Spurious solutions in few-body equations

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Abstract
After Faddeev and Yakubovskii showed how to write connected few-body equations which are free from discrete spurious solutions various authors have proposed different connected few-body scattering equations. Federbush first pointed out that Weinberg's formulation admits the existence of discrete spurious solutions. In this paper we investigate the possibility and consequence of the existence of spurious solutions in some of the few-body formulations. Contrary to a proof by Hahn, Kouri, and Levin and by Bencze and Tandy the channel coupling array scheme of Kouri, Levin, and Tobocman which is also the starting point of a formulation by Hahn is shown to admit spurious solutions. We can show that the set of six coupled four-body equations proposed independently by Mitra, Gillespie, Sugar, and Panchapakesan, by Rosenberg, by Alessandrini, and by Takahashi and Mishima and the seven coupled four-body equations proposed by Sloan are related by matrix multipliers to basic sets which correspond uniquely to the Schrödinger equation. These multipliers are likely to give spurious solutions to these equations. In all these cases spuriosities are shown to have no hazardous consequence if one is interested in studying the scattering problem.