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
c1​,c2​ such that −c1​a4≤ϵ(a)−(π/d)2≤−c2​a4. An analogous conclusion holds for a pair of Dirichlet strips, of
generally different widths, with a window of length 2a in the common
boundary.Comment: LaTeX file, 12 pages, no figure