Memories to Stanisław Kielich: My teacher, my mentor and my friend

A consistent scheme is proposed for quantizing the potential amplitude in the Schrödinger equation in the caseof negative energies (lying in the discrete-spectrum domain). The properties of the eigenfunctionsφν(r) and eigenvalues ανcorresponding to zero, small, and large absolute values of energy E < 0 are analyzed. Expansion in the set {φν(r)} is used todevelop a regular perturbation theory (for E < 0), and a general expression is found for the Green function associated with thetime-independent Schrödinger equation. A similar method is used to solve several physical problems: the polarizability of abound quantum-mechanical system, the two-center problem, and the elastic scattering of slow particles. The proposed approachis advantageous in that it does not require the use of continuum states (for E > 0).