Existence and asymptotics of poles with small imaginary part for the Helmholtz resonator

R. R. Gadyl'shin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 52 (1997) no. 1, pp. 1-72

Bibliographie

[1] Agranovich M. S., “Spektralnye svoistva zadach difraktsii”, Dopolnenie k kn. [9], 289–416

[2] Agranovich M. S., “O summiruemosti ryadov po kornevym vektoram nesamosopryazhennykh ellipticheskikh operatorov”, Funktsion. analiz i ego prilozh., 10:3 (1976), 1–12 | MR | Zbl

[3] Agranovich M. S., “O ryadakh po kornevym vektoram operatorov, ochen blizkikh k samosopryazhennym”, Funktsion. analiz i ego prilozh., 11:4 (1977), 65–67 | MR | Zbl

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

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

[6] Babich V. M., “Ob analiticheskom prodolzhenii rezolventy vneshnikh zadach dlya operatora Laplasa na vtoroi list”, Teoriya funktsii, funktsion. analiz i ikh prilozheniya, no. 3, Izd-vo Khark. un-ta, Kharkov, 1966, 151–157

[7] Bakhvalov N. S., Panasenko G. P., “Osrednenie protsessov v periodicheskikh sredakh”, Matematicheskie zadachi mekhaniki kompozitsionnykh materialov, Nauka, M., 1984 | Zbl

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

[9] Voitovich N. N., Katsenelenbaum B. Z., Sivov A. N., Obobschennyi metod sobstvennykh kolebanii v teorii difraktsii, Nauka, M., 1977 | Zbl

[10] Gadylshin R. R., “Asimptotika sobstvennogo znacheniya singulyarno vozmuschennoi ellipticheskoi zadachi s malym parametrom v granichnom uslovii”, Differents. uravneniya, 22 (1986), 640–652 | MR | Zbl

[11] Gadylshin R. R., “Rasscheplenie kratnogo sobstvennogo znacheniya zadachi Dirikhle dlya operatora Laplasa pri singulyarnom vozmuschenii granichnogo usloviya”, Matem. zametki, 52:4 (1992), 42–55 | MR | Zbl

[12] Gadylshin R. R., “Rasscheplenie kratnogo sobstvennogo znacheniya v kraevoi zadache dlya membrany, zakreplennoi na malom uchastke granitsy”, Sib. matem. zhurn., 33:3 (1993), 43–61 | MR

[13] Gadylshin R. R., “Asimptotika vtoroi sobstvennoi chastoty dlya sistemy dvukh tel, soedinennykh tonkoi peremychkoi”, TMF, 97:1 (1993), 68–77 | MR | Zbl

[14] Gadylshin R. R., “O sobstvennykh chastotakh tel s tonkimi otrostkami. I: Skhodimost i otsenki”, Matem. zametki, 54:6 (1993), 10–21 | MR | Zbl

[15] Gadylshin R. R., “Metod sraschivaniya asimptoticheskikh razlozhenii v zadache ob akusticheskom rezonatore Gelmgoltsa”, Prikl. matem. i mekh., 56 (1992), 412–418 | MR | Zbl

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

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

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

[19] Gadylshin R. R., “Singulyarno vozmuschennaya zadacha dlya operatora Gelmgoltsa”, Differentsialnye uravneniya s malym parametrom, Izd-vo UNTs AN SSSR, Sverdlovsk, 1984, 18–35 | MR

[20] Gadylshin R. R., “Ob amplitude kolebanii dlya rezonatora Gelmgoltsa”, Dokl. AN SSSR, 310 (1990), 1094–1097 | MR

[21] Gadylshin R. R., “O kvazisobstvennykh chastotakh rezonatora Gelmgoltsa”, Asimptoticheskie resheniya zadach matematicheskoi fiziki, Izd-vo BNTs UrO AN SSSR, Ufa, 1990, 33–49

[22] Gadylshin R. R., “O rasscheplenii polyusov funktsii Grina dlya elektromagnitnogo analoga rezonatora Gelmgoltsa”, Radiotekhnika i elektronika, 36 (1991), 2045–2047

[23] Gadylshin R. R., “Poverkhnostnye potentsialy i metod sraschivaniya asimptoticheskikh razlozhenii v zadache o rezonatore Gelmgoltsa”, UMN, 46:6 (1991), 175–176

