We pose the problem of finding an operator, defined for all relative distances of two particles, which plays the role of a Hamiltonian such that the subsequent eigenvalue problem has as its only solutions precisely the eigenfunctions for two hard spheres, each of core diameter $a$. This operator is explicitly constructed. Its crucial aspect is that, despite its being defined for all relative distances ${r}_{12}$, its eigenfunctions serve as a complete set for expressing a two-particle wave function only in the restricted interval ${r}_{12}\ensuremath{\ge}a$. Subsequent papers of this series will be devoted to the application of these results to the problem of quantum hard-sphere gases.