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Optics and Spectroscopy

Eigenvalue problem for bundle of operators and the sturm expansion method in quantum mechanics

S. I. Nikitin, A. I. Sherstyuk

Published Apr 1, 2009• 0 citations
Publisher

Abstract

Different variants of the generalized formulation of the eigenvalue problem related to a single-particle Schrödinger equation are considered. The nature of the spectrum of problems for a bundle of operators containing the Hamiltonian of the system is analyzed in relation to the form of the weighting operator. For rather general conditions, corresponding systems of eigenfunctions provide diagonal representations of the Green’s function and provide of a rather fast convergence of expansions over intermediate states. Cases of potentials with Coulomb asymptotics and short-range potentials are considered.

DOI
10.1134/s0030400x09040018

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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 ανcorresponding 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).
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 energ y 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).
Expansion of a wave function in a basis of eigenfunctions of a differential eigenvalue problem lies at the heart of the R-matrix methods for both the Schrödinger and Dirac particles. A central issue that should be carefully analyzed when functional series are applied is their convergence. In the present paper, we study the properties of the eigenfunction expansions appearing in nonrelativistic and relativistic R-matrix theories. In particular, we confirm the findings of Rosenthal [J. Phys. G 13, 491 (1987)] and Szmytkowski and Hinze [J. Phys. B 29, 761 (1996); Szmytkowski and Hinze J. Phys. A 29, 6125 (1996)] that in the most popular formulation of the R-matrix theory for Dirac particles, the functional series fails to converge to a limit claimed by other authors.