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 $\varepsilon\ll 1$ 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 $\varepsilon>0$, there exists a sequence $(2N-1)/\sqrt{3}+O(\varepsilon)$ of such distances; here $N=1,2,3,\dots$ . The result is obtained by the asymptotic analysis of an auxiliary object, namely, the augmented scattering matrix.