This thesis is concerned with the use of L-squared or square integrable functions in electron atom scattering at intermediate energies, and tests the success of various L-squared approximations in model problems of electron hydrogen atom scattering. The representation of part or all of the wave and Green's functions by a set of L-squared pseudostates, and the associated occurrence of unphysical pseudoresonances at the pseudostate thresholds is discussed. The original work of this thesis is in two parts. In the first, a model coupled channel problem is considered in which an L-squared optical potential is used to represent the effect of additional (Q space) channels on the first (P space) channel. A method of Bransden and Stelbovics used successfully for a two channel problem is extended to the case of several channels. Numerical results are presented for the cases of two and three channels and the success of the procedure is assessed. The rest of the research presented here concerns the use of the Schwinger variational method in a restricted model of electron hydrogen atom scattering in which all states are assumed to be spherically symmetric. The method is used successfully to solve coupled channel problems using L-squared pseudostates to represent the s-wave continuum. The origins of the pseudoresonances that occur in these problems are investigated and a method of removing pseudoresonances before T matrix elements are calculated is considered. The limitations and instabilities of the Schwinger method when applied to the full model problem with different representations of hydrogen states in the trial and Green’s functions are investigated, and various modifications are considered in attempts to stabilise results where necessary in these more general cases.