and using the exact scattering solutions there are many advantages to introducing the quasistationary concept of metastable localized eigenfunctions, by means of which the nonperturbative behavior near resonances can be described, whereas nonlocalized features are given by plane-wave functions. It is shown how metastable eigenfunctions can be defined in a natural way without arbitrariness. It is found that a number of general relations have to be satisfied by the metastable functions, which helps to approximate them in the case of a special problem. © 1964 The American Physical Society.