P. Exner, S.A. Vugalter
Asymptotic estimates for bound states in quantum
waveguides coupled laterally through a narrow window
(34K, LaTeX)
ABSTRACT. 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.