The dielectric response function of a strongly correlated superlattice
is calculated in the quasilocalized charge (QLC) approximation. The r
esulting QLC static local-field correction, which contains both intral
ayer and interlayer pair-correlational effects, is identical to the co
rrelational part of the third-frequency-moment sum-rule coefficient. T
his approximation treats the interlayer and intralayer couplings on an
equal footing. The resulting dispersion relation is first analyzed to
determine the effect of intralayer coupling on the out-of-phase acous
tic-mode dispersion; in this approximation the interlayer coupling is
suppressed and the mutual interaction of the layers is taken into acco
unt only through the average random-phase approximation (RPA) field. I
n the resulting mode dispersion, the onset of a finite-k (k being the
in-plane wave number) reentrant low-frequency excitation developing (w
ith decreasing d/a) into a dynamical instability is indicated (a being
the in-plane Wigner-Seitz radius and d the distance between adjacent
lattice planes). This dynamical instability parallels a static structu
ral instability reported earlier both for a bilayer electron system an
d a superlattice and presumably indicates a structural change in the e
lectron liquid. If one takes account of interlayer correlations beyond
the RPA, the acoustic excitation spectrum is dramatically modified by
the appearance of an energy gap which also has a stabilizing effect o
n the instability. We extend a previous energy gap study at k=0 [G. Ka
lman, Y. Ren, and K. I. Golden, Phys Rev. B 50, 2031 (1994)] to a calc
ulation of the dispersion of the gapped acoustic excitation spectrum i
n the long-wavelength domain.