Quadratic Schemes Research Articles

Accurate and efficient quadrature for volterra integral equations

Four quadrature schemes have been tested and compared in considerable detail to determine their usefulness in the noniterative integral equation method for single channel quantum mechanical calculations. They are two forms of linear approximation (Trapezoidal rule) and two forms of quadratic approximation (Simpson's rule). Their implementation in this method is shown, a formal discussion of error propagation is given, and tests are performed to determine actual operating characteristics on various bound and scattering problems in different potentials. The quadratic schemes are generally superior to the linear ones in terms of accuracy and efficiency. The previous implementation of Simpson's rule is shown to possess an inherent instability which requires testing on each problem for which it is used to assure its reliability. The alternative quadratic approximation which we propose does not suffer this deficiency, but still enjoys the advantages of higher order. In addition, the new scheme obeys very well an h 4 Richardson extrapolation, whereas the old one does so rather poorly.

Study of η-η′ mixing

We discuss the available experimental evidence on η-η′ mixing in an attempt to decide between the linear and quadratic schemes. The two-photon decays of pseudoscalar mesons tend to favour linear mixing, due to a recent lowering of the upper bound for the η′ width. A 2 decays are inconclusive because of a large uncertainty in the branching ratio to η′ π. Linear mixing is certainly not ruled out by high-energy production experiments.