Asymptotics of an eigenvalue on the continuous spectrum of two quantum waveguides coupled through narrow windows

Conditions under which two planar identical waveguides coupled through narrow windows of width ɛ ≪ 1 have an eigenvalue on the continuous spectrum are obtained. It is established that the eigenvalue appears only for certain values of the distance between the windows: for each sufficiently small ɛ > 0, there exists a sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(2N - 1)/\sqrt 3 + O(\varepsilon )$\end{document} of such distances; here N = 1, 2, 3, .... The result is obtained by the asymptotic analysis of an auxiliary object, namely, the augmented scattering matrix.