Geodesic
An augmented scattering matrix and an exponentially decreasing solution of elliptic boundary-value problem in the domain with cylindrical outlets
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 29, Tome 264 (2000), pp. 66-82

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Self-adjoint elliptic boundary-value problem in domain with cylindrical outlets to infinity is considered. The notion of augmented scattering matrix is introduced due to artificial radiation conditions. The properties of augmented scattering matrix are studied and the connection with classical scattering matrix is demonstrated. The central point is possibility to calculate the number of leaner independent solutions of homogeneous problem with fixed rate of decreasing at infinity by analyzing the spectrum of augmented scattering matrix. This property is applied to problem of diffraction on periodical boundary as example. 
@article{ZNSL_2000_264_a4,
     author = {I. V. Kamotskii and S. A. Nazarov},
     title = {An augmented scattering matrix and an exponentially decreasing solution of elliptic boundary-value problem  in the domain with cylindrical outlets},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {66--82},
     publisher = {mathdoc},
     volume = {264},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2000_264_a4/}
}
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I. V. Kamotskii; S. A. Nazarov. An augmented scattering matrix and an exponentially decreasing solution of elliptic boundary-value problem  in the domain with cylindrical outlets. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 29, Tome 264 (2000), pp. 66-82. http://geodesic.mathdoc.fr/item/ZNSL_2000_264_a4/