Open waveguides in a thin Dirichlet ladder: I. Asymptotic structure of the spectrum

S. A. Nazarov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 1, pp. 144-162

Résumé

The spectra of open angular waveguides obtained by thickening or thinning the links of a thin square lattice of quantum waveguides (the Dirichlet problem for the Helmholtz equation) are investigated. Asymptotics of spectral bands and spectral gaps (i.e., zones of wave transmission and wave stopping, respectively) for waveguides with variously shaped periodicity cells are found. It is shown that there exist eigenfunctions of two types: localized around nodes of a waveguide and on its links. Points of the discrete spectrum of a perturbed lattice with eigenfunctions concentrated about corners of the waveguide are found.

Bibliographie

[1] Bonnet-Bendhia A.-S., Starling F., “Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem”, Math. Meth. Appl. Sci., 77 (1994), 305–338 | DOI | MR

[2] Nazarov S. A., “Properties of spectra of boundary value problems in cylindrical and quasicylindrical domains”, Sobolev Spaces in Mathematics, v. II, International Mathematical Series, 9, ed. Maz'ya V., Springer, New York, 2008, 261–309 | DOI | MR

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

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

[5] 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

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

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

[8] Bonnet-Ben Dhia A.-S., Dakhia G., Hazard C., Chorfi L., “Doffraction by a defect in an open waveguide: a mathematical analysis based on a modal radiation condition”, SIAM J. Appl. Math., 70:3 (2009), 677–693 | DOI | MR | Zbl

[9] Bonnet-Ben Dhia A.-S., Goursaud B., Hazard C., “Mathematical analysis of the junction of two acoustic open waveguides”, SIAM J. Appl. Math., 71:6 (2001), 2048–2071 | MR

[10] 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

[11] 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

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

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

[14] Nazarov S. A., Shanin A. V., “Trapped modes in angular joints of 2D waveguides”, Applicable Analysis, 93:3 (2014), 572–582 | DOI | MR | Zbl

[15] Nazarov S. A., “Spektr pryamougolnykh reshetok kvantovykh volnovodov”, Izv. RAN. Ser. matem., 81:1 (2017) | DOI

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

[17] Nazarov S. A., “O spektre operatora Laplasa na beskonechnoi lestnitse Dirikhle”, Algebra i analiz, 23:6 (2011), 143–176

[18] Birman M. Sh., Solomyak M. Z., Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, Izd-vo Leningr. un-ta, L., 1980 | MR

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

[20] Nazarov S. A., “Variatsionnyi i asimptoticheskii metody poiska sobstvennykh chisel pod porogom nepreryvnogo spektra”, Sib. matem. zh., 51:5 (2010), 1086–1101 | Zbl

[21] Nazarov S. A., “Lokalizovannye volny v T-obraznom volnovode”, Akusticheskii zhurnal, 56:6 (2010), 747–758

[22] Nazarov S. A., “Asimptotika sobstvennykh znachenii zadachi Dirikhle na skoshennom $\mathcal{T}$-obraznom volnovode”, Zh. vychisl. matem. i matem. fiz., 54:5 (2014), 793–814 | DOI | Zbl

[23] Vishik M. I., Lyusternik L. A., “Regulyarnoe vyrozhdenie i pogranichnyi sloi dlya lineinykh differentsialnykh uravnenii s malym parametrom”, Uspekhi matem. nauk, 12:5 (1957), 3–122 | Zbl

[24] Nazarov S. A., “Stroenie spektra reshetki kvantovykh volnovodov i ogranichennye resheniya modelnoi zadachi na poroge”, Dokl. AN, 458:6 (2014), 636–640 | DOI | Zbl

[25] 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

[26] Nazarov S. A., “Lokalizatsiya uprugikh kolebanii v krestoobraznykh ploskikh ortotropnykh volnovodakh”, Dokl. AN, 458:1 (2014), 42–46 | DOI

[27] Van Daik M., Metody vozmuschenii v mekhanike zhidkostei, Mir, M., 1967

[28] Ilin A. M., Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989