THE FOURIER-GRID FORMALISM - PHILOSOPHY AND APPLICATION TO SCATTERING PROBLEMS USING R-MATRIX THEORY

The Fourier-grid (FG) method is a recent L2 variational treatment of the quantum mechanical eigenvalue problem that does not require the use of a set of basis functions; it is rather a discrete variable representation approach. In this article we restate the FG philosophy in more general terms, examine and compare this method with other approaches to the eigenvalue problem, and begin the development of an FG R-matrix method for scattering. The philosophy of the FG method is to use the simplest representation for each of the kinetic and potential energy operators of the Hamiltonian, and use a generalized Fourier transform to put the matrix elements of one of the above operators in the same representation as the other, so the Hamiltonian has a single representation. Thus, the Hamiltonian is represented at discrete points in either configuration or its reciprocal space. In examining the method, we find that its errors arise from the finite grid size and grid spacing; however, it is not encumbered by the symmetry of a reference Hamiltonian as are basis function methods so that it is possible for the FG method to give a better description of the Hamiltonian and a better fit of the wavefunctions when the symmetry of the reference Hamiltonian breaks down. This is borne out in this, the first detailed comparison of the FG and linear variational basis set (LVBS) method. The LVBS method, using an harmonic oscillator basis set, initially outperforms the FG method in obtaining the properties of low-lying Morse oscillator eigenstates. Once a sufficient grid spacing is employed, however, the FG method is able to obtain properties of even the highest-lying eigenstates and with very high precision, something which the LVBS is unable to do. Finally, we present a FG formulation of the Wigner-Eisenbud R-matrix theory of scattering, in which the FG formalism rather than a basis set formalism is employed to solve the eigenvalue problem inside the R-matrix boundaries. An example is given for potential scattering, and the resulting phase shifts are compared with the numerically exact quantities.