Abstract
A version of the $J$-matrix method for solving numerically the three-body Faddeev-Merkuriev differential equations is proposed. This version allows one to take into account the full spectrum of the two-body Coulomb subsystem. As a result, a discrete analog of the Lippmann-Schwinger equation is obtained which allows one to interpret correctly the three-body wave function in two-body domains. The scheme is applied to calculations of the fully resolved absolute differential cross sections for the $\mathrm{He}(e,2e){\mathrm{He}}^{+}$ reactions leaving the residual ion in an excited state, at small energy and momentum transfer. The results are in good agreement with the experiment both in shape and in absolute value.
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