Two-body bound state problem and nonsingular scattering equations.

Abstract

We present a new momentum space approach to the two-body problem in partial waves. In contrast to the usual momentum space approaches, we treat the bound state case with the help of an inhomogeneous integral equation which possesses solutions for all (negative) energies. The bound state energies and corresponding wave functions are identified by an additional condition. This procedure straightforwardly leads to a nonsingular formulation of the scattering problem in terms of essentially the same equation and thus unifies the descriptions of both energy regimes. We show that the properties of our momentum-space approach can be understood in terms of the so-called regular solution of the Schr\"odinger equation in position space. The unified description of the bound state and scattering energy regimes in terms of one single, real, and manifestly nonsingular equation allows us to construct an exact representation of the two-body off-shell T matrix in which all the bound state pole and scattering cut information is contained in one single separable term, the remainder being real, nonsingular, and vanishing half on-shell. Such a representation may be of considerable advantage as input in three-body Faddeev-type integral equations. We demonstrate the applicability of our method by calculating bound state and scattering data for the two-nucleon system with the s-wave Malfliet--Tjon III potential.