  In traditional semiconductors with large effective Bohr radius, an electron
donor creates a hydrogen-like bound state just below the conduction band edge.
The properties of the impurity band arising from such hydrogenic impurities
have been studied extensively during the last 70 years. In this paper we
consider whether a similar bound state and a similar impurity band can exist in
mixed-valence insulators, where the gap arises at low temperature due to strong
electron-electron interactions. We find that the structure of the hybridized
conduction band leads to an unusual bound state that can be described using the
physics of the one-dimensional hydrogen atom. The properties of the resulting
impurity band are also modified in a number of ways relative to the traditional
semiconductor case; most notably, the impurity band can hold a much larger
concentration without inducing an insulator-to-metal transition. We estimate
the critical doping associated with this transition, and then proceed to
calculate the dc and ac conductivities and the specific heat. We discuss our
results in light of recent measurements on the mixed-valence insualtor SmB$_6$,
and find them to be consistent with the experiments.
