Open waveguides in a thin Dirichlet lattice: II. Localized waves and radiation conditions

S. A. Nazarov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 2, pp. 237-254

Résumé

Wave processes localized near an angular open waveguide obtained by thickening two perpendicular semi-infinite rows of ligaments in a thin square lattice of quantum waveguides (Dirichlet problem for the Helmholtz equation) are investigated. Waves of two types are discovered: the first are observed near the lattice nodes and almost do not affect the ligaments, while the second, on the contrary, excite oscillations in the ligaments, whereas the nodes stay relatively at rest. Asymptotic representations of the wave fields are derived, and radiation conditions are imposed on the basis of the Umov-Mandelstam energy principle.

Bibliographie

[1] Carini J. P., Londergan J. T., Murdock D. P., Binding and scattering in two-dimensional systems: applications to quantum wires, waveguides, and photonic crystals, Lecture Notes in Phys., Springer-Verlag, Berlin, 1999 | Zbl

[2] Kuchment P. A., “Teoriya Floke dlya differentsialnykh uravnenii v chastnykh proizvodnykh”, Uspekhi matem. nauk, 37:4 (1982), 3–52

[3] Skriganov M. M., Geometricheskie i arifmeticheskie metody v spektralnoi teorii mnogomernykh periodicheskikh operatorov, Tr. matem. in-ta im. V.A. Steklova AN SSSR, 171, Nauka, L., 1985

[4] Kuchment P., Floquet theory for partial differential equations, Birchäuser, Basel, 1993 | MR | Zbl

[5] Grieser D., “Spectra of graph neighborhoods and scattering”, Proc. London Math. Soc., 97:3 (2008), 718–752 | DOI | MR | Zbl

[6] Nazarov S. A., “Diskretnyi spektr krestoobraznykh kvantovykh volnovodov”, Problemy matem. analiza, 73, Novosibirsk, 2013, 101–127

[7] Nazarov S. A., “Asimptotika matritsy rasseyaniya vblizi kraev spektralnoi lakuny”, Izv. AN. Ser. matem., 81:1 (2017)

[8] Nazarov S. A., “Ogranichennye resheniya v T-obraznom volnovode i spektralnye svoistva lestnitsy Dirikhle”, Zh. vychisl. matem. i matem. fiz., 54:8 (2014), 1299–1318 | DOI | Zbl

[9] Cardone G., Nazarov S. A., Taskinen J., “Spectra of open waveguides in periodic media”, J. Funct. Anal., 269:8 (2015), 2328–2364 | DOI | MR | Zbl

[10] Gelfand I. M., “Razlozhenie po sobstvennym funktsiyam uravneniya s periodicheskimi koeffitsientami”, Dokl. AN SSSR, 73 (1950), 1117–1120 | Zbl

[11] Nazarov S. A., “Ellipticheskie kraevye zadachi s periodicheskimi koeffitsientami v tsilindre”, Izv. AN SSSR. Ser. matem., 45:1 (1981), 101–112 | Zbl

[12] Nazarov S. A., Plamenevsky B. A., Elliptic problems in domains with piecewise smooth boundaries, Walter de Gruytei, Berlin–New York, 1994 | MR

[13] Nazarov S. A., “Diskretnyi spektr kolenchatykh, razvetvlyayuschikhsya i periodicheskikh volnovodov”, Algebra i analiz, 23:2 (2011), 206–247

[14] Nazarov S. A., “Otkrytye volnovody v tonkoi reshetke Dirikhle. I. Asimptoticheskoe stroenie spektra”, Zh. vychisl. matem. i matem. fiz., 57:1 (2017), 144–162 | DOI

[15] Nazarov S. A., “Properties of spectra of boundary value problems in cylindrical and quasicylindrical domains”, Sobolev Spaces in Mathematics, v. II, Internat. Math. Ser., 9, Springer, New York, 2008, 261–309 | DOI | MR

[16] Gokhberg I. Ts., Krein M. G., Vvedenie v teoriyu nesamosopryazhennykh operatorov, Nauka, M., 1965

[17] Umov N. A., Uravneniya dvizheniya energii v telakh, Tipogr. Ulrikha i Shultse, Odessa, 1874

[18] Mandelshtam L. I., Lektsii po optike teorii otnositelnosti i kvantovoi mekhanike, Sb. trudov, v. 2, Izd-vo AN SSSR, M., 1947

[19] Vorovich I. I., Vabeshko V. A., Dinamicheskie smeshannye zadachi teorii uprugosti dlya neklassicheskikh oblastei, Nauka, M., 1979

[20] Nazarov S. A., “Usloviya izlucheniya Umova–Mandelshtama v uprugikh periodicheskikh volnovodakh”, Matem. sbornik, 205:7 (2014), 43–72 | DOI | Zbl

[21] Nazarov S. A., Plamenevskii B. A., “Ob usloviyakh izlucheniya dlya samosopryazhennykh ellipticheskikh zadach”, Dokl. AN SSSR, 31:3 (1990), 532–536

[22] Nazarov S. A., Plamenevskii B. A., “Printsipy izlucheniya dlya samosopryazhennykh ellipticheskikh zadach”, Problemy matem. fiziki, 13, Izd-vo LGU, L., 1991, 192–244

[23] Nazarov S. A., “Asimptotika sobstvennykh chisel na nepreryvnom spektre regulyarno vozmuschennogo kvantovogo volnovoda”, Teoreticheskaya i matematicheskaya fizika, 167:2 (2011), 239–262 | DOI

[24] Poynting J. H., “On the transfer of energy in the electromagnetic field”, Phil. Trans. of the Royal Society of London, 175 (1884), 343–361 | DOI