Effective-charge tensor and acoustic sum rule for an insulating slab

Explicit expressions of the effective-charge tensor for a thin slab of an insulating material are presented. To obtain them we use a Ward identity for the surface which determines the behavior of the microscopic response function $\chi$ for $\overrightarrow{\mathrm{q}}\to 0$, where $\overrightarrow{\mathrm{q}}$ is a two-dimensional wave vector. The effective-charge tensor is shown to obey the acoustic sum rule.