- This is the first paper in a two part series aimed at placing the theory of Wigner's R matrix on a mathematically rigorous footing. In Paper I of the series, we will show that the eigenvalue problem associated with the R matrix can be solved for a large class of potentials, including Coulomb-like potentials. We will do this for the case in which the boundary of the internal region is a smooth surface-although the results remain true for a much larger class of surfaces. In Paper II of the series, we will show that the R matrix exists for the class of potentials mentioned, is a compact operator, and can be approximated uniformly (i.e., normwise) by the usual expansions associated with the R matrix. Copyright 1974 American Institute of Physics.