Abstract
A waveguide occupies a domain G with several cylindrical ends. The waveguide is described by a nonstationary equation of the form \(i{{\partial }_{t}}f = \mathcal{A}f\), where \(\mathcal{A}\) is a selfadjoint second order elliptic operator with variable coefficients (in particular, for \(\mathcal{A} = - \Delta \), where Δ stands for the Laplace operator, the equation coincides with the Schrödinger equation). For the corresponding stationary problem with spectral parameter, we define continuous spectrum eigenfunctions and a scattering matrix. The limiting absorption principle provides expansion in the continuous spectrum eigenfunctions. We also calculate wave operators and prove their completeness. Then we define a scattering operator and describe its connections with the scattering matrix.
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REFERENCES
S. A. Nazarov and B. A. Plamenevskii, Elliptic Problems in Domains with Piecewise Smooth Boundaries (Walter de Gruyter, Berlin, 1994).
B. A. Plamenevskii and A. S. Poretskii, SPb. Math. J. 30, 285 (2019). https://doi.org/10.1090/spmj/1543
W. C. Lyford, Math Ann. 218, 229 (1975). https://doi.org/10.1007/BF01349697
W. C. Lyford, Math. Ann. 217, 257 (1975). https://doi.org/10.1007/BF01436177
R. Picard and S. Seidler, Math. Ann. 269, 411 (1984). https://doi.org/10.1007/BF01450702
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The study was supported by project Russian Science Foundation no. 17-11-01126.
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Plamenevskii, B.A., Poretskii, A.S. & Sarafanov, O.V. Mathematical Scattering Theory in Quantum Waveguides. Dokl. Phys. 64, 430–433 (2019). https://doi.org/10.1134/S102833581911003X
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DOI: https://doi.org/10.1134/S102833581911003X