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Izv. RAN. Ser. Mat., 2001, Volume 65, Issue 5, Pages 33–72 (Mi izv356)  

This article is cited in 12 scientific papers (total in 12 papers)

Asymptotic solutions of Hartree equations concentrated near low-dimensional submanifolds. I. The model with logarithmic singularity

M. V. Karaseva, A. V. Pereskokovb

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

Abstract: We consider a two-dimensional model Schrödinger equation with logarithmic integral non-linearity. We find asymptotic expansions for its solutions (Airy polarons) that decay exponentially at the “semi-infinity” and oscillate along one direction. These solutions may be regarded as new special functions, which are somewhat similar to the Airy function. We use them to construct global asymptotic solutions of Schrödinger equations with a small parameter and with integral non-linearity of Hartree type.

DOI: https://doi.org/10.4213/im356

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English version:
Izvestiya: Mathematics, 2001, 65:5, 883–921

Bibliographic databases:

MSC: 45K05, 81Q05, 35Q99, 35P30
Received: 13.03.1998

Citation: M. V. Karasev, A. V. Pereskokov, “Asymptotic solutions of Hartree equations concentrated near low-dimensional submanifolds. I. The model with logarithmic singularity”, Izv. RAN. Ser. Mat., 65:5 (2001), 33–72; Izv. Math., 65:5 (2001), 883–921

Citation in format AMSBIB
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\by M.~V.~Karasev, A.~V.~Pereskokov
\paper Asymptotic solutions of Hartree equations concentrated near low-dimensional submanifolds.~I. The model with logarithmic singularity
\jour Izv. RAN. Ser. Mat.
\yr 2001
\vol 65
\issue 5
\pages 33--72
\mathnet{http://mi.mathnet.ru/izv356}
\crossref{https://doi.org/10.4213/im356}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1874353}
\zmath{https://zbmath.org/?q=an:1019.81018}
\transl
\jour Izv. Math.
\yr 2001
\vol 65
\issue 5
\pages 883--921
\crossref{https://doi.org/10.1070/IM2001v065n05ABEH000356}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33746728524}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. M. V. Karasev, A. V. Pereskokov, “Asymptotic solutions of Hartree equations concentrated near low-dimensional submanifolds. II. Localization in planar discs”, Izv. Math., 65:6 (2001), 1127–1168  mathnet  crossref  crossref  mathscinet  zmath
    2. 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
    3. 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
    4. 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
    5. 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
    6. 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
    7. 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
    8. 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
    9. 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
    10. 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
    11. 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
    12. 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
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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