  We consider the discrete spectrum of the Dirichlet Laplacian on a manifold
consisting of two adjacent parallel strips or planar layers coupled by a finite
number N of windows in the common boundary. If the windows are small enough,
there is just one isolated eigenvalue. We find upper and lower asymptotic
bounds on the gap between the eigenvalue and the essential spectrum in the
planar case, as well as for N=1 in three dimensions. Based on these results, we
formulate a conjecture on the weak-coupling asymptotic behaviour of such bound
states.
