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Gap opening and split band edges in waveguides coupled by a periodic system of small windows. (English. Russian original) Zbl 1288.81046
Math. Notes 93, No. 5, 660-675 (2013); translation from Mat. Zametki 93, No. 5, 665-683 (2013).
The authors provide a mathematical rigorous description of the split band wedge in coupled waveguides. More precisely, let \(H^{\varepsilon}\) be the Laplacian in an infinite straight strip, \(\mathbb{R} \times (-d,\pi), 0<d< \pi\), with Dirichlet boundary conditions at the lines \(x_{2}=-d, x_{2}=\pi\) and Neumann boundary condition at the line \(x_{2}=0\), except at the intervals \(\omega^{\varepsilon}_{n} = (2nh- \varepsilon, 2nh+ \varepsilon) \times \left\{ 0 \right\}, n \in \mathbb{Z}, \varepsilon \geq 0\). The Hamiltonian \(H^{\varepsilon}\) describes a system of two waveguides coupled by the \(2h\)-periodic system of windows \(\omega^{\varepsilon}_{n}\). The Hamiltonian \(H^{0}\) can be decomposed as \(H^{0}=H_{+} \oplus H_{-}\), where \(H_{\pm}\) are Laplacians with Dirichlet boundary conditions at \(x_{2}=\pi\), respectively at \(x_{2}=-d\) and Neumann conditions at \(x_{2}=0\). Let \(H_{\pm}=\int_{(-\pi,\pi]}^{\oplus} H_{\pm}(k)\, dk\) be the Bloch-Floquet decomposition of the operators \(H_{\pm}\), with elementary cell \((-h,h)\times (0,\pi)\), respectively \((-h,h)\times (-d,0)\). The main result of the paper gives sufficient conditions on the eigenvalues of \(H_{\pm}(k)\) in order that \(H^{\varepsilon}\) admits a spectral gap. Asymptotics of the endpoints of the spectral gap, for \(\varepsilon \rightarrow 0\), are also provided.

MSC:
81Q37 Quantum dots, waveguides, ratchets, etc.
47A10 Spectrum, resolvent
35J25 Boundary value problems for second-order elliptic equations
35P99 Spectral theory and eigenvalue problems for partial differential equations
82D77 Quantum waveguides, quantum wires
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[1] P. Kuchment, “The mathematics of photonic crystals”, Mathematical Modeling in Optical Science, Frontiers Appl. Math., 22, SIAM, Philadelphia, PA, 2001, 207 – 272 · Zbl 0986.78004 · adsabs.harvard.edu
[2] P. Exner, P. Kuchment, B. Winn, “On the location of spectral edges in \(\mathbb Z\)-periodic media”, J. Phys. A. Math. Theor., 43:47 (2010), 474022 · Zbl 1204.81063 · doi:10.1088/1751-8113/43/47/474022 · arxiv:1006.3001
[3] J. M. Harrison, P. Kuchment, A. Sobolev, B. Winn, “On occurrence of spectral edges for periodic operators inside the Brillouin zone”, J. Phys. A. Math. Theor., 40:27 (2007), 7597 – 7618 · Zbl 1141.82340 · doi:10.1088/1751-8113/40/27/011 · arxiv:math-ph/0702035
[4] A. Figotin, I. Vitebskiy, “Slow-wave resonance in periodic stacks of anisotropic layers”, Phys. Rev. A, 76:5 (2007), 053839 · doi:10.1103/PhysRevA.76.053839
[5] S. Mahmoodian, A. A. Sukhorukov, S. Ha, A. V. Lavrinenko, C. G. Poulton, K. B. Dossou, L. C. Botten, R. C. McPhedran, C. M. de Sterke, “Paired modes of heterostructure cavities in photonic crystal waveguides with split band edges”, Optics Express, 18:25 (2010), 25693 – 25701 · doi:10.1364/OE.18.025693 · adsabs.harvard.edu
[6] G. Mumcu, K. Sertel, J. L. Volakis, I. Vitebskiy, A. Figotin, “RF propagation in finite thickness unidirectional magnetic photonic crystals”, IEEE Trans. Antennas Propagation, 53:12 (2005), 4026 – 4034 · doi:10.1109/TAP.2005.859764 · adsabs.harvard.edu
[7] R. Hempel, O. Post, “Spectral gaps for periodic elliptic operators with high contrast: an overview”, Progress in Analysis, Vol. I (Berlin 2001), World Sci. Publ., River Edge, NJ, 2003, 577 – 587 · Zbl 1061.35060 · arxiv:math-ph/0207020
