Trapping of a wave in a curved cylindrical acoustic waveguide with constant cross-section

S. A. Nazarov

Algebra i analiz, Tome 31 (2019) no. 5, pp. 154-183.

Résumé

Cylindrical acoustic waveguides with constant cross-section  $ \omega $ are considered, specifically, a straight waveguide $ \Omega ={\mathbb{R}}\times \omega \subset {\mathbb{R}}^d$ and a locally curved waveguide $ \Omega ^\varepsilon $ that depends on a parameter $ \varepsilon \in (0,1]$. For $ d>2$, in two different settings ( $ \varepsilon =1$ and $ \varepsilon \ll 1$), the task is to find an eigenvalue $ \lambda ^\varepsilon $ that is embedded in the continuous spectrum $ [0,+\infty )$ of the waveguide $ \Omega ^\varepsilon $ and, hence, is inherently unstable. In other words, a solution of the Neumann problem for the Helmholtz operator $ \Delta +\lambda ^\varepsilon $ arises that vanishes at infinity and implies an eigenfunction in the Sobolev space $ H^1(\Omega ^\varepsilon )$. In the first case, it is assumed that the cross-section $ \omega $ has a double symmetry and an eigenvalue arises for any nontrivial curvature of the axis of the waveguide $ \Omega ^\varepsilon $. In the second case, under an assumption on the shape of an asymmetric cross-section $ \omega $, the eigenvalue $ \lambda ^\varepsilon $ is formed by scrupulous fitting of the curvature $ O(\varepsilon )$ for small $ \varepsilon >0$.

Détail

Keywords: continuous and point spectra, eigenvalue, Neumann problem for the Laplace operator, curved cylinder, asymptotics, expanded scattering matrix.

Bibliographie

[1] Birman M. Sh., Solomyak M. Z., Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, LGU, L., 1980

[2] Ladyzhenskaya O. A., Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973

[3] Maslov V. P., “Asimptotika sobstvennykh funktsii uravneniya $\Delta u+k^2u=0$ s kraevymi usloviyami na ekvidistantnykh krivykh i rasseyanie elekromagnitnykh voln v volnovode”, Dokl. AN SSSR, 123:4 (1958), 631–633 | Zbl

[4] Maslov V. P., Vorobev E. M., “Ob odnomodovykh otkrytykh rezonatorakh”, Dokl. AN SSSR, 179:3 (1968), 558–561

[5] Exner P., “Bound states in curved quantum waveguides”, J. Math. Phys., 30:11 (1989), 2574–2580 | DOI | MR | Zbl

[6] Duclos P., Exner P., “Curvature-induced bound states in quantum waveguides in two and three dimensions”, Rev. Math. Phys., 7:1 (1995), 73–102 | DOI | MR | Zbl

[7] Exner P., Kovarîk H., Quantum waveguides, Theoretical and Mathematical Physics, Springer, Cham, 2015 | DOI | MR | Zbl

[8] Nazarov S. A., “Zakhvachennye volny v kolenchatom volnovode s zhestkimi stenkami”, Akust. zh., 57:6 (2011), 746–754

[9] Evans D. V., Levitin M., Vasil'ev D., “Existence theorems for trapped modes”, J. Fluid Mech., 261 (1994), 21–31 | DOI | MR | Zbl

[10] Kamotskii I. V., Nazarov S. A., “Anomalii Vuda i poverkhnostnye volny v zadachakh rasseyaniya na periodicheskoi granitse. I; II”, Mat. sb., 190:1 (1999), 109–138 | DOI | MR | Zbl

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

[12] Nazarov S. A., “Prinuditelnaya ustoichivost prostogo sobstvennogo chisla na nepreryvnom spektre volnovoda”, Funkts. anal. i ego pril., 47:3 (2013), 37–53 | DOI | MR | Zbl

[13] Nazarov S. A., Plamenevskii B. A., “Samosopryazhennye ellipticheskie zadachi: operatory rasseyaniya i polyarizatsii na rebrakh granitsy”, Algebra i analiz, 6:4 (1994), 157–186

[14] Kamotskii I. V., Nazarov S. A., “Rasshirennaya matritsa rasseyaniya i eksponentsialno zatukhayuschie resheniya ellipticheskoi zadachi v tsilindricheskoi oblasti”, Zap. nauch. semin. POMI, 264, 2000, 66–82 | Zbl

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

[16] Djurgjevich I., Dauge M., Faou E., Rossle A., “Eigenmode asymptotics in thin elastic plates”, J. Math. Pures Appl., 78:9 (1999), 925–964 | DOI | MR

[17] Nazarov S. A., “Ob asimptotike spektra zadachi teorii uprugosti dlya tonkoi plastiny”, Sib. mat. zh., 41:4 (2000), 895–912 | MR | Zbl

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

[19] Mazja W. G., Nazarov S. A., Plamenewski B. A., Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten, v. I, II, Math. Monogr., 82, 83, Akademie-Verlag, Berlin, 1991 | MR

[20] Nazarov S. A., Plamenevsky B. A., Elliptic problems in domains with piecewise smooth boundaries, De Gruyter Expo. Math., 13, Walter de Gruyter, Berlin, 1994 | MR

[21] Mazya V. G., Plamenevskii B. A., “Otsenki v $L_p$ i v klassakh Geldera i printsip maksimuma Miranda–Agmona dlya reshenii ellipticheskikh kraevykh zadach v oblastyakh s osobymi tochkami na granitse”, Math. Nachr., 77 (1977), 25–82

[22] Kamotskii I. V., Nazarov S. A., “Eksponentsialno zatukhayuschie resheniya zadachi o difraktsii na zhestkoi periodicheskoi reshetke”, Mat. zametki, 73:1 (2003), 138–140 | DOI | MR | Zbl

[23] Nazarov S. A., “Properties of spectra of boundary value problems in cylindrical and quasicylindrical domains”, Sobolev Spaces in Math. II, Int. Math. Ser., 9, Springer, New York, 2009, 261–309 | DOI | MR | Zbl

[24] Kozlov V. A., Nazarov S. A., Orlof A., “Trapped modes supported by localized potentials in the zigzag graphene ribbon”, C. R. Math. Acad. Sci. Paris, 354:1 (2016), 63–67 | DOI | MR | Zbl

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

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

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

[28] Nazarov S. A., “Weighted spaces with detached asymptotics in application to the Navier–Stokes equations”, Advances in Mathematical Fluid Mechanics (Paseky, 1999), Springer-Verlag, Berlin, 2000, 159–191 | DOI | MR | Zbl

[29] Nazarov S. A., Plamenevskii B. A., “Ob usloviyakh izlucheniya dlya samosopryazhennykh ellipticheskikh zadach”, Dokl. AN SSSR, 311:3 (1990), 532–536 | Zbl

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

[31] Aleksandrov A. D., Netsvetaev N. Yu., Geometriya, Nauka, M., 1990 | MR

[32] Avishai Y., Bessis D., Giraud B. G., Mantica G., “Quantum bound states in open geometries”, Phys. Rev. B, 44:15 (1991), 8028–8034 | DOI

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