Exactly soluble model for crystal with spatial dispersion

We consider a discrete, semi-infinite, one-dimensional crystal exhibiting "spatial dispersion," with the interaction between crystal sites falling off exponentially with distance; a model which has also been treated by Sipe and Van Kranendonk. The problem of the interaction of such a crystal with the electromagnetic field is exactly soluble. Results for physical properties are compared with those obtained using the "dielectric approximation," in which the polariton Green's function for full translational invariance is used to approximate the true one; and also with those obtained from the "near-neighbor approximation," in which the interaction is cut off after $N$ nearest neighbors. In both cases, it is shown that the approximate results do not agree with the exact ones in all respects, even in the limit $N\to \infty$ for the near-neighbor approximation. Arguments are given to support the conclusion that these pessimistic results probably are general, and not merely artifacts of the particular model considered. The possibility of using this model, or generalizations of it, in practical calculations is briefly discussed.