Abstract
We propose a hyperspherical adiabatic formalism for the calculation of the 3-to-3S-matrix at low energy, for repulsive potentials, and use it then in a model calculation. That is for McGuire's model (3 particles in one dimension subject to repulsive delta-function interactions), we use analytical expressions for the hyperspherical adiabatic basis, the adiabatic coupling matrix elements, and eigenpotentials to obtain the first terms of the exactS-matrix analytically, in an expansion in powers of the wave number. We were able to associate the definite powers ofq in the expansion of theS-matrix to the corresponding inverse powers of ρ in the expansions of the adiabatic eigenpotentials and coupling matrix elements. We investigate the effect of making the “usual” approximations found in the literature (extreme and uncoupled adiabatic approximations), when calculating the diagonal and off-diagonalS-matrix elements. Finally, we show that the coupled adiabatic equations uncouple as the energy goes to zero.
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We gratefully acknowledge W. G. Gibson who, using Thacker's exact expression for the S-matrix, provided us with the first two terms in a low-energy expansion of the hyperspherical-harmonicS-matrix (private communication)
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Popiel, J.J., Larsen, S.Y. Low-energy three-body scattering in a hyperspherical adiabatic basis. Few-Body Systems 15, 129–143 (1993). https://doi.org/10.1007/BF01076429
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DOI: https://doi.org/10.1007/BF01076429