[24] Gadylshin R. R., “Ob analiticheskom prodolzhenii funktsii Grina rezonatora Gelmgoltsa”, Asimptoticheskie metody resheniya differentsialnykh uravnenii, Izd-vo BNTs UrO AN SSSR, Ufa, 1992, 18–26

[25] Gadylshin R. R., “Rasseyanie $E$-polyarizovannogo polya na idealno provodyaschei tsilindricheskoi poverkhnosti s uzkoi schelyu”, Pisma v ZhTF, 18:4 (1992), 10–13

[26] Gadylshin R. R., “Poverkhnostnye potentsialy i metod soglasovaniya asimptoticheskikh razlozhenii v zadache o rezonatore Gelmgoltsa”, Algebra i analiz, 4:2 (1992), 88–115 | MR

[27] Gadylshin R. R., “O slipayuschikhsya polyusakh akusticheskogo rezonatora”, Dokl. AN, 234 (1992), 773–776 | MR

[28] Gadylshin R. R., “Rezonator Gelmgoltsa s malym otverstiem”, Zhurn. vychisl. matem. i matem. fiziki, 32 (1992), 1446–1457 | MR

[29] Gadylshin R. R., “O sisteme vlozhennykh rezonatorov”, Dokl. AN, 326 (1992), 939–942 | Zbl

[30] Gadylshin R. R., “O polyusakh akusticheskogo rezonatora”, Funktsion. analiz i ego prilozh., 27:5 (1993), 3–16 | MR | Zbl

[31] Gadylshin R. R., “Dvumernyi analog rezonatora Gelmgoltsa s idealno zhestkimi stenkami”, Differents. uravneniya, 30 (1994), 221–229 | MR | Zbl

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

[33] Gotlib V. Yu., “Rasseyanie na kruglom rezonatore s uzkoi schelyu kak zadacha teorii vozmuschenii”, Dokl. AN SSSR, 287 (1986), 1109–1113 | MR

[34] Zhikov V. V., Kozlov S. M., Oleinik O. A., Kha Ten Ngoan, “Usrednenie i $G$-skhodimost differentsialnykh operatorov”, UMN, 34:5 (1979), 65–133 | MR | Zbl

[35] Zaionchkovskii V., Solonnikov V. A., “O zadache Neimana dlya ellipticheskikh uravnenii vtorogo poryadka v oblasti s rebrami na granitse”, Zap. nauchn. semin. LOMI, 127, 1983, 7–48 | MR

[36] Ilin A. M., “Kraevaya zadacha dlya ellipticheskogo uravneniya vtorogo poryadka v oblasti s uzkoi schelyu. I: Dvumernyi sluchai”, Matem. sb., 99 (1976), 514–537 | MR | Zbl

[37] Ilin A. M., “Issledovanie asimptotiki resheniya ellipticheskoi kraevoi zadachi v oblasti s malym otverstiem”, Trudy seminara im. I. G. Petrovskogo, 6, 1981, 57–82 | Zbl

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

[39] Ilin A. M., “Kraevaya zadacha dlya ellipticheskogo uravneniya vtorogo poryadka v oblasti s uzkoi schelyu. II: Oblast s malym otverstiem”, Matem. sb., 103 (1977), 265–284 | MR | Zbl

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

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

[42] Kondratev V. A., “O gladkosti resheniya zadachi Dirikhle dlya ellipticheskogo uravneniya vtorogo poryadka v kusochno-gladkoi oblasti”, Differents. uravneniya, 6 (1970), 1831–1843 | MR | Zbl

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

[44] Kondratev V. A., Oleinik O. A., “Kraevye zadachi dlya uravnenii s chastnymi proizvodnymi v negladkikh oblastyakh”, UMN, 38:2 (1983), 3–76 | MR | Zbl

[45] Krein S. G., Petunin Yu. I., “Shkaly banakhovykh prostranstv”, UMN, 21:2 (1966), 89–168 | MR | Zbl

