Scattering coefficients and threshold resonances in a waveguide with inflating resonator

S. A. Nazarov   ; K. M. Ruotsalainen   ; P. J. Uusitalo

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 51, Tome 506 (2021), pp. 175-209

Résumé

The spectral Dirichlet problem is considered in a waveguide formed from a semi-infinite cylinder $\Pi$ and the resonator $\Theta_R$ obtained by inflating $R$ times a fixed star-shaped domain $\Theta$. The behaviour of the scattering coefficient $s(R)$ is studied as the parameter $R$ grows, namely it is verified that this coefficient moves clockwise without stops along the unit circle in the complex plane. For $s(R)=-1$, the proper threshold resonance occurs which is accompanied by the appearance of an almost standing wave and provokes for various near threshold anomalies, in particular, splitting eigenvalues off from the threshold. It is shown that under the geometrical symmetry resonances of other type are generated by trapped waves at the threshold. The justification of asymptotics is made by applying the technique of weighted spaces with detached asymptotics and an analysis of the singularities of physical fields at the edge $\partial\Theta_R\cap \partial \Pi$.

Bibliographie

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

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

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

[4] S. Molchanov, B. Vainberg, “Scattering solutions in networks of thin fibers; small diameter asymptotics”, Comm. Math. Phys., 273:2 (2007), 533–559 | DOI | MR | Zbl

[5] S. A. Nazarov, “Porogovye rezonansy i virtualnye urovni v spektre tsilindricheskikh i periodicheskikh volnovodov”, Izv. RAN, Seriya matem., 84:6 (2020), 73–130 | MR | Zbl

[6] S. A. Nazarov, “Raznoobraznye proyavleniya anomalii Vuda v lokalno iskrivlennykh kvantovykh volnovodakh”, Zh. vychisl. matem. i matem. fiz., 58:11 (2018), 1911–1930

[7] S. A. Nazarov, “Anomalii rasseyaniya akusticheskikh voln vblizi tochek otsechki nepreryvnogo spektra (obzor)”, Akust. zh., 66:5 (2020), 489–508

[8] S. A. Nazarov, “Volnovod s dvoinym porogovym rezonansom na prostom poroge”, Matem. sb., 211:8 (2020), 20–67 | MR | Zbl

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

[10] K. Pankrashkin, “Eigenvalue inequalities and absence of threshold resomamnces for waveguide junctions”, J. Math. Anal., Appl., 449:1 (2017), 907–925 | DOI | MR | Zbl

[11] F. L. Bakharev, S. A. Nazarov, “Kriterii nalichiya i otsutstviya ogranichennykh reshenii na poroge nepreryvnogo spektra v ob'edinenii kvantovykh volnovodov”, Algebra i analiz, 32:6 (2020), 1–23 | MR

[12] Maz'ya V., Nazarov S., Plamenevskij B., Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, v. 1, 2, Birkhäuser Verlag, Basel, 2000

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

[14] T. Kato, Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972

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

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

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

[18] R. Mitra, S. Li, Analiticheskie metody teorii volnovodov, Mir, M., 1974

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

[20] J. H. Poynting, “On the transfer of energy in the electromagnetic field”, Phil. Trans. of the Royal Society of London, 175 (1984), 343–361

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

[22] L. Bers, F. John, M. Schehter, Partial differential equations, Interscience, New York, 1964 | Zbl

[23] V. A. Kondratev, “O gladkosti reshenii zadachi Dirikhle dlya ellipticheskogo uravneniya vtorogo poryadka v okrestnosti rebra”, Diff. uravneniya, 6:10 (1970), 1831–1843 | Zbl

[24] V. A. Kondratev, “Osobennosti reshenii zadachi Dirikhle dlya ellipticheskogo uravneniya vtorogo poryadka v okrestnosti rebra”, Diff. uravneniya, 13:11 (1977), 2026–2032 | MR | Zbl

[25] V. A. Nikishkin, “Osobennosti reshenii zadachi Dirikhle dlya ellipticheskogo uravneniya vtorogo poryadka v okrestnosti rebra”, Vestn. Mosk. un-ta, 1979, no. 2, 51–62 | MR | Zbl

[26] V. G. Mazya, B. A. Plamenevskii, “Ob elliptichnosti kraevykh zadach v oblastyakh s kusochno gladkoi granitsei”, Tr. simp. po mekh. sploshnykh sred i rodstvennym probl. analiza, v. 1, Metsniereba, Tbilisi, 1973, 171–181

[27] V. G. Mazya, B. A. Plamenevskii, “Otsenki funktsii Grina i shauderovskie otsenki reshenii ellipticheskikh kraevykh zadach v dvugrannom ugle”, Sib. matem. zh., 19:5 (1978), 1065–1082 | MR

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

[29] M. I. Vishik, L. A. Lyusternik, “Regulyarnoe vyrozhdenie i pogranichnyi sloi dlya lineinykh differentsialnykh uravnenii s malym parametrom”, Uspekhi matem. nauk, 12:5 (1953), 3–122