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Teoreticheskaya i Matematicheskaya Fizika, 2002, Volume 131, Number 3, Pages 389–406
DOI: https://doi.org/10.4213/tmf336 (Mi tmf336) 		 
This article is cited in 7 scientific papers (total in 7 papers)

Asymptotic Solutions of Two-Dimensional Hartree-Type Equations Localized in the Neighborhood of Line Segments

A. V. Pereskokov

Moscow Power Engineering Institute (Technical University)
Full-text PDF (286 kB) Citations (7)
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DOI: https://doi.org/10.4213/tmf336
Abstract: 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.
Received: 11.10.2001
English version:
Theoretical and Mathematical Physics, 2002, Volume 131, Issue 3, Pages 775–790
DOI: https://doi.org/10.1023/A:1015923406662
Bibliographic databases:
Language: Russian
Citation: A. V. Pereskokov, “Asymptotic Solutions of Two-Dimensional Hartree-Type Equations Localized in the Neighborhood of Line Segments”, TMF, 131:3 (2002), 389–406; Theoret. and Math. Phys., 131:3 (2002), 775–790
Citation in format AMSBIB
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  • https://doi.org/10.4213/tmf336
  • https://www.mathnet.ru/eng/tmf/v131/i3/p389
    • This publication is cited in the following 7 articles:
    1. A. V. Pereskokov, “Asymptotic Solutions to the Hartree Equation Near a Sphere. Asymptotics of Self-Consistent Potentials”, J Math Sci, 276:1 (2023), 154  crossref
    2. A. V. Pereskokov, “Semiclassical Asymptotics of Solutions to Hartree Type Equations Concentrated on Segments”, J Math Sci, 226:4 (2017), 462  crossref
    3. 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
    4. 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
    5. 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
    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 Asymptotic solution of the one-dimensional hartree equation with the non-smooth interaction potential. asymtotics of quantum averages  elib
    7. 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
    Citing articles in Google Scholar: Russian citations, English citations
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    		Теоретическая и математическая физика Theoretical and Mathematical Physics
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