  Wightman function and vacuum expectation value of the field square are
evaluated for a massive scalar field with general curvature coupling parameter
subject to Robin boundary conditions on two codimension one parallel branes
located on $(D+1)$-dimensional background spacetime $AdS_{D_1+1}\times \Sigma $
with a warped internal space $\Sigma $. The general case of different Robin
coefficients on separate branes is considered. The application of the
generalized Abel-Plana formula for the series over zeros of combinations of
cylinder functions allows us to extract manifestly the part due to the bulk
without boundaries. Unlike to the purely AdS bulk, the vacuum expectation value
of the field square induced by a single brane, in addition to the distance from
the brane, depends also on the position of the brane in the bulk. The brane
induced part in this expectation value vanishes when the brane position tends
to the AdS horizon or AdS boundary. The asymptotic behavior of the vacuum
densities near the branes and at large distances is investigated. The
contribution of Kaluza-Klein modes along $\Sigma $ is discussed in various
limiting cases. As an example the case $\Sigma =S^1$ is considered,
corresponding to the $AdS_{D+1}$ bulk with one compactified dimension. An
application to the higher dimensional generalization of the Randall-Sundrum
brane model with arbitrary mass terms on the branes is discussed.
