Geodesic
Asymptotics of an Eigenvalue on the Continuous Spectrum of Two Quantum Waveguides Coupled through Narrow Windows
Matematičeskie zametki, Tome 93 (2013) no. 2, pp. 227-245

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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.
Keywords: planar waveguide, window-coupled quantum waveguides, augmented scattering matrix, Laplace operator, Dirichlet boundary condition, Neumann boundary condition, Helmholtz equation, Wood's anomalies. 
S. A. Nazarov. Asymptotics of an Eigenvalue on the Continuous Spectrum of Two Quantum Waveguides Coupled through Narrow Windows. Matematičeskie zametki, Tome 93 (2013) no. 2, pp. 227-245. http://geodesic.mathdoc.fr/item/MZM_2013_93_2_a7/
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