[8] K. Yoshitomi, “Band gap of the spectrum in periodically curves quantum waveguides”, J. Differential Equations, 142:1 (1998), 123 – 166 · Zbl 0901.35066 · doi:10.1006/jdeq.1997.3337
[9] L. Friedlander, M. Solomyak, “On the spectrum of narrow periodic waveguides”, Russ. J. Math. Phys., 15:2 (2008), 238 – 242 · Zbl 1180.35392 · doi:10.1134/S1061920808020076
[10] С. А. Назаров, “Лакуна в существенном спектре задачи Неймана для эллиптической системы на периодической области”, Функц. анализ и его прил., 43:3 (2009), 92 – 95 · doi:10.4213/faa2966 · mi.mathnet.ru
[11] С. А. Назаров, “Пример множественности лакун в спектре периодического волновода”, Матем. сб., 201:4 (2010), 99 – 124 · Zbl 1193.35038 · doi:10.1070/SM2010v201n04ABEH004082 · mi.mathnet.ru · adsabs.harvard.edu
[12] K. Pankrashkin, “On the spectrum of a waveguide with periodic cracks”, J. Phys. A. Math. Theor., 43:47 (2010), 474030 · Zbl 1204.81069 · doi:10.1088/1751-8113/43/47/474030 · arxiv:1007.3656
[13] K. Yoshitomi, “Band spectrum of the Laplacian on a slab with the Dirichlet boundary condition on a grid”, Kyushu J. Math., 57:1 (2003), 87 – 116 · Zbl 1067.35058 · doi:10.2206/kyushujm.57.87
[14] F. L. Bakharev, S. A. Nazarov, K. M. Ruotsalainen, A Gap in the spectrum of the Neumann – Laplacian on a Periodic Waveguide, math.SP/1110.5990 · Zbl 1302.35266 · doi:10.1080/00036811.2012.711819 · arxiv:1110.5990
[15] G. Cardone, S. A. Nazarov, C. Perugia, “A gap in the essential spectrum of a cylindrical waveguide with a periodic perturbation of the surface”, Math. Nachr., 283:9 (2010), 1222 – 1244 · Zbl 1213.35327 · doi:10.1002/mana.200910025 · arxiv:0910.5679
[16] С. А. Назаров, “Открытие лакуны в непрерывном спектре периодически возмущенного волновода”, Матем. заметки, 87:5 (2010), 764 – 786 · doi:10.4213/mzm8719 · mi.mathnet.ru
[17] О. П. Мельничук, И. Ю. Попов, “Квантовые волноводы, связанные через периодическую систему малых отверстий: оценка запрещенной зоны”, Письма в ЖТФ, 28:8 (2002), 69 – 73
[18] И. Ю. Попов, А. И. Трифанов, Е. С. Трифанова, “Связанные диэлектрические волноводы со свойствами фотонного кристалла”, Ж. вычисл. матем. и матем. физ., 50:11 (2010), 1931 – 1937 · Zbl 1224.82034 · doi:10.1134/S0965542510110072 · www.maik.ru
[19] I. Yu. Popov, “Asymptotics of bound states and bands for laterally coupled waveguides and layers”, J. Math. Phys., 43:1 (2002), 215 – 234 · Zbl 1059.78025 · doi:10.1063/1.1425081 · adsabs.harvard.edu
[20] V. Maz/ya, S. Nazarov, B. Plamenevskij, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, Vol. I, Birkhaüser Verlag, Basel, 2000 · Zbl 1127.35300
[21] Vol. II, Birkhaüser Verlag, Basel, 2000 · Zbl 1127.35301
[22] В. Г. Мазья, С. А. Назаров, Б. А. Пламеневский, “Асимптотические разложения собственных чисел краевых задач для оператора Лапласа в областях с малыми отверстиями”, Изв. АН СССР. Сер. матем., 48:2 (1984), 347 – 371 · Zbl 0566.35031 · doi:10.1070/IM1985v024n02ABEH001237 · mi.mathnet.ru
[23] Р. Р. Гадыльшин, “Расщепление кратного собственного значения задачи Дирихле для оператора Лапласа при сингулярном возмущении граничного условия”, Матем. заметки, 52:4 (1992), 42 – 55 · Zbl 0816.35093 · doi:10.1007/BF01210435 · mi.mathnet.ru
[24] L. Hillairet, C. Judge, “The eigenvalues of the Laplacian on domains with small slits”, Trans. Amer. Math. Soc., 362:12 (2010), 6231 – 6259 · Zbl 1208.35095 · doi:10.1090/S0002-9947-2010-04943-8 · arxiv:0802.2597
[25] M. Reed, B. Simon, Methods of Modern Mathematical Physics. Vol. 4. Analysis of Operators, Academic Press, New York, 1978 · Zbl 0401.47001
[26] A. M. Ильин, Согласование асимптотических разложений решений краевых задач, Наука, М., 1989 · Zbl 0671.35002
[27] Т. Като, Теория возмущений линейных операторов, Мир, М., 1972 · Zbl 0247.47009
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