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 ? ? 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 ? > 0, there exists a sequence $(2N - 1)/sqrt 3 + O(varepsilon )$ 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.