Asymptotic estimates for bound states in quantum waveguides coupled laterally through a narrow window

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 $ϵ(a)$; 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.