Application of the $J$-matrix Method to Faddeev-Merkuriev equation: beyond pseudostates

A version of the $J$-matrix method for solving numerically the three-body Faddeev-Merkuriev differential equations is proposed. This version allows 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 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 He$(e,2e)$He$^+$ and He$(e,3e)$He$^{++}$ reactions at small energy and momentum transfers. The results are in good agreement with the experiment both in shape and in absolute value.

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