Schwartz singularities

Various techniques have been devised for coping with Schwartz singularities. They may either be ignored, or they can be avoided by using a modified form of the Kohn variational method in which the asymptotic form of the trial function has an amplitude different from that of equation (3.42) ... 

Schwartz singularities are avoided using the Harris method, but results can only be obtained at the discrete energies Ep (although the values of Ep can be altered by changing the values of the non-linear parameters in the trial function). Furthermore, the error in the phase shift is only of first order in the error in the trial wave function, and the results may therefore be less accurate than those of a well-behaved Kohn calculation. 

Probably the most accurate positron-hydrogen s-wave phase shifts are those obtained by Bhatia et al. (4974), who avoided the possibility of Schwartz singularities by using a bounded variational method based on the optical potential formalism described previously. These authors chose their basis functions spanning the closed-channel Q-space, see equation (3.44), to be of essentially the same Hylleraas form as those used in the Kohn trial function, equation (3.42), and their most accurate results were obtained with 84 such terms. By extrapolating to infinite u in a somewhat similar way to that described in equation (3.54), they obtained phase shifts which are believed to be accurate to within 0.0002 rad. They also established that there are no Feshbach resonances below the positronium formation threshold. 

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