Geodesic
Asymptotics of the spectrum of a two-dimensional Hartree-type operator with a Coulomb self-action potential near the lower boundaries of spectral clusters
Teoretičeskaâ i matematičeskaâ fizika, Tome 199 (2019) no. 3, pp. 445-459 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the eigenvalue problem for a perturbed two-dimensional oscillator where the perturbation is an integral Hartree-type nonlinearity with a Coulomb self-action potential. We obtain asymptotic eigenvalues and asymptotic eigenfunctions near the lower boundaries of spectral clusters formed in a neighborhood of the eigenvalues of the unperturbed operator and construct an asymptotic expansion near a circle where the solution is localized.
Keywords: self-consistent field, spectral cluster, spectrum splitting, asymptotic eigenvalue, asymptotic eigenfunction. 
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     title = {Asymptotics of the~spectrum of a~two-dimensional {Hartree-type} operator with {a~Coulomb} self-action potential near the~lower boundaries of spectral clusters},
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D. A. Vakhrameeva; A. V. Pereskokov. Asymptotics of the spectrum of a two-dimensional Hartree-type operator with a Coulomb self-action potential near the lower boundaries of spectral clusters. Teoretičeskaâ i matematičeskaâ fizika, Tome 199 (2019) no. 3, pp. 445-459. http://geodesic.mathdoc.fr/item/TMF_2019_199_3_a6/

[1] N. N. Bogolyubov, “Ob odnoi novoi forme adiabaticheskoi teorii vozmuschenii v zadache o vzaimodeistvii chastitsy s kvantovym polem”, Ukr. matem. zhurn., 2:2 (1950), 3–24

[2] L. P. Pitaevskii, “Kondensatsiya Boze–Einshteina v magnitnykh lovushkakh. Vvedenie v teoriyu”, UFN, 168:6 (1998), 641–653 | DOI | DOI

[3] S. A. Achmanov, R. V. Hocklov, A. P. Suchorukov, “Self-defocusing and self-modulation in nonlinear media”, Laserhandbuch, v. 2, eds. F. T. Arecchi, E. O. Schulz-Dubois, North-Holland, Amsterdam, 1972, 5–108

[4] A. S. Davydov, Solitony v molekulyarnykh sistemakh, Naukova dumka, Kiev, 1984 | DOI | MR

[5] V. P. Maslov, Kompleksnye markovskie tsepi i kontinualnyi integral Feimana, Nauka, M., 1976 | MR

[6] M. V. Karasev, V. P. Maslov, “Algebry s obschimi perestanovochnymi sootnosheniyami i ikh prilozheniya. II. Operatornye unitarno-nelineinye uravneniya”, Itogi nauki i tekhn. Ser. Sovrem. probl. matem., 13 (1979), 145–267 | DOI | MR | Zbl

[7] S. I. Chernykh, “Kvaziklassicheskaya chastitsa v odnomernom samosoglasovannom pole”, TMF, 52:3 (1982), 491–494 | DOI | MR

[8] I. V. Simenog, “Ob asimptotike resheniya statsionarnogo nelineinogo uravneniya Khartri”, TMF, 30:3 (1977), 408–414 | DOI | MR

[9] V. V. Belov, F. N. Litvinets, A. Yu. Trifonov, “Kvaziklassicheskie spektralnye serii operatora tipa Khartri, otvechayuschie tochke pokoya klassicheskoi sistemy Gamiltona–Erenfesta”, TMF, 150:1 (2007), 26–40 | DOI | DOI | MR | Zbl

[10] I. Bryuning, S. Yu. Dobrokhotov, R. V. Nekrasov, A. I. Shafarevich, “Rasprostranenie gaussovykh volnovykh paketov v kvantovykh tonkikh periodicheskikh volnovodakh s nelokalnoi nelineinostyu”, TMF, 155:2 (2008), 215–235 | DOI | DOI | MR | Zbl

[11] V. M. Babich, V. S. Buldyrev, Asimptoticheskie metody v difraktsii korotkikh voln. Metod etalonnykh zadach, Nauka, M., 1972 | MR | Zbl

[12] V. P. Maslov, Kompleksnyi metod VKB v nelineinykh uravneniyakh, Nauka, M., 1977 | DOI | MR | MR | Zbl

[13] A. V. Pereskokov, “Kvaziklassicheskaya asimptotika spektra operatora tipa Khartri vblizi verkhnikh granits spektralnykh klasterov”, TMF, 178:1 (2014), 88–106 | DOI | DOI | MR | Zbl

[14] A. V. Pereskokov, “Kvaziklassicheskaya asimptotika spektra vblizi nizhnikh granits spektralnykh klasterov dlya operatora tipa Khartri”, Matem. zametki, 101:6 (2017), 894–910 | DOI | DOI | MR

[15] A. V. Pereskokov, “Asimptotika spektra operatora Khartri vblizi verkhnikh granits spektralnykh klasterov. Asimptoticheskie resheniya, lokalizovannye vblizi okruzhnosti”, TMF, 183:1 (2015), 78–89 | DOI | DOI | MR

[16] A. V. Pereskokov, “Kvaziklassicheskaya asimptotika spektra dvumernogo operatora Khartri vblizi verkhnikh granits spektralnykh klasterov”, TMF, 187:1 (2016), 74–87 | DOI | DOI | MR

[17] A. V. Pereskokov, “Asymptotics of the spectrum of a two-dimensional Hartree type operator near upper boundaries of spectral clusters. Asymptotic solutions located near a circle”, J. Math. Sci. (N. Y.), 226:4 (2017), 517–530 | DOI | MR

[18] Z. Flyugge, Zadachi po kvantovoi mekhanike, v. 1, Mir, M., 1974 | MR | MR | Zbl

[19] A. F. Nikiforov, V. B. Uvarov, Osnovy teorii spetsialnykh funktsii, Nauka, M., 1974 | MR

[20] G. Beitmen, A. Eirdeii, Vysshie transtsendentnye funktsii, v. 3, Ellipticheskie i avtomorfnye funktsii. Funktsii Lame i Mate, Nauka, M., 1967 | MR | Zbl

[21] P. K. Suetin, Klassicheskie ortogonalnye mnogochleny, Fizmatlit, M., 2007 | MR | Zbl

[22] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integraly i ryady. Elementarnye funktsii, Nauka, M., 1981 | MR | MR | Zbl

[23] V. P. Maslov, Operatornye metody, Nauka, M., 1973 | MR

[24] M. V. Karasev, A. V. Pereskokov, “Asimptoticheskie resheniya uravnenii Khartri, sosredotochennye vblizi malomernykh podmnogoobrazii. I. Model s logarifmicheskoi osobennostyu”, Izv. RAN. Ser. matem., 65:5 (2001), 33–72 | DOI | DOI | MR | Zbl

[25] M. V. Karasev, A. V. Pereskokov, “Asimptoticheskie resheniya uravnenii Khartri, sosredotochennye vblizi malomernykh podmnogoobrazii. II. Lokalizatsiya v ploskikh diskakh”, Izv. RAN. Ser. matem., 65:6 (2001), 57–98 | DOI | DOI | MR | Zbl

[26] A. V. Pereskokov, “Semiclassical asymptotics of solutions to Hartree type equations concentrated on segments”, J. Math. Sci., 226:4 (2017), 462–516 | DOI | MR