On the quantization of the potential amplitude in the Schrödinger equation

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