On the quantization of the potential amplitude in the Schrödinger equation

Issue title: Atomic and molecular nonlinear optics: Theory, Experiment and Computation A homage to the pioneering work of Stanisław Kielich (1925-1993)

Guest editors: G. Maroulis, T. Bancewicz, B. Champagne and A.D. Buckingham

Article type: Research Article

Authors: Balagurov, B.Ya.^{; *}

Affiliations: Emanuel Institute of Biochemical Physics, Russian Academy of Sciences, ul.Kosygina 4, Moscow, 119334, Russia

Correspondence: [*] Corresponding author. E-mail: [email protected]

Abstract: 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 φν(r) and eigenvalues αν corresponding to zero, small, and large absolute values of energy E<0 are analyzed. Expansion in the set φν(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).

Keywords: Potential amplitude, eigenfunction, eigenvalue

DOI: 10.3233/JCM-2010-0313

Journal: Journal of Computational Methods in Sciences and Engineering, vol. 10, no. 3-6, pp. 105-127, 2010