Noniterative Solutions of Integral Equations for Scattering. III. Coupled Open and Closed Channels and Eigenvalue Problems

Abstract

The homogeneous integral solution formalism developed by Sams and Kouri is applied to the problems of coupled open- and closed-channel radial equations and coupled radial eigenvalue equations. The problem of coupled eigenvalue equations is considered first. Because we deal with the integral equation form of the Schrödinger equation, the proper solutions may be obtained by integrating from the origin outward, rather than both outward from and inward toward the origin. For the case of eigenvalue radial equations, quantization arises from the fact that exponentially decaying solutions are obtained only when the eigenvalue of a certain matrix equals one. Next, we consider the coupled open- and closed-channel radial equations. Unlike the case of coupled open channels, we find it convenient to consider a column vector solution rather than a matrix. Just as in the open-channel case, however, the Volterra integral equations generated in solving the coupled open- and closed-channel equations are the same for the homogeneous and inhomogeneous integral solutions. Thus, the number of equations which must be solved is not as large as first appears. The method necessitates at most two matrix inversions, one for a n0 by n0 matrix and one for a nc by nc matrix, at the very end of the calculation. Here n0 is the number of open channels and nc the number of closed channels. Finally, application of the eigenvalue procedure is made to a Lennard-Jones (12–6) potential in order to illustrate the method.

New full-wave phase-shift approach to solve the Helmholtz acoustic wave equation for modeling

“…The Feshbach projection operator approach was first introduced in nuclear physics (Feshbach, 1962) and was later used in molecular quantum scattering (Sams and Kouri, 1970). We have adapted this approach for the acoustic problem.…”

“…(Kosloff and Kessler, 1987) removed all evanescent components (growing and decaying) to make their solution stable, but they thereby lose the decaying evanescent waves that make a contribution to large dip scattering and can be very important. We have used the Feshbach projection method, developed in nuclear physics and adapted to chemical physics (Feshbach, 1962;Sams and Kouri, 1970) to remove only growing evanescent waves whereas the decaying evanescent waves and all propagating waves ("up" and "down") are retained.…”

Electron scattering fromH2+: Resonances in the Σ and Π symmetries

“…(15) and (16), and integrating over all target and angular-continuum coordinates convert Eq. (13) into a radial set of coupled integral equations of the general form (1 -G U)f=G where the matrices have the form (21) [U]pq --V p(r, )cu;t " (22) radial coordinate from r; i to r, . We have found that a standard mesh of 60 points and five zones of the form (10,20, 10,15,5/O.…”

“…Therefore, by closing the channels, we convert to an eigenvalue problem [21] whose solution exists only at a discrete set of allowed values of E. These values represent the bound energies of the composite system. Sams and Kouri [22] showed that at these discrete energies E;, the determinant of the matrix M defined by…”