Non-superposition effects in the Dirichlet Casimir effect

We study non-superposition effects in the Dirichlet Casimir interaction energy for N boundaries in d spatial dimensions, quantifying its departure from the case of an interaction where a superposition principle is valid. We first derive some general results about those effects, and then show that they only become negligible when the distances between surfaces are larger than the sizes of each individual surface. We consider different examples in one, two and three spatial dimensions.