  Consider the Laplacian in a straight planar strip of width $\,d\,$, with the
Neumann boundary condition at a segment of length $\,2a\,$ of one of the
boundaries, and Dirichlet otherwise. For small enough $\,a\,$ this operator has
a single eigenvalue $\,\epsilon(a)\,$; we show that there are positive
$\,c_1,c_2\,$ such that $\,-c_1 a^4 \le \epsilon(a)- \left(\pi/ d\right)^2 \le
-c_2 a^4\,$. An analogous conclusion holds for a pair of Dirichlet strips, of
generally different widths, with a window of length $\,2a\,$ in the common
boundary.
