Measuring the accuracy of the solution subspace obtained by numerical integration of the Schrödinger equation

Abstract

In general, the quantum-mechanical description of inelastic scattering processes requires the numerical solution of the radial Schrödinger equation. To investigate the accuracy of the numerical integration process, a method has been used succesfully for measuring the accuracy of the regular solution subspace spanned by the solution vectors, rather than the accuracy of the solution vectors themselves. This method computes the principal angles between two solution subspaces obtained under different numerical conditions. One of the subspaces is constructed under optimal conditions so that it is considered to be the reference subspace, the other being the subspace to be investigated. In this method, the quality of a solution subspace obtained by a numerical procedure, can be measured, e.g., the extent to which solution vectors, as a basis of the solution subspace, remain linearly independent in the range from the origin to the matching radius R m during the integration. The computation of the principal angles can be used to inspect the loss of accuracy in the integration range originating from the truncation error inherent in the difference formula employed and to detect possible sources of deficiencies in the numerical process for solving the Schrödinger equation. A method has been developed and applied with which deficiencies caused by discontinuities in the potential matrix can be avoided. The loss of accuracy due to the tendency of the solution vectors to become nearly linearly dependent during the integration through a classically forbidden region as an effect of round-off errors, can be examined by determining the principal angles, as well. This loss of accuracy requires stabilization of the set of solution vectors. We found that the stabilization in only a few well chosen mesh points in our nuclear physics test cases of alpha scattering from 28Si, proved to be sufficient for obtaining an S-matrix accuracy satisfactory for practical purposes.