Geodesic
Resonator systems
Izvestiya. Mathematics , Tome 64 (2000) no. 3, pp. 487-529.

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper deals with a system of embedded resonators and a chain of two resonators. We prove that the Green functions of the corresponding Neumann boundary-value problems have poles with small imaginary parts. We find complete asymptotics for these poles and the corresponding eigenfunctions by the method of matched asymptotic expansions. We consider the cases when the limit value of the pole is an eigenfrequency either of a single limit volume or of two such volumes simultaneously. We show that the orders of smallness of the imaginary parts of the poles for systems are quite different from those for the classical Helmholtz resonator. We apply the asymptotics obtained to the scattering problem. 
@article{IM2_2000_64_3_a1,
     author = {R. R. Gadyl'shin},
     title = {Resonator systems},
     journal = {Izvestiya. Mathematics },
     pages = {487--529},
     publisher = {mathdoc},
     volume = {64},
     number = {3},
     year = {2000},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2000_64_3_a1/}
}
TY  - JOUR
AU  - R. R. Gadyl'shin
TI  - Resonator systems
JO  - Izvestiya. Mathematics 
PY  - 2000
SP  - 487
EP  - 529
VL  - 64
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2000_64_3_a1/
LA  - en
ID  - IM2_2000_64_3_a1
ER  - 
%0 Journal Article
%A R. R. Gadyl'shin
%T Resonator systems
%J Izvestiya. Mathematics 
%D 2000
%P 487-529
%V 64
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2000_64_3_a1/
%G en
%F IM2_2000_64_3_a1
R. R. Gadyl'shin. Resonator systems. Izvestiya. Mathematics , Tome 64 (2000) no. 3, pp. 487-529. http://geodesic.mathdoc.fr/item/IM2_2000_64_3_a1/

[1] Lord Rayleigh, “The theory of Helmholtz resonator”, Proc. Roy. Soc. London. A, 92 (1916), 265–275 | DOI | Zbl

[2] Miles J. W., “Scattering by a spherical cap”, J. Acoust. Soc. Amer., 50:2 (1971), 892–903 | DOI

[3] Hubert-Palencia J., Sanchez-Palencia E., Coupling of Continious Systems. Asymptotic Methods, Springer-Verlag, Berlin, 1989 | MR | Zbl

[4] Arsenev A. A., “Ob osobennostyakh analiticheskogo prodolzheniya i rezonansnykh svoistvakh resheniya zadachi rasseyaniya dlya uravneniya Gelmgoltsa”, ZhVMiMF, 12 (1971), 112–138 | MR

[5] Beale J. T., “Scattering frequencies of resonators”, Comm. Pure and Applied Math., 26 (1973), 549–564 | DOI | MR

[6] Arsenev A. A., “O suschestvovanii rezonansnykh polyusov i rezonansov pri rasseyanii v sluchae kraevykh uslovii II i III roda”, ZhVMiMF, 16 (1976), 718–724 | MR | Zbl

[7] Petras S. V., “O rasscheplenii serii rezonansov na “nefizicheskom” liste”, Funktsion. analiz i ego prilozh., 9:2 (1975), 89–90 | MR | Zbl

[8] Petras S. V., “O rasscheplenii serii rezonansov na “nefizicheskom liste””, Zapiski nauch. sem. LOMI, 51, Nauka, L., 1975, 155–169 | MR | Zbl

[9] Brown R. M., Hislop P. D., Martinez A., Eigenvalues and resonances for domains with tubes: Neumann boundary conditions, Research report 92-06, University of Kentucky, 1992

[10] Hislop P. D., Martinez A., “Scattering resonances of Helmholtz resonator”, Indiana Univ. Math. J., 40 (1991), 767–788 | DOI | MR | Zbl

[11] Vainberg B. R., “Ob analiticheskikh svoistvakh rezolventy dlya odnogo klassa puchkov operatorov”, Matem. sb., 77 (1968), 259–296 | MR | Zbl

[12] Sanches-Palensia E., Neodnorodnye sredy i teoriya kolebanii, Mir, M., 1984 | MR

[13] Laks P., Fillips R., Teoriya rasseyaniya, Mir, M., 1971 | MR

[14] Gadylshin R. R., “Suschestvovanie i asimptotiki polyusov s maloi mnimoi chastyu dlya rezonatora Gelmgoltsa”, UMN, 52:1 (313) (1997), 3–77 | MR

[15] Gadylshin R. R., “Rasscheplenie polyusov rezonatora Gelmgoltsa”, Izv. RAN. Ser. matem., 57:5 (1993), 44–74 | MR | Zbl

[16] Gadylshin R. R., “Metod sraschivaniya asimptoticheskikh razlozhenii v zadache ob akusticheskom rezonatore Gelmgoltsa”, Prikladnaya matematika i mekhanika, 56 (1992), 412–418 | MR | Zbl

[17] Gadylshin R. R., “O kvazistatsionarnom rezhime rezonatora Gelmgoltsa”, Prikladnaya matematika i mekhanika, 57:5 (1993), 54–61 | MR

[18] Gadylshin R. R., “O vliyanii vybora mesta otverstiya i ego formy na svoistva akusticheskogo rezonatora Gelmgoltsa”, TMF, 93:1 (1992), 107–118 | MR

[19] Gadylshin R. R., “O polyusakh akusticheskogo rezonatora”, Funktsion. analiz i ego prilozh., 27:4 (1993), 3–16 | MR | Zbl

[20] Kolton D., Kress R., Metody integralnykh uravnenii v teorii rasseyaniya, Mir, M., 1987 | MR

[21] Ladyzhenskaya O. A., Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973 | MR

[22] Mikhailov V. P., Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1983 | MR

[23] Smirnov M. M., Differentsialnye uravneniya v chastnykh proizvodnykh vtorogo poryadka, Nauka, M., 1964 | MR

[24] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR | Zbl

[25] Van-Daik M., Metody vozmuschenii v mekhanike zhidkosti, Mir, M., 1967

[26] Naife A. Kh., Metody vozmuschenii, Mir, M., 1986

[27] Ilin A. M., Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989 | MR

[28] Mazya V. G., Nazarov S. A., Plamenevskii B. A., Asimptotika reshenii ellipticheskikh kraevykh zadach pri singulyarnom vozmuschenii oblasti, Izd-vo Tbil. un-ta, Tbilisi, 1981 | Zbl

[29] Eskin G. I., Kraevye zadachi dlya ellipticheskikh psevdodifferentsialnykh uravnenii, Nauka, M., 1973 | MR

[30] Polia G., Sege G., Izoperimetricheskie neravenstva v matematicheskoi fizike, Fizmatgiz, M., 1962

[31] Landkof N. S., Osnovy sovremennoi teorii potentsiala, Nauka, M., 1966 | MR | Zbl

[32] Gadylshin R. R., “O sisteme akusticheskikh rezonatorov v kvazistatsionarnom rezhime”, Prikladnaya matematika i mekhanika, 58:3 (1994), 104–112 | MR | Zbl