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. 
@article{MZM_2013_93_2_a7,
     author = {S. A. Nazarov},
     title = {Asymptotics of an {Eigenvalue} on the {Continuous} {Spectrum} of {Two} {Quantum} {Waveguides} {Coupled} through {Narrow} {Windows}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {227--245},
     publisher = {mathdoc},
     volume = {93},
     number = {2},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2013_93_2_a7/}
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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/