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

[47] Laks P., Fillips R., Teoriya rasseyaniya, Mir, M., 1971

[48] Mazya V. G., Nazarov S. A., Plamenevskii B. A., “Asimptoticheskie razlozheniya sobstvennykh chisel kraevykh zadach dlya operatora Laplasa v oblastyakh s malymi otverstiyami”, Izv. AN SSSR. Ser. matem., 48 (1984), 347–371 | MR

[49] Mazya V. G., Nazarov S. A., Plamenevskii B. A., “Asimptotika resheniya zadachi Dirikhle v oblasti s vyrezannoi tonkoi trubkoi”, Matem. sb., 116 (1981), 187–217 | MR | Zbl

[50] Marchenko V. A., Khruslov E. Ya., Kraevye zadachi v oblastyakh s melkozernistoi granitsei, Naukova dumka, Kiev, 1974

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

[52] Nazarov S. A., Plamenevskii B. A., Ellipticheskie zadachi v oblastyakh s kusochno gladkoi granitsei, Nauka, M., 1991

[53] Oleinik O. A., Chechkin G. A., “O kraevykh zadachakh dlya ellipticheskikh uravnenii s bystromenyayuschimsya tipom granichnykh uslovii”, UMN, 48:6 (1993), 163–164 | MR | Zbl

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

[55] Petras S. V., “O rasscheplenii serii rezonansov na “nefizicheskom liste””, Zap. nauchn. semin. LOMI, 51, 1975, 155–169 | MR | Zbl

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

[57] Relei Dzh., Teoriya zvuka, Gostekhizdat, M., 1955

[58] Samarskii A. A., “O vliyanii zakrepleniya na sobstvennye chastoty zamknutykh ob'emov”, Dokl. AN SSSR, 63 (1948), 631–634 | MR | Zbl

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

[60] Smirnov M. M., Differentsialnye uravneniya v chastnykh proizvodnykh vtorogo poryadka, Nauka, M., 1964

[61] Fedoryuk M. V., “Asimptotika resheniya zadachi Dirikhle dlya uravnenii Laplasa i Gelmgoltsa vo vneshnosti tonkogo tsilindra”, Izv. AN SSSR. Ser. matem., 45:1 (1981), 167–186 | MR | Zbl

[62] Fedoryuk M. V., “Zadacha Dirikhle dlya operatora Laplasa vo vneshnosti tonkogo tela vrascheniya”, Trudy seminara S. L. Soboleva, no. 1, Novosibirsk, 1980, 113–131 | MR | Zbl

[63] Fedoryuk M. V., “Asimptotika resheniya zadachi rasseyaniya na tsilindre s bolshim vozmuscheniem”, Trudy MMO, 48, 1985, 150–162 | MR | Zbl

[64] Chechkin G. A., “Usrednenie kraevykh zadach s singulyarnym vozmuscheniem granichnykh uslovii”, Matem. sb., 184:6 (1993), 99–150 | MR | Zbl

[65] Shabat B. V., Vvedenie v kompleksnyi analiz, Chast 1, Nauka, M., 1985

[66] Shestopalov V. P., Summatornye uravneniya v sovremennoi teorii diffraktsii, Naukova dumka, Kiev, 1983

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

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

[69] 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

[70] Fernandez C., “Resonances in scattering by a resonator”, Indiana Univ. Math. J., 34 (1985), 115–125 | DOI | MR | Zbl

[71] Fang Q., “Asymptotic behavior and domain-dependency of solutions to a class of reaction-diffusion systems with large diffusion coefficients”, Hiroshima Math. J., 20 (1990), 549–571 | MR | Zbl

[72] Gadyl'shin R. R., “On acoustic Helmholtz resonator and on its electromagnetic analogue”, J. Math. Phys., 35 (1994), 3464–3481 | DOI | MR | Zbl

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

[74] Jimbo S., “The singularly perturbed domain and the characterization for the eigenfunctions with Neumann boundary condition”, J. Differential Equations, 77 (1989), 322–350 | DOI | MR | Zbl

[75] Majda A., “Outgoing solutions for perturbation of $-\Delta$ with applications to spectral and scattering theory”, J. Differential Equations, 16 (1974), 515–547 | DOI | MR | Zbl

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

[77] Ozawa S., “Singular variation of domains and eigenvalues of the Laplacian”, Duke Math. J., 48 (1981), 767–778 | DOI | MR | Zbl

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