Semiphenomenological Solutions of Pion-Nucleon Partial-Wave Dispersion Relations

A variational method of solving $\pi -N$ partial-wave dispersion relations is developed. Analytic trial functions are used, and their parameters are varied to obtain a unitary solution. A good solution for the ($\frac{3}{2},\frac{3}{2}$) resonance is obtained by using the Layson function. The shape of the resonance, as well as its position, is obtained. The only problem about the solution is the validity of the short-range interaction term which is used. Crossing of the real part of the amplitude verifies that it is accurate, but it is not obvious why the variation method should give such a good result. The explanation appears to be that in many cases the low-energy behavior of a partial wave is dominated by the long-range interactions, and a comparatively simple analytic function will give a good solution. An application of the variational method to confirm an earlier analysis of $s$-wave $\pi -N$ scattering is also given.