Geodesic
Asymptotic Solutions of Two-Dimensional Hartree-Type Equations Localized in the Neighborhood of Line Segments
Teoretičeskaâ i matematičeskaâ fizika, Tome 131 (2002) no. 3, pp. 389-406 
A. V. Pereskokov. Asymptotic Solutions of Two-Dimensional Hartree-Type Equations Localized in the Neighborhood of Line Segments. Teoretičeskaâ i matematičeskaâ fizika, Tome 131 (2002) no. 3, pp. 389-406. http://geodesic.mathdoc.fr/item/TMF_2002_131_3_a2/
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     title = {Asymptotic {Solutions} of {Two-Dimensional} {Hartree-Type} {Equations} {Localized} in the {Neighborhood} of {Line} {Segments}},
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We consider the eigenvalue problem for the two-dimensional Schrödinger equation containing an integral Hartree-type nonlinearity with an interaction potential having a logarithmic singularity. Global asymptotic solutions localized in the neighborhood of a line segment in the plane are constructed using the matching method for asymptotic expansions. The Bogoliubov and Airy polarons are used as model functions in these solutions. An analogue of the Bohr–Sommerfeld quantization rule is established to find the related series of eigenvalues.

[1] D. R. Hartree, Proc. Cambridge Philos. Soc., 24 (1928), 89 ; 111 ; 426 | DOI | Zbl

[2] S. I. Pekar, Issledovaniya po elektronnoi teorii kristallov, Gostekhizdat, M., 1951

[3] P. Ring, P. Schuck, The Nuclear Many-Body Problem, Springer, Berlin, 1980 | MR

[4] G. V. Gadiyak, K. A. Nasyrov, Chisl. met. mekh. splosh. sredy (Novosibirsk), 12:5 (1981), 3 | Zbl

[5] J. Kurlandski, Bull. Acad. Polon. Sci., 30:3–4 (1982), 135

[6] V. M. Olkhov, TMF, 51:1 (1982), 150 | MR

[7] D. V. Chudnovsky, “Infinite component two-dimensional completely integrable systems of KdV type”, The Riemann Problem, Complete Integrability and Arithmetic Application, Proc. of a Seminar (IHES and Columbia Univ., 1979–1980), Lect. Notes Math., 925, eds. D. Chudnovsky and G. Chudnovsky, Springer, Berlin, 1982, 71 | DOI | MR

[8] Yu. N. Karamzin, I. L. Tsvetkova, ZhVMiMF, 22:1 (1982), 235 | MR | Zbl

[9] A. A. Bogolyubskaya, I. L. Bogolyubskii, TMF, 54:2 (1983), 258

[10] D. Gogny, P. L. Lions, Model. Math. Anal. Numer., 20:4 (1986), 571 | DOI | MR | Zbl

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

[12] V. P. Maslov, “Uravneniya samosoglasovannogo polya”, Itogi nauki i tekhniki. Sovremennye problemy matematiki, 11, ed. R. V. Gamkrelidze, VINITI, M., 1978, 153 | MR

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

[14] I. V. Simenog, TMF, 30:3 (1977), 408 | MR

[15] M. V. Karasev, V. P. Maslov, “Algebry s obschimi perestanovochnymi sootnosheniyami i ikh prilozheniya. II: Operatornye unitarno-nelineinye uravneniya”, Itogi nauki i tekhniki. Sovremennye problemy matematiki, 13, ed. R. V. Gamkrelidze, VINITI, M., 1979, 145

[16] M. V. Karasev, A. V. Pereskokov, TMF, 97:1 (1993), 78 | MR

[17] M. V. Karasev, Yu. V. Osipov, TMF, 52:2 (1982), 263 | MR

[18] M. V. Karasev, Kvantovaya reduktsiya na orbity algebr simmetrii i zadacha Erenfesta, Preprint ITF-87-157R, ITF AN SSSR, Kiev, 1987

[19] R. R. Gabdullin, Malomernaya approksimatsiya reshenii uravneniya polyarona Pekara, Preprint, NTsBI AN SSSR, Puschino, 1991

[20] M. V. Karasev, A. V. Pereskokov, Izv. RAN. Ser. matem., 65:5 (2001), 33 | DOI | MR | Zbl

[21] M. V. Karasev, A. V. Pereskokov, Izv. RAN. Ser. matem., 65:6 (2001), 57 | DOI | MR | Zbl