Analysis of the Harris Variational Method in Scattering Theory

The variational method recently proposed by Harris for calculation of scattering phase shifts is analyzed in the context of the standard methods of Kohn and Hulthén. The Hulthén method is shown to give a result identical to the Harris method at the energy eigenvalues characteristic of the latter. The Kohn method and its analog for the cotangent of the phase shift, the second Hulthén or Rubinow method, lead to somewhat different limiting values of the phase shift at a Harris eigenvalue. The Harris method is an application to nonresonant situations of a formalism appropriate to scattering resonances. A new method is proposed to deal with stationary states, true resonances, and nonresonant scattering within a common formalism, which defines effective width and shift functions for all values of the energy in terms of a Breit-Wigner formula. The proposed resonance formalism is expressed in forms equivalent to the Rubinow method and to the Hulthén method. A method of smoothing out irregular behavior of computed Hulthén phase shifts, due to branch points at Harris eigenvalues, is proposed. The Kohn and Rubinow methods are shown to give smooth results near Harris eigenvalues. Spurious fluctuations encountered in computations by the Kohn method are shown to be due to poles not present in the Hulthén method. Similar poles occur in the Rubinow method, but at different energy values. A criterion is suggested for avoiding the irregular behavior due to these poles by making an appropriate choice of either the Kohn or the Rubinow formula at a given energy. The formal discussion is illustrated by some computed results for $S$-wave scattering by an attractive exponential potential.