Bounded solutions in a $\mathrm{T}$-shaped waveguide and the spectral properties of the Dirichlet ladder

S. A. Nazarov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 54 (2014) no. 8, pp. 1299-1318

Résumé

The Dirichlet problem is considered on the junction of thin quantum waveguides (of thickness $h\ll1$) in the shape of an infinite two-dimensional ladder. Passage to the limit as $h\to+\infty$ is discussed. It is shown that the asymptotically correct transmission conditions at nodes of the corresponding one-dimensional quantum graph are Dirichlet conditions rather than the conventional Kirchhoff transmission conditions. The result is obtained by analyzing bounded solutions of a problem in the $\mathrm{T}$-shaped waveguide that the boundary layer phenomenon.

Bibliographie

[1] Birman M. Sh., Skvortsov G. E., “O kvadratichnoi summiruemosti starshikh proizvodnykh resheniya zadachi Dirikhle v oblasti s kusochno-gladkoi granitsei”, Izv. vuzov. Matem., 1962, no. 5, 11–21

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

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

[4] Nazarov S. A., Shanin A. V., “Raschet kharakteristik zakhvachennykh voln v T-obraznykh volnovodakh”, Zh. vychisl. matem. i matem. fiz., 51:1 (2011), 96–110

[5] Kuchment P., “Graph models for waves in thin structures”, Waves in Random Media, 12:12 (2002), R1–R24 | DOI

[6] Kuchment P., Post O., “On the spectrum of carbon nano-structures”, Commun. Math. Phys., 275:3 (2007), 805–826 | DOI

[7] Kuchment P., “Quantum graphs and their applications”, Waves in Random media, 14:1, special issue (2004)

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

[9] Nazarov S. A., “Dvuchlennaya asimptotika reshenii spektralnykh zadach s singulyarnymi vozmuscheniyami”, Matem. sbornik, 181:3 (1990), 291–320

[10] Nazarov S. A., “Asimptoticheskii analiz i modelirovanie sochleneniya massivnogo tela s tonkimi sterzhnyami”, Tr. seminara im. I. G. Petrovskogo, 24, Izd-vo MGU, M., 2004, 95–214

[11] Nazarov S. A., “Otsenki tochnosti modelirovaniya kraevykh zadach na sochlenenii oblastei s razlichnymi predelnymi razmernostyami”, Izv. RAN. Seriya matem., 68:6 (2004), 119–156 | DOI

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

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

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

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

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

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

[18] Nazarov S. A., “Energeticheskie usloviya izlucheniya Mandelshtama i vektor Umova–Pointinga v uprugikh volnovodakh”, Probl. matem. analiza, 72, Novosibirsk, 2013, 101–146

[19] Nazarov S. A., “Polinomialnoe svoistvo samosopryazhennykh ellipticheskikh kraevykh zadach i algebraicheskoe opisanie ikh atributov”, Uspekhi matem. nauk, 54:5 (1999), 77–142 | DOI

[20] Nazarov S. A., “Asimptotika sobstvennykh chisel na nepreryvnom spektre regulyarno vozmuschennogo kvantovogo volnovoda”, Teor. i matem. fiz., 167:2 (2011), 239–262 | DOI

[21] Leis R., Initial boundary value problems of mathematical physics, B. G. Teubner, Stuttgart, 1986

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

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

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

[25] Nazarov S. A., “Variatsionnyi i asimptoticheskii metody poiska sobstvennykh chisel pod porogom nepreryvnogo spektra”, Sibirsk. matem. zhurnal, 51:5 (2010), 1086–1101

[26] Nazarov S. A., “Prinuditelnaya ustoichivost prostogo sobstvennogo chisla na nepreryvnom spektre volnovoda”, Funktsionalnyi analiz i ego prilozheniya, 47:3 (2013), 37–53 | DOI

[27] Grushin V. V., “O sobstvennykh znacheniyakh finitno vozmuschennogo operatora Laplasa v beskonechnykh tsilindricheskikh oblastyakh”, Matem. zametki, 75:3 (2004), 360–371 | DOI

[28] Gadylshin R. R., “O lokalnykh vozmuscheniyakh kvantovykh volnovodov”, Teor. i matem. fiz., 145:3 (2005), 358–371 | DOI

[29] Nazarov S. A., Asimptoticheskaya teoriya tonkikh plastin i sterzhnei. Ponizhenie razmernosti i integralnye otsenki, Nauchn. kniga, Novosibirsk, 2002

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

[31] Kuchment P., Floquet theory for partial differential equations, Birchhäuser, Basel, 1993

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

[33] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, Mir, M., 1971

[34] 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

[35] Nazarov S. A., “Proyavlenie prostranstvennoi struktury polya napryazhenii v okrestnosti uglovoi tochki tonkoi plastiny”, Prikl. matem. i mekhan., 55:4 (1991), 653–661

[36] Nazarov S. A., “Asimptotika reshenii i modelirovanie zadach teorii uprugosti v oblasti s bystroostsilliruyuschei granitsei”, Izv. RAN. Ser. matem., 72:3 (2008), 103–158 | DOI

[37] Kuchment P., Zeng H., “Asymptotics of spectra of Neumann Laplacian in thin domains”, Advances in Differen. Equat. Math. Phys., Contemporary Math., 387, eds. Yu. Karpeshin etc., AMS, 2003, 199–213 | DOI

[38] Nazarov S. A., “Obschaya skhema osredneniya samosopryazhennykh ellipticheskikh sistem v mnogomernykh oblastyakh, v tom chisle tonkikh”, Algebra i analiz, 7:5 (1995), 1–92

[39] 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

[40] Jones D. S., “The eigenvalues of $\nabla^2u+\lambda u=0$ when the boundary conditions are given on semi-infinite domains”, Proc. Camb. Phil., 49 (1953), 668–684 | DOI