Semiclassical asymptotic spectrum of the two-dimensional Hartree operator near a local maximum of the eigenvalues in a spectral cluster

Abstract

We consider the eigenvalue problem for the two-dimensional Hartree operator with a small nonlinearity coefficient. We find the asymptotic eigenvalues and asymptotic eigenfunctions near a local maximum of the eigenvalues in spectral clusters formed near the eigenvalues of the unperturbed operator.

This is a preview of subscription content, access via your institution.

References

  1. 1

    N. N. Bogoliubov, “On a new form of the adiabatic theory of disturbances in the problem of interaction of particles with a quantum field [in Russian],” Ukr. Math. J., 2, No. 2, 3–24 (1950).

    Google Scholar 

  2. 2

    L. P. Pitaevskii, “Bose–Einstein condensation in magnetic traps: Introduction to the theory,” Phys. Usp., 41, 569–580 (1998).

    Article  Google Scholar 

  3. 3

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

    Google Scholar 

  4. 4

    A. S. Davydov, Solitons in Molecular Systems [in Russian], Naukova Dumka, Kiev (1984); English transl., Springer, Dordrecht (1985).

    MATH  Google Scholar 

  5. 5

    V. P. Maslov, Complex Markov Chains and the Continual Feynman Integral [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  6. 6

    M. V. Karasev and V. P. Maslov, “Algebras with general commutation relations and their applications: II. Operator unitary-nonlinear equations,” J. Soviet Math., 15, 273–368 (1981).

    Article  Google Scholar 

  7. 7

    A. V. Pereskokov, “Semiclassical asymptotic approximation of the two-dimensional Hartree operator spectrum near the upper boundaries of spectral clusters,” Theor. Math. Phys., 187, 511–524 (2016).

    MathSciNet  Article  Google Scholar 

  8. 8

    A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series: Elementary Functions [in Russian], Nauka, Moscow (1981); English transl., Gordon and Breach, New York (1986).

    MATH  Google Scholar 

  9. 9

    I. S. Gradshtein and I. M. Ryzhik, Table of Integrals, Sums, Series, and Products [in Russian], Fizmatgiz, Moscow (1963); English transl.: Table of Integrals, Series, and Products, Acad. Press, New York (1980).

    Google Scholar 

  10. 10

    M. V. Fedoryuk, Asymptotic Methods for Linear Ordinary Differential Equations [in Russian], Nauka, Moscow (1983); English transl.: Asymptotic Analysis: Linear Ordinary Differential Equations, Springer, Berlin (1993).

    MATH  Google Scholar 

  11. 11

    M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Natl. Bur. Stds. Appl. Math. Ser., Vol. 55), Dover,New York (1972).

    MATH  Google Scholar 

  12. 12

    G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, Cambridge (1944).

    MATH  Google Scholar 

  13. 13

    A. V. Pereskokov, “Semiclassical asymptotic spectrum of a Hartree-type operator near the upper boundary of spectral clusters,” Theor. Math. Phys., 178, 76–92 (2014).

    MathSciNet  Article  Google Scholar 

  14. 14

    A. V. Pereskokov, “Semiclassical asymptotics of the spectrum near the lower boundary of spectral clusters for a Hartree-type operator,” Math. Notes, 101, 1009–1022 (2017).

    MathSciNet  Article  Google Scholar 

  15. 15

    A. V. Pereskokov, “Asymptotics of the Hartree operator spectrum near the upper boundaries of spectral clusters: Asymptotic solutions localized near a circle,” Theor. Math. Phys., 183, 516–526 (2015).

    MathSciNet  Article  Google Scholar 

  16. 16

    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, 517–530 (2017).

    MathSciNet  Article  Google Scholar 

  17. 17

    D. A. Vakhrameeva and 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,” Theor. Math. Phys., 199, 864–877 (2019).

    MathSciNet  Article  Google Scholar 

  18. 18

    M. V. Karasev and A. V. Pereskokov, “Logarithmic corrections in a quantization rule: The polaron spectrum,” Theor. Math. Phys., 97, 1160–1170 (1993).

    Article  Google Scholar 

Download references

Funding

This research was performed in the framework of a state task of the Ministry of Education and Science of the Russian Federation (Project No. FSWF-2020-0022).

Author information

Affiliations

Authors

Corresponding author

Correspondence to A. V. Pereskokov.

Ethics declarations

The author declares no conflicts of interest.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pereskokov, A.V. Semiclassical asymptotic spectrum of the two-dimensional Hartree operator near a local maximum of the eigenvalues in a spectral cluster. Theor Math Phys 205, 1652–1665 (2020). https://doi.org/10.1134/S0040577920120077

Download citation

Keywords

  • spectral cluster
  • WKB approximation
  • asymptotic eigenvalue
  • asymptotic eigenfunction
  • logarithmic singularity