Bound states of the coupled-channel Schr�dinger equation: General eigenvalue function

Abstract

The eigenvalue problem for a system of N coupled one‐dimensional Schrödinger equations, arising in bound state in quantum mechanics, is considered. A canonical approach for the calculation of the energy eigenvalues of this system is presented. This method replaces the use of the wave functions by 2N canonical functions having well‐defined initial values at an arbitrary point r0. An eigenvalue function D(E) is associated with the system, where the energy E is considered as a variable. It is shown that the energy eigenvalues are the zeros of this function. This method is new in the sense that it reduces the eigenvalues search to that of finding the zeros of the function D(E), which is defined and constructed from the canonical functions independently from the wave function. A numerical application to a model problem proposed by Freidman et al. [Freidman, R. S.; Jamieson, M. J Comput Phys Commun 1989, 55, 137] is presented. It is shown that the eigenvalues computed by the present method are highly accurate for low and high levels; the average relative discrepancy between computed and exact one is about 1.5×10−11. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 325–332, 1999

Distorted waves with exact nonlocal exchange: A canonical function approach

“…In this case the static potential is given by: To integrate the coupled equation systems (7) and (9) preference is given in the present work to the "integral superposition" (I.S.) method which was shown by Kobeissi et al [8,10,13] to be highly accurate in the case of both single and coupled differential equations. This requires potentials to be expressed in analytical form.…”

“…Essentially, we need only to develop a single algorithm for the solution of a system of two coupled equations of the form (7) or (9) and then apply it to the different cases represented by the intial conditions (8) or (10).…”

“…In section 3 we describe the use of the the Canonical Fuction approach to solve the resulting coupled differential equations. It is interesting to note that the present treatment of exchange can ultimately be combined with the Canonical Function approach to the solution of coupled equations without exchange [7,8,12] so as to get a full solution of the Close Coupling equations including all potentials and non-local exchange terms (diagonal and off-diagonal).…”