Optimization of Nuclear Resonance Reaction Calculations

An important current problem is to find the most economical and physically significant method of calculating nuclear resonant cross sections. Tobocman and co-workers have discussed an $R$-matrix type of approach in which the wave function is restricted to be a linear combination of a given finite set of states associated with the resonances. That work presented several different formulas for calculation, leaving uncertainty as to which, if any, was the best one. This paper gives systematic reasons why one specific formulat should be more accurate than all others. This result applies to any set of states, and is not restricted to the special set used in the previous work. A curious incidental result in this paper is that the variational principle for scattering problems introduced by Kohn fails to give a stationary result when applied to the present problem.