Abstract:
In this paper one analyses the discrete spectrum of an asymmetric pair of two-dimensional quantum waveguides with common boundary in which a window of finite size is made. The phenomenon of new eigenvalues arising at the boundary of the essential spectrum as the
length of the window passes over critical values is considered. For the newly arising eigenvalues one constructs asymptotic expansions with respect to the small parameter equal to the difference between the window length and the closest critical value. The behaviour of the spectrum under an unrestricted growth of the length of the window is also under investigation; asymptotic expansions for eigenvalues with respect to the large parameter, the length of the window, are constructed.
Bibliography: 22 titles.
\Bibitem{Bor06}
\by D.~I.~Borisov
\paper Discrete spectrum of an asymmetric pair of waveguides coupled through a~window
\jour Sb. Math.
\yr 2006
\vol 197
\issue 4
\pages 475--504
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\crossref{https://doi.org/10.1070/SM2006v197n04ABEH003767}
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https://doi.org/10.1070/SM2006v197n04ABEH003767
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This publication is cited in the following 52 articles:
D. I. Borisov, A. I. Mukhametrakhimova, “Asymptotics for problems in perforated domains with Robin nonlinear condition on the boundaries of cavities”, Sb. Math., 213:10 (2022), 1318–1371
S. A. Nazarov, “Asimptoticheskii analiz spektra kvantovogo volnovoda s shirokim “oknom” Neimana v svete mekhaniki treschin”, Matematicheskie voprosy teorii rasprostraneniya voln. 52, Zap. nauchn. sem. POMI, 516, POMI, SPb., 2022, 176–237
Borisov I D., Golovina A.M., “On Finitely Many Resonances Emerging Under Distant Perturbations in Multi-Dimensional Cylinders”, J. Math. Anal. Appl., 496:2 (2021), 124809
Borisov I D., Zezyulin D.A., Znojil M., “Bifurcations of Thresholds in Essential Spectra of Elliptic Operators Under Localized Non-Hermitian Perturbations”, Stud. Appl. Math., 146:4 (2021), 834–880
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
D. I. Borisov, A. I. Mukhametrakhimova, “Uniform convergence and asymptotics for problems in domains finely perforated along a prescribed manifold in the case of the homogenized Dirichlet condition”, Sb. Math., 212:8 (2021), 1068–1121
Vorobiev A.M., “Resonance Asymptotics For Quantum Waveguides With Semitransparent Multi-Perforated Wall”, Nanosyst.-Phys. Chem. Math., 12:4 (2021), 462–471
Borisov I D., Zezyulin D.A., “Bifurcations of Essential Spectra Generated By a Small Non-Hermitian Hole. i. Meromorphic Continuations”, Russ. J. Math. Phys., 28:4 (2021), 416–433
D. I. Borisov, A. M. Golovina, “On Occurrence of Resonances from Multiple Eigenvalues of the Schrödinger Operator in a Cylinder with Distant Perturbations”, J Math Sci, 258:1 (2021), 1
Borisov D., Cardone G., “Spectra of Operator Pencils With Small P & Xdcab;& X1D4Af;& Xdcaf;-Symmetric Periodic Perturbation”, ESAIM-Control OPtim. Calc. Var., 26 (2020), UNSP 21
S. A. Nazarov, “Threshold resonances and virtual levels in the spectrum of cylindrical and periodic waveguides”, Izv. Math., 84:6 (2020), 1105–1160
Vorobiev A.M., Trifanova E.S., Popov I.Y., “Resonance Asymptotics For a Pair Quantum Waveguides With Common Semitransparent Perforated Wall”, Nanosyst.-Phys. Chem. Math., 11:6 (2020), 619–627
Bagmutov A.S., Popov I.Y., “Window-Coupled Nanolayers: Window Shape Influence on One-Particle and Two-Particle Eigenstates”, Nanosyst.-Phys. Chem. Math., 11:6 (2020), 636–641
D. I. Borisov, A. M. Golovina, A. I. Mukhametrakhimova, “Analytic Continuation of Resolvents of Elliptic Operators in a Multidimensional Cylinder”, J Math Sci, 250:2 (2020), 260
D. I. Borisov, A. M. Golovina, “O vozniknovenii rezonansov iz kratnogo sobstvennogo znacheniya operatora Shredingera v tsilindre s razbegayuschimisya vozmuscheniyami”, Differentsialnye uravneniya, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 163, VINITI RAN, M., 2019, 3–14
Vorobiev A.M., Bagmutov A.S., Popov I A., “On Formal Asymptotic Expansion of Resonance For Quantum Waveguide With Perforated Semitransparent Barrier”, Nanosyst.-Phys. Chem. Math., 10:4 (2019), 415–419
Piat V.Ch., Nazarov S.A., Taskinen J., “Embedded Eigenvalues Forwater-Waves in Athree-Dimensional Channel With Athin Screen”, Q. J. Mech. Appl. Math., 71:2 (2018), 187–220
S. A. Nazarov, “Transmission of waves through a small aperture in the cross-wall in an acoustic waveguide”, Siberian Math. J., 59:1 (2018), 85–101
S. A. Nazarov, “Asymptotics of eigenvalues in spectral gaps of periodic waveguides with small singular perturbations”, J. Math. Sci. (N. Y.), 243:5 (2019), 746–773
D. I. Borisov, “Perturbations of the Continuous Spectrum of a Certain Nonlinear Two-Dimensional Operator Sheaf”, J. Math. Sci. (N. Y.), 252:2 (2021), 135–146
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