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Izv. RAN. Ser. Mat., 2001, Volume 65, Issue 6, Pages 57–98 (Mi izv365)  

This article is cited in 11 scientific papers (total in 11 papers)

Asymptotic solutions of Hartree equations concentrated near low-dimensional submanifolds. II. Localization in planar discs

M. V. Karaseva, A. V. Pereskokovb

a Moscow State Institute of Electronics and Mathematics
b Moscow Power Engineering Institute (Technical University)

Abstract: We consider the eigenvalue problem for the three-dimensional Hartree equation in an external field and construct asymptotic (quasi-classical) solutions concentrated near two-dimensional planar discs. The rate of decrease of these solutions along the normal to the disc is determined by the Bogolyubov polaron, and near the edge of the disc it is defined by the Airy analogue of the polaron. To find the related series of eigenvalues, an analogue of the Bohr–Sommerfeld quantization rule is found from which is derived a simpler algebraic equation determining the main terms in the asymptotics of the eigenvalues.


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English version:
Izvestiya: Mathematics, 2001, 65:6, 1127–1168

Bibliographic databases:

MSC: 45K05, 81Q05, 34D05, 35Q99, 35C20, 35P30, 58F19, 81V70
Received: 13.03.1998

Citation: M. V. Karasev, A. V. Pereskokov, “Asymptotic solutions of Hartree equations concentrated near low-dimensional submanifolds. II. Localization in planar discs”, Izv. RAN. Ser. Mat., 65:6 (2001), 57–98; Izv. Math., 65:6 (2001), 1127–1168

Citation in format AMSBIB
\by M.~V.~Karasev, A.~V.~Pereskokov
\paper Asymptotic solutions of Hartree equations concentrated near low-dimensional submanifolds.~II. Localization in planar discs
\jour Izv. RAN. Ser. Mat.
\yr 2001
\vol 65
\issue 6
\pages 57--98
\jour Izv. Math.
\yr 2001
\vol 65
\issue 6
\pages 1127--1168

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    This publication is cited in the following articles:
    1. V. V. Belov, A. Yu. Trifonov, A. V. Shapovalov, “Semiclassical Trajectory-Coherent Approximations of Hartree-Type Equations”, Theoret. and Math. Phys., 130:3 (2002), 391–418  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. A. V. Pereskokov, “Asymptotic Solutions of Two-Dimensional Hartree-Type Equations Localized in the Neighborhood of Line Segments”, Theoret. and Math. Phys., 131:3 (2002), 775–790  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. M. V. Karasev, A. V. Pereskokov, “Asymptotic solutions for Hartree equations and logarithmic obstructions for higher corrections of semiclassical approximation”, Proc. Steklov Inst. Math. (Suppl.), 2003no. , suppl. 1, S123–S128  mathnet  mathscinet  zmath  elib
    4. V. V. Belov, F. N. Litvinets, A. Yu. Trifonov, “Semiclassical spectral series of a Hartree-type operator corresponding to a rest point of the classical Hamilton–Ehrenfest system”, Theoret. and Math. Phys., 150:1 (2007), 21–33  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. Lipskaya A.V., Pereskokov A.V., “Ob asimptoticheskikh resheniyakh uravneniya tipa khartri s potentsialom vzaimodeistviya yukavy, sosredotochennykh v share”, Vestnik Moskovskogo energeticheskogo instituta, 2011, no. 6, 30–38  elib
    6. Lipskaya A.V., Pereskokov A.V., “Asimptoticheskie resheniya odnomernogo uravneniya khartri s negladkim potentsialom vzaimodeistviya. asimptotika kvantovykh srednikh”, Vestnik moskovskogo energeticheskogo instituta, 2012, no. 6, 105–116  elib
    7. A. V. Pereskokov, “Semiclassical asymptotic spectrum of a Hartree-type operator near the upper boundary of spectral clusters”, Theoret. and Math. Phys., 178:1 (2014), 76–92  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    8. A. V. Pereskokov, “Asymptotics of the Hartree operator spectrum near the upper boundaries of spectral clusters: Asymptotic solutions localized near a circle”, Theoret. and Math. Phys., 183:1 (2015), 516–526  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    9. A. V. Pereskokov, “Semiclassical asymptotic approximation of the two-dimensional Hartree operator spectrum near the upper boundaries of spectral clusters”, Theoret. and Math. Phys., 187:1 (2016), 511–524  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    10. A. V. Pereskokov, “Semiclassical Asymptotics of the Spectrum near the Lower Boundary of Spectral Clusters for a Hartree-Type Operator”, Math. Notes, 101:6 (2017), 1009–1022  mathnet  crossref  crossref  mathscinet  isi  elib
    11. 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”, Theoret. and Math. Phys., 199:3 (2019), 864–877  mathnet  crossref
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