  We consider the Dirichlet Laplacian in a domain two three-dimensional
parallel layers having common boundary and coupled by a window. The window
produces the bound states below the essential spectrum; we obtain two-sided
estimates for them. It is also shown that the eigenvalues emerge from the
threshold of essential spectrum as the window passes through certain critical
shapes. We prove the necessary condition for the window to be of critical
shape. Under an additional assumption we show that this condition is sufficient
and obtain the asymptotic expansion for the emerging eigenvalue as well as for
the associated eigenfunction.
