Abstract:
For the Helmholtz equation Δu+k2u=0 in a domain Ω with a cylindrical outlet Q+=ω×R+ to infinity, we construct a fictitious scattering operator S that is unitary in L2(ω) and establish a bijection between the lineal of decaying solutions of the Dirichlet problem in Ω and the subspace of eigenfunctions of S corresponding to the eigenvalue 1 and orthogonal to the eigenfunctions with eigenvalues λn⩽k2 of the Dirichlet problem for the Laplace operator on the cross-section ω.
Citation:
S. A. Nazarov, “A Criterion for the Existence of Decaying Solutions in the Problem on a Resonator with a Cylindrical Waveguide”, Funktsional. Anal. i Prilozhen., 40:2 (2006), 20–32; Funct. Anal. Appl., 40:2 (2006), 97–107
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\by S.~A.~Nazarov
\paper A Criterion for the Existence of Decaying Solutions in the Problem on a Resonator with a Cylindrical Waveguide
\jour Funktsional. Anal. i Prilozhen.
\yr 2006
\vol 40
\issue 2
\pages 20--32
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\jour Funct. Anal. Appl.
\yr 2006
\vol 40
\issue 2
\pages 97--107
\crossref{https://doi.org/10.1007/s10688-006-0016-1}
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Linking options:
https://www.mathnet.ru/eng/faa4
https://doi.org/10.4213/faa4
https://www.mathnet.ru/eng/faa/v40/i2/p20
This publication is cited in the following 29 articles:
Sergei A. Nazarov, Keijo M. Ruotsalainen, “Curved channels with constant cross sections may support trapped surface waves”, Z. Angew. Math. Phys., 74:4 (2023)
S. A. Nazarov, “The preservation of threshold resonances and the splitting off of eigenvalues from the threshold of the continuous spectrum of quantum waveguides”, Sb. Math., 212:7 (2021), 965–1000
S. A. Nazarov, “Threshold resonances and virtual levels in the spectrum of cylindrical and periodic waveguides”, Izv. Math., 84:6 (2020), 1105–1160
Nazarov S.A., “Anomalies of Acoustic Wave Scattering Near the Cut-Off Points of Continuous Spectrum (a Review)”, Acoust. Phys., 66:5 (2020), 477–494
F. L. Bakharev, S. A. Nazarov, “Criteria for the absence and existence of bounded solutions at the threshold frequency in a junction of quantum waveguides”, St. Petersburg Math. J., 32:6 (2021), 955–973
Chesnel L., Pagneux V., “From Zero Transmission to Trapped Modes in Waveguides”, J. Phys. A-Math. Theor., 52:16 (2019), 165304
S. A. Nazarov, “Finite-dimensional approximations to the Poincaré–Steklov operator for general elliptic boundary value problems in domains with cylindrical and periodic exits to infinity”, Trans. Moscow Math. Soc., 80 (2019), 1–51
Chesnel L., Pagneux V., “Simple Examples of Perfectly Invisible and Trapped Modes in Waveguides”, Q. J. Mech. Appl. Math., 71:3 (2018), 297–315
S. A. Nazarov, “Finite-Dimensional Approximations of the Steklov–Poincaré Operator for the Helmholtz Equation in Periodic Waveguides”, J Math Sci, 232:4 (2018), 461
S. A. Nazarov, “Almost standing waves in a periodic waveguide with a resonator and near-threshold eigenvalues”, St. Petersburg Math. J., 28:3 (2017), 377–410
Nazarov S.A., Ruotsalainen K.M., “A Rigorous Interpretation of Approximate Computations of Embedded Eigenfrequencies of Water Waves”, Z. Anal. ihre. Anwend., 35:2 (2016), 211–242
S. A. Nazarov, “Transmission Conditions in One-Dimensional Model of a Rectangular Lattice of Thin Quantum Waveguides”, J Math Sci, 219:6 (2016), 994
Kemppainen J.T., Nazarov S.A., Ruotsalainen K.M., “Perturbation Analysis of Embedded Eigenvalues For Water-Waves”, J. Math. Anal. Appl., 427:1 (2015), 399–427
Nazarov S.A., Ruotsalainen K.M., “Criteria For Trapped Modes in a Cranked Channel With Fixed and Freely Floating Bodies”, Z. Angew. Math. Phys., 65:5 (2014), 977–1002
S. A. Nazarov, “Scheme for interpretation of approximately computed eigenvalues embedded in a continuous spectrum”, Comput. Math. Math. Phys., 53:6 (2013), 702–720
S. A. Nazarov, “Enforced Stability of a Simple Eigenvalue in the Continuous Spectrum of a Waveguide”, Funct. Anal. Appl., 47:3 (2013), 195–209
S. A. Nazarov, “The Mandelstam Energy Radiation Conditions and the Umov–Poynting Vector in Elastic Waveguides”, J Math Sci, 195:5 (2013), 676
S. A. Nazarov, “Enforced stability of an eigenvalue in the continuous spectrum of a waveguide with an obstacle”, Comput. Math. Math. Phys., 52:3 (2012), 448–464
S. A. Nazarov, “Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide”, Theoret. and Math. Phys., 167:2 (2011), 606–627
S. A. Nazarov, “Discrete spectrum of cranked, branchy, and periodic waveguides”, St. Petersburg Math. J., 23:2 (2012), 351–379
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